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<title>Probability current</title>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation>

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</math></span><img src="./1799e4a910c7d26396922a20ef5ceec25ca1871c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }" loading="lazy"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;">

<ul><li><a href="Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li>

<li><a href="Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li>

<li><a href="History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Background</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;">

<ul><li><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li>

<li><a href="Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li>

<li><a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul>

<ul><li><a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li>

<li><a href="Wave_interference" title="Wave interference">Interference</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Fundamentals</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li>

<li><a href="Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li>

<li><a href="Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li>

<li><a href="Energy_level" title="Energy level">Energy level</a></li>

<li><a href="Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li>

<li><a href="Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li>

<li><a href="Quantum_number" title="Quantum number">Quantum number</a></li>

<li><a href="Quantum_state" title="Quantum state">State</a></li>

<li><a href="Quantum_superposition" title="Quantum superposition">Superposition</a></li>

<li><a href="Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li>

<li><a href="Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li>

<li><a href="Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li>

<li><a href="Wave_function" title="Wave function">Wave function</a>

<ul><li><a href="Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Experiments</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Bell_test" title="Bell test">Bell's inequality</a></li>

<li><a href="CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li>

<li><a href="Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li>

<li><a href="Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li>

<li><a href="Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li>

<li><a href="Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li>

<li><a href="Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li>

<li><a href="Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li>

<li><a href="Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li>

<li><a href="Popper's_experiment" title="Popper's experiment">Popper</a></li></ul>

</div>

<ul><li><a href="Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a>

<ul><li><a href="Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul>

<ul><li><a href="Schr%C3%B6dinger's_cat" title="Schrödinger's cat">Schrödinger's cat</a></li>

<li><a href="Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li>

<li><a href="Wheeler's_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Formulations</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;">

<ul><li><a href="Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul>

<ul><li><a href="Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li>

<li><a href="Interaction_picture" title="Interaction picture">Interaction</a></li>

<li><a href="Matrix_mechanics" title="Matrix mechanics">Matrix</a></li>

<li><a href="Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li>

<li><a href="Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li>

<li><a href="Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Equations</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Dirac_equation" title="Dirac equation">Dirac</a></li>

<li><a href="Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li>

<li><a href="Pauli_equation" title="Pauli equation">Pauli</a></li>

<li><a href="Rydberg_formula" title="Rydberg formula">Rydberg</a></li>

<li><a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li>

<li><a href="Consciousness_causes_collapse" title="Consciousness causes collapse">Consciousness causes collapse</a></li>

<li><a href="Consistent_histories" title="Consistent histories">Consistent histories</a></li>

<li><a href="Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li>

<li><a href="De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li>

<li><a href="Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li>

<li><a href="Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a></li>

<li><a href="Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li>

<li><a href="Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li>

<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>

<li><a href="Superdeterminism" title="Superdeterminism">Superdeterminism</a></li>

<li><a href="Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li>

<li><a href="Transactional_interpretation" title="Transactional interpretation">Transactional</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Advanced topics</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;">

<ul><li><a href="Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li>

<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li>

<li><a href="Quantum_information_science" title="Quantum information science">Quantum information science</a></li>

<li><a href="Quantum_computing" title="Quantum computing">Quantum computing</a></li>

<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>

<li><a href="Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li>

<li><a href="Density_matrix" title="Density matrix">Density matrix</a></li>

<li><a href="Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li>

<li><a href="Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li>

<li><a href="Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li>

<li><a href="Hans_Bethe" title="Hans Bethe">Bethe</a></li>

<li><a href="Patrick_Blackett" title="Patrick Blackett">Blackett</a></li>

<li><a href="Felix_Bloch" title="Felix Bloch">Bloch</a></li>

<li><a href="Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li>

<li><a href="Arthur_Compton" title="Arthur Compton">Compton</a></li>

<li><a href="Paul_Dirac" title="Paul Dirac">Dirac</a></li>

<li><a href="Clinton_Davisson" title="Clinton Davisson">Davisson</a></li>

<li><a href="Peter_Debye" title="Peter Debye">Debye</a></li>

<li><a href="Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>

<li><a href="Hugh_Everett_III" title="Hugh Everett III">Everett</a></li>

<li><a href="Enrico_Fermi" title="Enrico Fermi">Fermi</a></li>

<li><a href="Richard_Feynman" title="Richard Feynman">Feynman</a></li>

<li><a href="Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li>

<li><a href="Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li>

<li><a href="Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li>

<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>

<li><a href="Pascual_Jordan" title="Pascual Jordan">Jordan</a></li>

<li><a href="Hans_Kramers" title="Hans Kramers">Kramers</a></li>

<li><a href="Lev_Landau" title="Lev Landau">Landau</a></li>

<li><a href="Henry_Moseley" title="Henry Moseley">Moseley</a></li>

<li><a href="Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li>

<li><a href="Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li>

<li><a href="Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li>

<li><a href="Max_Planck" title="Max Planck">Planck</a></li>

<li><a href="C._V._Raman" title="C. V. Raman">Raman</a></li>

<li><a href="Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li>

<li><a href="Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li>

<li><a href="Michelle_Simmons" title="Michelle Simmons">Simmons</a></li>

<li><a href="Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">von Neumann</a></li>

<li><a href="Eugene_Wigner" title="Eugene Wigner">Wigner</a></li>

<li><a href="Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li>

<li><a href="Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul>

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<p>In <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, the <b>probability current</b> (sometimes called <b>probability <a href="Flux" title="Flux">flux</a></b>) is a mathematical quantity describing the flow of <a href="Probability" title="Probability">probability</a>. Specifically, if one thinks of probability as a <a href="Heterogeneous" class="mw-redirect" title="Heterogeneous">heterogeneous</a> fluid, then the probability current is the rate of flow of this fluid. It is a <a href="Real_number" title="Real number">real</a> <a href="Euclidean_vector" title="Euclidean vector">vector</a> that changes with space and time. Probability currents are analogous to <a href="Mass_flow_(physics)" class="mw-redirect" title="Mass flow (physics)">mass currents</a> in <a href="Hydrodynamic" class="mw-redirect" title="Hydrodynamic">hydrodynamics</a> and <a href="Electric_currents" class="mw-redirect" title="Electric currents">electric currents</a> in <a href="Electromagnetism" title="Electromagnetism">electromagnetism</a>. As in those fields, the probability current (i.e. the probability current density) is related to the <a href="Probability_density_function" title="Probability density function">probability density function</a> via a <a href="Continuity_equation" title="Continuity equation">continuity equation</a>. The probability current is <a href="Invariant_(mathematics)" title="Invariant (mathematics)">invariant</a> under <a href="Gauge_transformation" class="mw-redirect" title="Gauge transformation">gauge transformation</a>.

</p><p>The concept of probability current is also used outside of quantum mechanics, when dealing with probability density functions that change over time, for instance in <a href="Brownian_motion" title="Brownian motion">Brownian motion</a> and the <a href="Fokker%E2%80%93Planck_equation" title="Fokker–Planck equation">Fokker–Planck equation</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>The relativistic equivalent of the probability current is known as the probability four-current.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition_(non-relativistic_3-current)">Definition (non-relativistic 3-current)</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Free_spin-0_particle">Free spin-0 particle</h3></div>

<p>In non-relativistic quantum mechanics, the probability current <span class="texhtml"><b>j</b></span> of the <a href="Wave_function" title="Wave function">wave function</a> <span class="texhtml">Ψ</span> of a particle of mass <span class="texhtml mvar" style="font-style:italic;">m</span> in one dimension is defined as<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle j={\frac {\hbar }{2mi}}\left(\Psi ^{*}{\frac {\partial \Psi }{\partial x}}-\Psi {\frac {\partial \Psi ^{*}}{\partial x}}\right)={\frac {\hbar }{m}}\Re \left\{\Psi ^{*}{\frac {1}{i}}{\frac {\partial \Psi }{\partial x}}\right\}={\frac {\hbar }{m}}\Im \left\{\Psi ^{*}{\frac {\partial \Psi }{\partial x}}\right\},}</annotation>

</semantics>

</math></span></span>

where

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>

</semantics>

</math></span><img src="./de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"></span> is the reduced <a href="Planck_constant" title="Planck constant">Planck constant</a>;</li>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \Psi ^{*}}</annotation>

</semantics>

</math></span><img src="./849b5fc60c04296df0ca9dff8c83bc3d97a4b478.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.862ex; height:2.343ex;" alt="{\displaystyle \Psi ^{*}}" loading="lazy"></span> denotes the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> of the <a href="Wave_function" title="Wave function">wave function</a>;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Re }</annotation>

</semantics>

</math></span><img src="./2cc5a2cb7aa22f6d765976edb1daebefaf408142.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle \Re }" loading="lazy"></span> denotes the <a href="Real_part" class="mw-redirect" title="Real part">real part</a>;</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Im }</annotation>

</semantics>

</math></span><img src="./d3e0312a4871a615cbdae168be102907f0a51e95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.288ex; height:2.176ex;" alt="{\displaystyle \Im }" loading="lazy"></span> denotes the <a href="Imaginary_part" class="mw-redirect" title="Imaginary part">imaginary part</a>.</li></ul>

<p>Note that the probability current is proportional to a <a href="Wronskian" title="Wronskian">Wronskian</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(\Psi ,\Psi ^{*}).}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W(\Psi ,\Psi ^{*}).}</annotation>

</semantics>

</math></span><img src="./f80dd4bf29b57d6418402ddf3f1097cbe0582ed5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.596ex; height:2.843ex;" alt="{\displaystyle W(\Psi ,\Psi ^{*}).}" loading="lazy"></span>

</p><p>In three dimensions, this generalizes to

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} ={\frac {\hbar }{2mi}}\left(\Psi ^{*}\mathbf {\nabla } \Psi -\Psi \mathbf {\nabla } \Psi ^{*}\right)={\frac {\hbar }{m}}\Re \left\{\Psi ^{*}{\frac {\nabla }{i}}\Psi \right\}={\frac {\hbar }{m}}\Im \left\{\Psi ^{*}\nabla \Psi \right\}\,,}</annotation>

</semantics>

</math></span></span>

where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation>

</semantics>

</math></span><img src="./a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy"></span> denotes the <a href="Del" title="Del">del</a> or <a href="Gradient" title="Gradient">gradient</a> <a href="Operator_(mathematics)" title="Operator (mathematics)">operator</a>. This can be simplified in terms of the <a href="Momentum_operator" title="Momentum operator">kinetic momentum operator</a>,

<mover>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }</annotation>

</semantics>

</math></span></span>

to obtain

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} ={\frac {1}{2m}}\left(\Psi ^{*}\mathbf {\hat {p}} \Psi +\Psi \left(\mathbf {\hat {p}} \Psi \right)^{*}\right)\,.}</annotation>

</semantics>

</math></span></span>

</p><p>These definitions use the position basis (i.e. for a wavefunction in <a href="Position_space" class="mw-redirect" title="Position space">position space</a>), but <a href="Momentum_space" class="mw-redirect" title="Momentum space">momentum space</a> is possible. In fact, one can write the probability current operator as

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {j}} (\mathbf {r} )={\frac {\mathbf {\hat {p}} |\mathbf {r} \rangle \langle \mathbf {r} |+|\mathbf {r} \rangle \langle \mathbf {r} |\mathbf {\hat {p}} }{2m}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {j}} (\mathbf {r} )={\frac {\mathbf {\hat {p}} |\mathbf {r} \rangle \langle \mathbf {r} |+|\mathbf {r} \rangle \langle \mathbf {r} |\mathbf {\hat {p}} }{2m}}}</annotation>

</semantics>

</math></span><img src="./f25310f68a19ed46f42210e3d2047c8ff2d9a744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.608ex; height:5.676ex;" alt="{\displaystyle \mathbf {\hat {j}} (\mathbf {r} )={\frac {\mathbf {\hat {p}} |\mathbf {r} \rangle \langle \mathbf {r} |+|\mathbf {r} \rangle \langle \mathbf {r} |\mathbf {\hat {p}} }{2m}}}" loading="lazy"></span>

</p><p>which do not depend on a particular choice of basis. The probability current is then the expectation of this operator,

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} (\mathbf {r} ,t)=\langle \Psi (t)|{\hat {\mathbf {j} }}(\mathbf {r} )|\Psi (t)\rangle .}">

</mrow>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} (\mathbf {r} ,t)=\langle \Psi (t)|{\hat {\mathbf {j} }}(\mathbf {r} )|\Psi (t)\rangle .}</annotation>

</semantics>

</math></span><img src="./de215011a24514707e72b3748da35208eb294adb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.164ex; width:25.601ex; height:3.343ex;" alt="{\displaystyle \mathbf {j} (\mathbf {r} ,t)=\langle \Psi (t)|{\hat {\mathbf {j} }}(\mathbf {r} )|\Psi (t)\rangle .}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Spin-0_particle_in_an_electromagnetic_field">Spin-0 particle in an electromagnetic field</h3></div>

/* start https://en.wikipedia.org/ */

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</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> and <a href="Kinetic_momentum" class="mw-redirect" title="Kinetic momentum">kinetic momentum</a></div>

<p>The above definition should be modified for a system in an external <a href="Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>. In <a href="SI_units" class="mw-redirect" title="SI units">SI units</a>, a <a href="Charged_particle" title="Charged particle">charged particle</a> of mass <span class="texhtml mvar" style="font-style:italic;">m</span> and <a href="Electric_charge" title="Electric charge">electric charge</a> <span class="texhtml mvar" style="font-style:italic;">q</span> includes a term due to the interaction with the electromagnetic field;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} ={\frac {1}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-2q\mathbf {A} |\Psi |^{2}\right]}</annotation>

</semantics>

</math></span></span>

where <span class="texhtml"><b>A</b> = <b>A</b>(<b>r</b>, <i>t</i>)</span> is the <a href="Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a>. The term <span class="texhtml"><i>q</i><b>A</b></span> has dimensions of momentum. Note that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }">

<mover>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }</annotation>

</semantics>

</math></span><img src="./50e668b83274ecf09084e1e4762438b3b4dfdfbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.437ex; height:2.676ex;" alt="{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }" loading="lazy"></span> used here is the <a href="Canonical_momentum" class="mw-redirect" title="Canonical momentum">canonical momentum</a> and is not <a href="Gauge_invariance" class="mw-redirect" title="Gauge invariance">gauge invariant</a>, unlike the <a href="Momentum_operator" title="Momentum operator">kinetic momentum operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {P}} =-i\hbar \nabla -q\mathbf {A} }">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {P}} =-i\hbar \nabla -q\mathbf {A} }</annotation>

</semantics>

</math></span><img src="./5cd4c2ffdab4158b5ccf91b0c5fd56f9b9c0dd19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.708ex; height:3.176ex;" alt="{\displaystyle \mathbf {\hat {P}} =-i\hbar \nabla -q\mathbf {A} }" loading="lazy"></span>.

</p><p>In <a href="Gaussian_units" title="Gaussian units">Gaussian units</a>:

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where <span class="texhtml mvar" style="font-style:italic;">c</span> is the <a href="Speed_of_light" title="Speed of light">speed of light</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Spin-s_particle_in_an_electromagnetic_field">Spin-<i>s</i> particle in an electromagnetic field</h3></div>

<p>If the particle has <a href="Spin_(physics)" title="Spin (physics)">spin</a>, it has a corresponding <a href="Magnetic_moment" title="Magnetic moment">magnetic moment</a>, so an extra term needs to be added incorporating the spin interaction with the electromagnetic field.

</p><p>According to Landau-Lifschitz's <i><a href="Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Course of Theoretical Physics</a></i> the electric current density is in Gaussian units:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} _{e}={\frac {q}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-{\frac {2q}{c}}\mathbf {A} |\Psi |^{2}\right]+{\frac {\mu _{S}c}{s\hbar }}\nabla \times (\Psi ^{*}\mathbf {S} \Psi )}</annotation>

</semantics>

</math></span></span>

</p><p>And in SI units: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} _{e}={\frac {q}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-2q\mathbf {A} |\Psi |^{2}\right]+{\frac {\mu _{S}}{s\hbar }}\nabla \times (\Psi ^{*}\mathbf {S} \Psi )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} _{e}={\frac {q}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-2q\mathbf {A} |\Psi |^{2}\right]+{\frac {\mu _{S}}{s\hbar }}\nabla \times (\Psi ^{*}\mathbf {S} \Psi )}</annotation>

</semantics>

</math></span></span>

</p><p>Hence the probability current (density) is in SI units: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} =\mathbf {j} _{e}/q={\frac {1}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-2q\mathbf {A} |\Psi |^{2}\right]+{\frac {\mu _{S}}{qs\hbar }}\nabla \times (\Psi ^{*}\mathbf {S} \Psi )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} =\mathbf {j} _{e}/q={\frac {1}{2m}}\left[\left(\Psi ^{*}\mathbf {\hat {p}} \Psi -\Psi \mathbf {\hat {p}} \Psi ^{*}\right)-2q\mathbf {A} |\Psi |^{2}\right]+{\frac {\mu _{S}}{qs\hbar }}\nabla \times (\Psi ^{*}\mathbf {S} \Psi )}</annotation>

</semantics>

</math></span></span>

</p><p>where <span class="texhtml"><b>S</b></span> is the <a href="Spin_(physics)" title="Spin (physics)">spin</a> vector of the particle with corresponding <a href="Spin_magnetic_moment" class="mw-redirect" title="Spin magnetic moment">spin magnetic moment</a> <span class="texhtml">μ<sub><i>S</i></sub></span> and <a href="Spin_quantum_number" title="Spin quantum number">spin quantum number</a> <span class="texhtml mvar" style="font-style:italic;">s</span>.

</p><p>It is doubtful if this formula is valid for particles with an interior structure. The <a href="Neutron" title="Neutron">neutron</a> has zero charge but non-zero magnetic moment, so <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mu _{S}}{qs\hbar }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mu _{S}}{qs\hbar }}}</annotation>

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</math></span><img src="./cb5d46c482e33c95d79f35572c9cdf4e905f8d0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:4.303ex; height:5.343ex;" alt="{\displaystyle {\frac {\mu _{S}}{qs\hbar }}}" loading="lazy"></span> would be impossible (except <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times (\Psi ^{*}\mathbf {S} \Psi )}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times (\Psi ^{*}\mathbf {S} \Psi )}</annotation>

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</math></span><img src="./7f72db0e543137ffa5744b1b72f6cfa6ba37ec05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.741ex; height:2.843ex;" alt="{\displaystyle \nabla \times (\Psi ^{*}\mathbf {S} \Psi )}" loading="lazy"></span> would also be zero in this case). For composite particles with a non-zero charge – like the <a href="Proton" title="Proton">proton</a> which has spin quantum number s=1/2 and μ<sub>S</sub>= 2.7927·<a href="Nuclear_magneton" title="Nuclear magneton">μ<sub>N</sub></a> or the <a href="Deuterium#Nuclear_properties_(the_deuteron)" title="Deuterium">deuteron</a> (H-2 nucleus) which has s=1 and μ<sub>S</sub>=0.8574·μ<sub>N</sub> <sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> – it is mathematically possible but doubtful.

</p>

<div class="mw-heading mw-heading2"><h2 id="Connection_with_classical_mechanics">Connection with classical mechanics</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Madelung_equations" title="Madelung equations">Madelung equations</a></div>

<p>The wave function can also be written in the <a href="Complex_number" title="Complex number">complex</a> <a href="Exponential_function" title="Exponential function">exponential</a> (<a href="Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar</a>) form:

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<annotation encoding="application/x-tex">{\displaystyle \Psi =Re^{iS/\hbar }}</annotation>

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</math></span></span>

where <span class="texhtml mvar" style="font-style:italic;">R, S</span> are real functions of <span class="texhtml"><b>r</b></span> and <span class="texhtml">t</span>.

</p><p>Written this way, the probability density is <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\Psi ^{*}\Psi =R^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \rho =\Psi ^{*}\Psi =R^{2}}</annotation>

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</math></span></span> and the probability current is:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {j} &={\frac {\hbar }{2mi}}\left(\Psi ^{*}\mathbf {\nabla } \Psi -\Psi \mathbf {\nabla } \Psi ^{*}\right)\\[5pt]&={\frac {\hbar }{2mi}}\left(Re^{-iS/\hbar }\mathbf {\nabla } Re^{iS/\hbar }-Re^{iS/\hbar }\mathbf {\nabla } Re^{-iS/\hbar }\right)\\[5pt]&={\frac {\hbar }{2mi}}\left[Re^{-iS/\hbar }\left(e^{iS/\hbar }\mathbf {\nabla } R+{\frac {i}{\hbar }}Re^{iS/\hbar }\mathbf {\nabla } S\right)-Re^{iS/\hbar }\left(e^{-iS/\hbar }\mathbf {\nabla } R-{\frac {i}{\hbar }}Re^{-iS/\hbar }\mathbf {\nabla } S\right)\right].\end{aligned}}}</annotation>

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</math></span></span>

</p><p>The exponentials and <span class="texhtml"><i>R</i>∇<i>R</i></span> terms cancel:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} ={\frac {\hbar }{2mi}}\left[{\frac {i}{\hbar }}R^{2}\mathbf {\nabla } S+{\frac {i}{\hbar }}R^{2}\mathbf {\nabla } S\right].}</annotation>

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</math></span></span>

</p><p>Finally, combining and cancelling the constants, and replacing <span class="texhtml"><i>R</i><sup>2</sup></span> with <span class="texhtml mvar" style="font-style:italic;">ρ</span>,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} =\rho {\frac {\mathbf {\nabla } S}{m}}.}</annotation>

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</math></span></span>Hence, the spatial variation of the phase of a wavefunction is said to characterize the probability flux of the wavefunction. If we take the familiar formula for the mass flux in hydrodynamics:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} =\rho \mathbf {v} ,}</annotation>

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</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math></span><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> is the mass density of the fluid and <span class="texhtml"><b>v</b></span> is its velocity (also the <a href="Group_velocity" title="Group velocity">group velocity</a> of the wave). In the classical limit, we can associate the velocity with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\nabla S}{m}},}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\nabla S}{m}},}</annotation>

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</math></span><img src="./6efa8ed42ad2d1f9cc6aa9d8dedd70aa2f8284de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.912ex; height:3.509ex;" alt="{\displaystyle {\tfrac {\nabla S}{m}},}" loading="lazy"></span> which is the same as equating <span class="texhtml">∇<i>S</i></span> with the classical momentum <span class="texhtml"><b>p</b> = <i>m</i><b>v</b></span> however, it does not represent a physical velocity or momentum at a point since simultaneous measurement of position and velocity violates <a href="Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>. This interpretation fits with <a href="Hamilton%E2%80%93Jacobi_theory" class="mw-redirect" title="Hamilton–Jacobi theory">Hamilton–Jacobi theory</a>, in which

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\nabla S}</annotation>

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in Cartesian coordinates is given by <span class="texhtml">∇<i>S</i></span>, where <span class="texhtml mvar" style="font-style:italic;">S</span> is <a href="Hamilton's_principal_function" class="mw-redirect" title="Hamilton's principal function">Hamilton's principal function</a>.

</p><p>The de Broglie-Bohm theory equates the velocity with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\nabla S}{m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\nabla S}{m}}}</annotation>

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</math></span><img src="./f085185b97fd4634b53b6c1ab76ac3b473414ee6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.265ex; height:3.509ex;" alt="{\displaystyle {\tfrac {\nabla S}{m}}}" loading="lazy"></span> in general (not only in the classical limit) so it is always well defined. It is an interpretation of quantum mechanics.

</p>

<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Continuity_equation_for_quantum_mechanics">Continuity equation for quantum mechanics</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Continuity_equation" title="Continuity equation">continuity equation</a></div>

<p>The definition of probability current and Schrödinger's equation can be used to derive the <a href="Continuity_equation#Quantum_mechanics" title="Continuity equation">continuity equation</a>, which has <i>exactly</i> the same forms as those for <a href="Hydrodynamics" class="mw-redirect" title="Hydrodynamics">hydrodynamics</a> and <a href="Electromagnetism" title="Electromagnetism">electromagnetism</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>For some <a href="Wave_function" title="Wave function">wave function</a> <span class="texhtml">Ψ</span>, let:

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<annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} ,t)=|\Psi |^{2}=\Psi ^{*}(\mathbf {r} ,t)\Psi (\mathbf {r} ,t).}</annotation>

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</math></span></span>be the <a href="Probability_density_function" title="Probability density function">probability density</a> (probability per unit volume, <span class="texhtml">*</span> denotes <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a>). Then,

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dt}}\int _{\mathcal {V}}dV\,\rho &=\int _{\mathcal {V}}dV\,\left({\frac {\partial \psi }{\partial t}}\psi ^{*}+\psi {\frac {\partial \psi ^{*}}{\partial t}}\right)\\&=\int _{\mathcal {V}}dV\,\left[-{\frac {i}{\hbar }}\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi \right)\psi ^{*}+{\frac {i}{\hbar }}\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi ^{*}+V\psi ^{*}\right)\psi \right]\\&=\int _{\mathcal {V}}dV\,{\frac {i\hbar }{2m}}\left[\left(\nabla ^{2}\psi \right)\psi ^{*}-\psi \left(\nabla ^{2}\psi ^{*}\right)\right]\\&=\int _{\mathcal {V}}dV\,\nabla \cdot \left({\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})\right)\\&=\int _{\mathcal {S}}d\mathbf {a} \cdot \left({\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})\right)\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dt}}\int _{\mathcal {V}}dV\,\rho &=\int _{\mathcal {V}}dV\,\left({\frac {\partial \psi }{\partial t}}\psi ^{*}+\psi {\frac {\partial \psi ^{*}}{\partial t}}\right)\\&=\int _{\mathcal {V}}dV\,\left[-{\frac {i}{\hbar }}\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi \right)\psi ^{*}+{\frac {i}{\hbar }}\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi ^{*}+V\psi ^{*}\right)\psi \right]\\&=\int _{\mathcal {V}}dV\,{\frac {i\hbar }{2m}}\left[\left(\nabla ^{2}\psi \right)\psi ^{*}-\psi \left(\nabla ^{2}\psi ^{*}\right)\right]\\&=\int _{\mathcal {V}}dV\,\nabla \cdot \left({\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})\right)\\&=\int _{\mathcal {S}}d\mathbf {a} \cdot \left({\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})\right)\end{aligned}}}</annotation>

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</math></span></span>

</p><p>where <span class="texhtml mvar" style="font-style:italic;">V</span> is any volume and <span class="texhtml mvar" style="font-style:italic;">S</span> is the boundary of <span class="texhtml mvar" style="font-style:italic;">V</span>.

</p><p>This is the <a href="Conservation_law_(physics)" class="mw-redirect" title="Conservation law (physics)">conservation law</a> for probability in quantum mechanics. The integral form is stated as:

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<annotation encoding="application/x-tex">{\displaystyle \int _{V}\left({\frac {\partial |\Psi |^{2}}{\partial t}}\right)\mathrm {d} V+\int _{V}\left(\mathbf {\nabla } \cdot \mathbf {j} \right)\mathrm {d} V=0}</annotation>

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</math></span></span>where<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} ={\frac {1}{2m}}\left(\Psi ^{*}{\hat {\mathbf {p} }}\Psi -\Psi {\hat {\mathbf {p} }}\Psi ^{*}\right)=-{\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})={\frac {\hbar }{m}}\operatorname {Im} (\psi ^{*}\nabla \psi )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} ={\frac {1}{2m}}\left(\Psi ^{*}{\hat {\mathbf {p} }}\Psi -\Psi {\hat {\mathbf {p} }}\Psi ^{*}\right)=-{\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})={\frac {\hbar }{m}}\operatorname {Im} (\psi ^{*}\nabla \psi )}</annotation>

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</math></span></span>is the probability current or probability flux (flow per unit area).

</p><p>Here, equating the terms inside the integral gives the <a href="Continuity_equation#Quantum_mechanics" title="Continuity equation">continuity equation</a> for probability:<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial t}}\rho \left(\mathbf {r} ,t\right)+\nabla \cdot \mathbf {j} =0,}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}\rho \left(\mathbf {r} ,t\right)+\nabla \cdot \mathbf {j} =0,}</annotation>

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</p>

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</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}\int _{V}|\Psi |^{2}\mathrm {d} V+}</annotation>

</semantics>

</math></span><img src="./639c0b20e355000fc2391bcce4d96657667bc9f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.601ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}\int _{V}|\Psi |^{2}\mathrm {d} V+}" loading="lazy"></span> <span class="mw-default-size" typeof="mw:File"><span></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle S}</annotation>

</semantics>

</math></span><img src="./62e91bf22a258bad807b5a9ae8068ef38202ca30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.06ex; height:1.676ex;" alt="{\displaystyle \scriptstyle S}" loading="lazy"></span></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} \cdot \mathrm {d} \mathbf {S} =0}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} \cdot \mathrm {d} \mathbf {S} =0}</annotation>

</semantics>

</math></span><img src="./9b33253ebb1434ed2e1d3242e6d076cf38a85f83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.164ex; width:9.698ex; height:2.509ex;" alt="{\displaystyle \mathbf {j} \cdot \mathrm {d} \mathbf {S} =0}" loading="lazy"></span></span>.</div>

<p>In particular, if <span class="texhtml mvar" style="font-style:italic;">Ψ</span> is a wavefunction describing a single particle, the integral in the first term of the preceding equation, sans time derivative, is the probability of obtaining a value within <span class="texhtml mvar" style="font-style:italic;">V</span> when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume <span class="texhtml mvar" style="font-style:italic;">V</span>. Altogether the equation states that the time derivative of the probability of the particle being measured in <span class="texhtml mvar" style="font-style:italic;">V</span> is equal to the rate at which probability flows into <span class="texhtml mvar" style="font-style:italic;">V</span>.

</p><p>By taking the limit of volume integral to include all regions of space, a well-behaved wavefunction that goes to zero at infinities in the surface integral term implies that the time derivative of total probability is zero ie. the normalization condition is conserved.<sup id="cite_ref-:1_7-0" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This result is in agreement with the unitary nature of time evolution operators which preserve length of the vector by definition.

</p>

<div class="mw-heading mw-heading3"><h3 id="Conserved_current_for_Klein–Gordon_fields">Conserved current for Klein–Gordon fields</h3></div>

<p>The probability (4-)current arises from <a href="Noether's_theorem" title="Noether's theorem">Noether's theorem</a> as applied to the <a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a> the <a href="Klein-Gordon_equation" class="mw-redirect" title="Klein-Gordon equation">Klein-Gordon Lagrangian density</a>

</mrow>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=\partial _{\mu }\phi ^{*}\,\partial ^{\mu }\phi +V(\phi ^{*}\,\phi )}</annotation>

</semantics>

</math></span></span>

of the complex scalar field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\phi :\mathbb {R} ^{n+1}\mapsto \mathbb {C} ">

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<annotation encoding="application/x-tex">\phi :\mathbb {R} ^{n+1}\mapsto \mathbb {C}</annotation>

</semantics>

</math></span><img src="./1d7de609d2110f0b59e505d295abcb3159445a1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.612ex; height:3.009ex;" alt="\phi :\mathbb {R} ^{n+1}\mapsto \mathbb {C} " loading="lazy"></span>. This is invariant under the symmetry transformation<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \mapsto \phi '=\phi \,e^{i\alpha }\,.}">

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi \mapsto \phi '=\phi \,e^{i\alpha }\,.}</annotation>

</semantics>

</math></span></span>

Defining <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\delta \phi =\phi '-\phi ">

<mrow>

<msup>

</msup>

</mrow>

<annotation encoding="application/x-tex">\delta \phi =\phi '-\phi</annotation>

</semantics>

</math></span><img src="./101100b758bf5a4adaa7f9b1c81a8ada13eaf9e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.828ex; height:2.843ex;" alt="\delta \phi =\phi '-\phi " loading="lazy"></span> we find the Noether current

<msup>

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</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

<mover>

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</mover>

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</msub>

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<mrow>

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<msup>

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<msup>

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</msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle j^{\mu }:={\frac {d{\mathcal {L}}}{d{\dot {\mathbf {q} }}}}\cdot \mathbf {Q} _{r}={\frac {d{\mathcal {L}}}{d(\partial _{\mu })\phi }}\,{\frac {d(\delta \phi )}{d\alpha }}{\bigg |}_{\alpha =0}+{\frac {d{\mathcal {L}}}{d(\partial _{\mu })\phi ^{*}}}\,{\frac {d(\delta \phi ^{*})}{d\alpha }}{\bigg |}_{\alpha =0}=i\,\phi \,(\partial ^{\mu }\phi ^{*})-i\,\phi ^{*}\,(\partial ^{\mu }\phi )}</annotation>

</semantics>

</math></span></span>which satisfies the continuity equation. Here <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} _{r}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} _{r}}</annotation>

</semantics>

</math></span><img src="./ab07d2bb828db60094005c0fc477f3765b7fa7c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.982ex; height:2.676ex;" alt="{\displaystyle \mathbf {Q} _{r}}" loading="lazy"></span> is the generator of the symmetry, which is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d(\delta \mathbf {q} )}{d\alpha _{r}}}}">

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d(\delta \mathbf {q} )}{d\alpha _{r}}}}</annotation>

</semantics>

</math></span><img src="./d7abc928633be79e4a1c8c1e741f880d0862d45e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.326ex; height:6.176ex;" alt="{\displaystyle {\frac {d(\delta \mathbf {q} )}{d\alpha _{r}}}}" loading="lazy"></span> in the case of a single parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>.

</p><p>The continuity equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }j^{\mu }=0}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }j^{\mu }=0}</annotation>

</semantics>

</math></span><img src="./fdaff79f0a778ba2868d12fa3a4435af326f8534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.9ex; height:3.009ex;" alt="{\displaystyle \partial _{\mu }j^{\mu }=0}" loading="lazy"></span> is satisfied. However, note that now, the analog of the probability density is not <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \phi ^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi \phi ^{*}}</annotation>

</semantics>

</math></span><img src="./e250f33ab70304e255aeeea5cde949739a2dcf17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.825ex; height:2.676ex;" alt="{\displaystyle \phi \phi ^{*}}" loading="lazy"></span> but rather <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{*}\partial _{t}\phi -\phi \partial _{t}\phi ^{*}}">

<msup>

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</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi ^{*}\partial _{t}\phi -\phi \partial _{t}\phi ^{*}}</annotation>

</semantics>

</math></span><img src="./f50ce318442a606c5c672a189dd2ca01f8a04358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.611ex; height:2.676ex;" alt="{\displaystyle \phi ^{*}\partial _{t}\phi -\phi \partial _{t}\phi ^{*}}" loading="lazy"></span>. As this quantity can now be negative, we must interpret it as a charge density, with an associated current density and <a href="4-current" class="mw-redirect" title="4-current">4-current</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Transmission_and_reflection_through_potentials">Transmission and reflection through potentials</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Transmission_coefficient#Quantum_mechanics" title="Transmission coefficient">Transmission coefficient</a> and <a href="Reflection_coefficient" title="Reflection coefficient">Reflection coefficient</a></div>

<p>In regions where a <a href="Step_potential" title="Step potential">step potential</a> or <a href="Potential_barrier" class="mw-redirect" title="Potential barrier">potential barrier</a> occurs, the probability current is related to the transmission and reflection coefficients, respectively <span class="texhtml mvar" style="font-style:italic;">T</span> and <span class="texhtml mvar" style="font-style:italic;">R</span>; they measure the extent the particles reflect from the potential barrier or are transmitted through it. Both satisfy:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T+R=1\,,}</annotation>

</semantics>

</math></span></span>

where <span class="texhtml mvar" style="font-style:italic;">T</span> and <span class="texhtml mvar" style="font-style:italic;">R</span> can be defined by:

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle T={\frac {|\mathbf {j} _{\mathrm {trans} }|}{|\mathbf {j} _{\mathrm {inc} }|}}\,,\quad R={\frac {|\mathbf {j} _{\mathrm {ref} }|}{|\mathbf {j} _{\mathrm {inc} }|}}\,,}</annotation>

</semantics>

</math></span></span>

where <span class="texhtml"><i>j</i><sub>inc</sub>, <i>j</i><sub>ref</sub>, <i>j</i><sub>trans</sub></span> are the incident, reflected and transmitted probability currents respectively, and the vertical bars indicate the <a href="Norm_(mathematics)" title="Norm (mathematics)">magnitudes</a> of the current vectors. The relation between <span class="texhtml mvar" style="font-style:italic;">T</span> and <span class="texhtml mvar" style="font-style:italic;">R</span> can be obtained from probability conservation:

<msub>

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<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} _{\mathrm {trans} }+\mathbf {j} _{\mathrm {ref} }=\mathbf {j} _{\mathrm {inc} }\,.}</annotation>

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</math></span></span>

</p><p>In terms of a <a href="Unit_vector" title="Unit vector">unit vector</a> <span class="texhtml"><b>n</b></span> <a href="Surface_normal" class="mw-redirect" title="Surface normal">normal</a> to the barrier, these are equivalently:

<mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle T=\left|{\frac {\mathbf {j} _{\mathrm {trans} }\cdot \mathbf {n} }{\mathbf {j} _{\mathrm {inc} }\cdot \mathbf {n} }}\right|\,,\qquad R=\left|{\frac {\mathbf {j} _{\mathrm {ref} }\cdot \mathbf {n} }{\mathbf {j} _{\mathrm {inc} }\cdot \mathbf {n} }}\right|\,,}</annotation>

</semantics>

</math></span></span>

where the absolute values are required to prevent <span class="texhtml mvar" style="font-style:italic;">T</span> and <span class="texhtml mvar" style="font-style:italic;">R</span> being negative.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Plane_wave">Plane wave</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Plane_wave" title="Plane wave">plane wave</a></div>

<p>For a <a href="Plane_wave" title="Plane wave">plane wave</a> propagating in space:

</mrow>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} ,t)=\,Ae^{i(\mathbf {k} \cdot {\mathbf {r} }-\omega t)}}</annotation>

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</math></span></span>

the probability density is constant everywhere;

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<annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} ,t)=|A|^{2}\rightarrow {\frac {\partial |\Psi |^{2}}{\partial t}}=0}</annotation>

</semantics>

</math></span></span>

(that is, plane waves are <a href="Stationary_state" title="Stationary state">stationary states</a>) but the probability current is nonzero – the square of the absolute amplitude of the wave times the particle's speed;

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {j} \left(\mathbf {r} ,t\right)=\left|A\right|^{2}{\hbar \mathbf {k} \over m}=\rho {\frac {\mathbf {p} }{m}}=\rho \mathbf {v} }</annotation>

</semantics>

</math></span></span>

</p><p>illustrating that the particle may be in motion even if its spatial probability density has no explicit time dependence.

</p>

<div class="mw-heading mw-heading3"><h3 id="Particle_in_a_box">Particle in a box</h3></div>

<p>For a <a href="Particle_in_a_box" title="Particle in a box">particle in a box</a>, in one spatial dimension and of length <span class="texhtml mvar" style="font-style:italic;">L</span>, confined to the region <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x<L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<x<L}</annotation>

</semantics>

</math></span><img src="./ee96942aa06606f2143e632b11ea773e3604b165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.272ex; height:2.176ex;" alt="{\displaystyle 0<x<L}" loading="lazy"></span>, the energy eigenstates are

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{n}={\sqrt {\frac {2}{L}}}\sin \left({\frac {n\pi }{L}}x\right)}</annotation>

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</math></span></span>

and zero elsewhere. The associated probability currents are

<msub>

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<annotation encoding="application/x-tex">{\displaystyle j_{n}={\frac {i\hbar }{2m}}\left(\Psi _{n}^{*}{\frac {\partial \Psi _{n}}{\partial x}}-\Psi _{n}{\frac {\partial \Psi _{n}^{*}}{\partial x}}\right)=0}</annotation>

</semantics>

</math></span></span>

since <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{n}=\Psi _{n}^{*}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{n}=\Psi _{n}^{*}}</annotation>

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</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Discrete_definition">Discrete definition</h2></div>

<p>For a particle in one dimension on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(\mathbb {Z} ),}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(\mathbb {Z} ),}</annotation>

</semantics>

</math></span><img src="./64f1f3f10f2cbe9b78d2fde3a40d2f0210cf59be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.03ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(\mathbb {Z} ),}" loading="lazy"></span> we have the Hamiltonian <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=-\Delta +V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=-\Delta +V}</annotation>

</semantics>

</math></span><img src="./c335255369928d5c202cdaa83b2edab4706ff2da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.534ex; height:2.343ex;" alt="{\displaystyle H=-\Delta +V}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\Delta \equiv 2I-S-S^{\ast }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\Delta \equiv 2I-S-S^{\ast }}</annotation>

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</math></span><img src="./95040fe88911dd14d081a824bb2e4a84572a7a4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.933ex; height:2.509ex;" alt="{\displaystyle -\Delta \equiv 2I-S-S^{\ast }}" loading="lazy"></span> is the discrete Laplacian, with <span class="texhtml mvar" style="font-style:italic;">S</span> being the right shift operator on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(\mathbb {Z} ).}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(\mathbb {Z} ).}</annotation>

</semantics>

</math></span><img src="./f5c55446ba2f502a54fc34dc24eec7842ddde295.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.03ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(\mathbb {Z} ).}" loading="lazy"></span> Then the probability current is defined as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\equiv 2\Im \left\{{\bar {\Psi }}iv\Psi \right\},}">

<mrow>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j\equiv 2\Im \left\{{\bar {\Psi }}iv\Psi \right\},}</annotation>

</semantics>

</math></span><img src="./7b2f14200ec990f33b5c73b5710103519fd68078.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:16.211ex; height:3.176ex;" alt="{\displaystyle j\equiv 2\Im \left\{{\bar {\Psi }}iv\Psi \right\},}" loading="lazy"></span> with <span class="texhtml mvar" style="font-style:italic;">v</span> the velocity operator, equal to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\equiv -i[X,\,H]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\equiv -i[X,\,H]}</annotation>

</semantics>

</math></span><img src="./ff86da0eb15dec9bdec0748133ae8b8fdd59d7ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.595ex; height:2.843ex;" alt="{\displaystyle v\equiv -i[X,\,H]}" loading="lazy"></span> and <span class="texhtml mvar" style="font-style:italic;">X</span> is the position operator on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}\left(\mathbb {Z} \right).}">

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}\left(\mathbb {Z} \right).}</annotation>

</semantics>

</math></span><img src="./dda9db548f398695b57aa30f6375804c0bce401d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.805ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}\left(\mathbb {Z} \right).}" loading="lazy"></span> Since <span class="texhtml mvar" style="font-style:italic;">V</span> is usually a multiplication operator on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(\mathbb {Z} ),}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(\mathbb {Z} ),}</annotation>

</semantics>

</math></span><img src="./64f1f3f10f2cbe9b78d2fde3a40d2f0210cf59be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.03ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(\mathbb {Z} ),}" loading="lazy"></span> we get to safely write <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i[X,\,H]=-i[X,\,-\Delta ]=-i\left[X,\,-S-S^{\ast }\right]=iS-iS^{\ast }.}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -i[X,\,H]=-i[X,\,-\Delta ]=-i\left[X,\,-S-S^{\ast }\right]=iS-iS^{\ast }.}</annotation>

</semantics>

</math></span></span>

</p><p>As a result, we find:

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</mover>

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<mtd>

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<mrow>

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<mrow>

<mrow>

<mrow>

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<mtr>

<mtd>

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}j\left(x\right)\equiv 2\Im \left\{{\bar {\Psi }}(x)iv\Psi (x)\right\}&=2\Im \left\{{\bar {\Psi }}(x)\left((-S\Psi )(x)+\left(S^{\ast }\Psi \right)(x)\right)\right\}\\&=2\Im \left\{{\bar {\Psi }}(x)\left(-\Psi (x-1)+\Psi (x+1)\right)\right\}\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFPaulBaschnagel1999" class="citation book cs1">Paul, Wolfgang; Baschnagel, Jörg (1999). <i>Stochastic Processes : From Physics to Finance</i>. Berlin: Springer. p. 84. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-66560-9</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFMcMahon2008" class="citation book cs1">McMahon, D. (2008). <i>Quantum Field Theory</i>. McGraw Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-154382-8</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFBallentine1990" class="citation book cs1">Ballentine, Leslie E. (1990). <i>Quantum Mechanics</i>. Prentice Hall Advanced Reference Series. Vol. 280. Englewood Cliffs: Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-747932-8</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">see page 473, equation 115.4, <cite id="CITEREFL.D._Landau,_E.M._Lifschitz" class="citation web cs1">L.D. Landau, E.M. Lifschitz. <a rel="nofollow" class="external text" href="https://ia803206.us.archive.org/4/items/landau-and-lifshitz-physics-textbooks-series/Vol%203%20-%20Landau%2C%20Lifshitz%20-%20Quantum%20mechanics%20-%20non-relativistic%20theory%20%283ed.%2C%201991%29.pdf">"COURSE OF THEORETICAL PHYSICS Vol. 3 – Quantum Mechanics"</a> <span class="cs1-format">(PDF)</span>. <i>ia803206.us.archive.org</i> (3rd ed.)<span class="reference-accessdate">. Retrieved <span class="nowrap">29 April</span> 2023</span>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/spectrpy/nmr/nmr2.htm">"Spin Properties of Nuclei"</a>. <i>www2.chemistry.msu.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">29 April</span> 2023</span>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-146100-0</bdi></span>

</li>

<li id="cite_note-:1-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-:1_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFSakuraiNapolitano2021" class="citation book cs1">Sakurai, Jun John; Napolitano, Jim (2021). <i>Modern quantum mechanics</i> (3rd ed.). Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-108-47322-4</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFResnickEisberg1985" class="citation book cs1">Resnick, R.; Eisberg, R. (1985). <i>Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles</i> (2nd ed.). John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-87373-X</bdi>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-02" href="https://en.wikipedia.org/wiki/?title=Probability_current&oldid=1293557729">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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