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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>
<div class="hlist">
<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>
<div class="hlist">
<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>
</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">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>
<div class="hlist">
<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>
</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">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>
</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"><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="John_Stewart_Bell" title="John Stewart Bell">Bell</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="David_Bohm" title="David Bohm">Bohm</a></li>
<li><a href="Niels_Bohr" title="Niels Bohr">Bohr</a></li>
<li><a href="Max_Born" title="Max Born">Born</a></li>
<li><a href="Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</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="Vladimir_Fock" title="Vladimir Fock">Fock</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="Willis_Lamb" title="Willis Lamb">Lamb</a></li>
<li><a href="Lev_Landau" title="Lev Landau">Landau</a></li>
<li><a href="Max_von_Laue" title="Max von Laue">Laue</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="Isidor_Isaac_Rabi" class="mw-redirect" title="Isidor Isaac Rabi">Rabi</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="Hermann_Weyl" title="Hermann Weyl">Weyl</a></li>
<li><a href="Wilhelm_Wien" title="Wilhelm Wien">Wien</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>
</div></div></div></td>
</tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><style data-mw-deduplicate="TemplateStyles:r1239400231">
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<p>In <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and <a href="Scattering_theory" class="mw-redirect" title="Scattering theory">scattering theory</a>, the one-dimensional <b>step potential</b> is an idealized system used to model incident, reflected and transmitted <a href="Matter_waves" class="mw-redirect" title="Matter waves">matter waves</a>. The problem consists of solving the time-independent <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> for a particle with a step-like <a href="Potential" title="Potential">potential</a> in one dimension. Typically, the potential is modeled as a <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Calculation">Calculation</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Schrödinger_equation_and_potential_function">Schrödinger equation and potential function</h3></div>
<p>The time-independent Schrödinger equation for the <a href="Wave_function" title="Wave function">wave function</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 \psi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \psi (x)}</annotation>
</semantics>
</math></span><img src="./a596a1fb4130a47f6b88c66150497338bd6cbccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}" loading="lazy"></span> 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 {\hat {H}}\psi (x)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)\right]\psi (x)=E\psi (x),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<mover>
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<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
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<mo>[</mo>
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<mo>−<!-- − --></mo>
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<mfrac>
<msup>
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<mi>m</mi>
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<mfrac>
<msup>
<mi>d</mi>
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<mn>2</mn>
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<mo>+</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
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<mo>]</mo>
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<mi>ψ<!-- ψ --></mi>
<mo stretchy="false">(</mo>
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<mi>ψ<!-- ψ --></mi>
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<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}\psi (x)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)\right]\psi (x)=E\psi (x),}</annotation>
</semantics>
</math></span></span>
where <i>Ĥ</i> is the <a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, <i>ħ</i> is the reduced <a href="Planck_constant" title="Planck constant">Planck constant</a>, <i>m</i> is the <a href="Mass" title="Mass">mass</a>, <i>E</i> the energy of the particle. The step potential is simply the product of <i>V</i><sub>0</sub>, the height of the barrier, and the <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>:
<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 V(x)={\begin{cases}0,&x<0\\V_{0},&x\geq 0\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mn>0</mn>
<mo>,</mo>
</mtd>
<mtd>
<mi>x</mi>
<mo><</mo>
<mn>0</mn>
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</mtr>
<mtr>
<mtd>
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>,</mo>
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<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle V(x)={\begin{cases}0,&x<0\\V_{0},&x\geq 0\end{cases}}}</annotation>
</semantics>
</math></span></span>
</p><p>The barrier is positioned at <i>x</i> = 0, though any position <i>x</i><sub>0</sub> may be chosen without changing the results, simply by shifting position of the step by −<i>x</i><sub>0</sub>.
</p><p>The first term in 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="{\textstyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>−<!-- − --></mo>
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<mfrac>
<msup>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mrow>
<mn>2</mn>
<mi>m</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mrow>
<mi>d</mi>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
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<mi>ψ<!-- ψ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi }</annotation>
</semantics>
</math></span><img src="./a4737894fd1c2158ff9da44f736b78a120dfab38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.89ex; height:4.343ex;" alt="{\textstyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi }" loading="lazy"></span> is the <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a> of the particle.
</p>
<div class="mw-heading mw-heading3"><h3 id="Solution">Solution</h3></div>
<p>The step divides space in two parts: <i>x</i> < 0 and <i>x</i> > 0. In any of these parts the potential is constant, meaning the particle is quasi-free, and the solution of the Schrödinger equation can be written as a <a href="Quantum_superposition" title="Quantum superposition">superposition</a> of left and right moving waves (see <a href="Free_particle" title="Free particle">free particle</a>)
</p>
<dl><dd><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 \psi _{1}(x)=\left(A_{\rightarrow }e^{ik_{1}x}+A_{\leftarrow }e^{-ik_{1}x}\right)\quad x<0,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">→<!-- → --></mo>
</mrow>
</msub>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo>+</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">←<!-- ← --></mo>
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<msup>
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<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
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<mn>1</mn>
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<mo>)</mo>
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<mspace width="1em"></mspace>
<mi>x</mi>
<mo><</mo>
<mn>0</mn>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi _{1}(x)=\left(A_{\rightarrow }e^{ik_{1}x}+A_{\leftarrow }e^{-ik_{1}x}\right)\quad x<0,}</annotation>
</semantics>
</math></span><img src="./49fd1e8e313dd043a6007b69b0e7c25e04b144d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.261ex; height:3.343ex;" alt="{\displaystyle \psi _{1}(x)=\left(A_{\rightarrow }e^{ik_{1}x}+A_{\leftarrow }e^{-ik_{1}x}\right)\quad x<0,}" loading="lazy"></span></dd>
<dd><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 \psi _{2}(x)=\left(B_{\rightarrow }e^{ik_{2}x}+B_{\leftarrow }e^{-ik_{2}x}\right)\quad x>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">→<!-- → --></mo>
</mrow>
</msub>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mi>x</mi>
</mrow>
</msup>
<mo>+</mo>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">←<!-- ← --></mo>
</mrow>
</msub>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
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<msub>
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<mn>2</mn>
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<mspace width="1em"></mspace>
<mi>x</mi>
<mo>></mo>
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<annotation encoding="application/x-tex">{\displaystyle \psi _{2}(x)=\left(B_{\rightarrow }e^{ik_{2}x}+B_{\leftarrow }e^{-ik_{2}x}\right)\quad x>0}</annotation>
</semantics>
</math></span><img src="./68acc426be3825becb5769462e199bf94c102898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.656ex; height:3.343ex;" alt="{\displaystyle \psi _{2}(x)=\left(B_{\rightarrow }e^{ik_{2}x}+B_{\leftarrow }e^{-ik_{2}x}\right)\quad x>0}" loading="lazy"></span></dd></dl>
<p>where subscripts 1 and 2 denote the regions <i>x</i> < 0 and <i>x</i> > 0 respectively, the subscripts (→) and (←) on the amplitudes <i>A</i> and <i>B</i> denote the direction of the particle's velocity vector: right and left respectively.
</p><p>The <a href="Wave_vector" title="Wave vector">wave vectors</a> in the respective regions being
</p>
<dl><dd><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 k_{1}={\sqrt {2mE/\hbar ^{2}}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
<mi>m</mi>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msup>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k_{1}={\sqrt {2mE/\hbar ^{2}}},}</annotation>
</semantics>
</math></span><img src="./c1aee36c0afdc9253d83801f631c290a7d20f252.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.852ex; height:4.843ex;" alt="{\displaystyle k_{1}={\sqrt {2mE/\hbar ^{2}}},}" loading="lazy"></span></dd>
<dd><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 k_{2}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
<mi>m</mi>
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msup>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k_{2}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}}</annotation>
</semantics>
</math></span><img src="./1b954cbbd34f111729e87129bb5f614930df1223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:23.264ex; height:4.843ex;" alt="{\displaystyle k_{2}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}}" loading="lazy"></span></dd></dl>
<p>both of which have the same form as the <a href="De_Broglie_relation" class="mw-redirect" title="De Broglie relation">De Broglie relation</a> (in one dimension)
</p>
<dl><dd><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 p=\hbar k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p=\hbar k}</annotation>
</semantics>
</math></span><img src="./24fee69175538303b28ac54e907baf53d0a58dbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.875ex; height:2.509ex;" alt="{\displaystyle p=\hbar k}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Boundary_conditions">Boundary conditions</h3></div>
<p>The coefficients <i>A</i>, <i>B</i> have to be found from the <a href="Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a> of the wave function at <i>x</i> = 0. The wave function and its derivative have to be <a href="Continuous_function" title="Continuous function">continuous</a> everywhere, so:
</p>
<dl><dd><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 \psi _{1}(0)=\psi _{2}(0),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo stretchy="false">(</mo>
<mn>0</mn>
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<mo>=</mo>
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<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \psi _{1}(0)=\psi _{2}(0),}</annotation>
</semantics>
</math></span><img src="./ae0e64e9d9d278ca2bdc4ea059feccfa712dee9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.824ex; height:2.843ex;" alt="{\displaystyle \psi _{1}(0)=\psi _{2}(0),}" loading="lazy"></span></dd>
<dd><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 \left.{\frac {d\psi _{1}}{dx}}\right|_{x=0}=\left.{\frac {d\psi _{2}}{dx}}\right|_{x=0}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left.{\frac {d\psi _{1}}{dx}}\right|_{x=0}=\left.{\frac {d\psi _{2}}{dx}}\right|_{x=0}.}</annotation>
</semantics>
</math></span><img src="./48ac01c1aadbd9778d001b4b602e8f3de16b0633.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.824ex; height:6.009ex;" alt="{\displaystyle \left.{\frac {d\psi _{1}}{dx}}\right|_{x=0}=\left.{\frac {d\psi _{2}}{dx}}\right|_{x=0}.}" loading="lazy"></span></dd></dl>
<p>Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients
</p>
<dl><dd><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 (A_{\rightarrow }+A_{\leftarrow })=(B_{\rightarrow }+B_{\leftarrow })}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">→<!-- → --></mo>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">←<!-- ← --></mo>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">→<!-- → --></mo>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">←<!-- ← --></mo>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (A_{\rightarrow }+A_{\leftarrow })=(B_{\rightarrow }+B_{\leftarrow })}</annotation>
</semantics>
</math></span><img src="./72d5eb8b848067a16b557f12e5f760b2458cbe35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.914ex; height:2.843ex;" alt="{\displaystyle (A_{\rightarrow }+A_{\leftarrow })=(B_{\rightarrow }+B_{\leftarrow })}" loading="lazy"></span></dd>
<dd><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 k_{1}(A_{\rightarrow }-A_{\leftarrow })=k_{2}(B_{\rightarrow }-B_{\leftarrow })}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">→<!-- → --></mo>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">←<!-- ← --></mo>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">→<!-- → --></mo>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">←<!-- ← --></mo>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k_{1}(A_{\rightarrow }-A_{\leftarrow })=k_{2}(B_{\rightarrow }-B_{\leftarrow })}</annotation>
</semantics>
</math></span><img src="./46cfd3ad6ef9247a931c3260592da7c497e47fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.445ex; height:2.843ex;" alt="{\displaystyle k_{1}(A_{\rightarrow }-A_{\leftarrow })=k_{2}(B_{\rightarrow }-B_{\leftarrow })}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Transmission_and_reflection">Transmission and reflection</h2></div>
<p>It is useful to compare the situation to the <a href="Classical_mechanics" title="Classical mechanics">classical</a> case. In both cases, the particle behaves as a free particle outside of the barrier region. A classical particle with energy <i>E</i> larger than the barrier height <i>V</i><sub>0</sub> will be slowed down but never reflected by the barrier, while a classical particle with <i>E</i> < <i>V</i><sub>0</sub> incident on the barrier from the left would always be reflected. Once we have found the quantum-mechanical result we will return to the question of how to recover the classical limit.
</p><p>To study the quantum case, consider the following situation: a particle incident on the barrier from the left side <i>A</i><sub>→</sub>. It may be reflected (<i>A</i><sub>←</sub>) or transmitted (<i>B</i><sub>→</sub>). Here and in the following assume <i>E</i> > <i>V</i><sub>0</sub>.
</p><p>To find the amplitudes for reflection and transmission for incidence from the left, we set in the above equations <i>A</i><sub>→</sub> = 1 (incoming particle), <i>A</i><sub>←</sub> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>R</i></span></span> (reflection), <i>B</i><sub>←</sub> = 0 (no incoming particle from the right) and <i>B</i><sub>→</sub> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>Tk</i><sub>1</sub>/<i>k</i><sub>2</sub></span></span> (transmission <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>). We then solve for <i>T</i> and <i>R</i>.
</p><p>The result is:
</p>
<dl><dd><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 {\sqrt {T}}={\frac {2{\sqrt {k_{1}k_{2}}}}{k_{1}+k_{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>T</mi>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</msqrt>
</mrow>
</mrow>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {T}}={\frac {2{\sqrt {k_{1}k_{2}}}}{k_{1}+k_{2}}}}</annotation>
</semantics>
</math></span><img src="./ae2791a89a2c238c95c985ca30de3ce40eb8b567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.524ex; height:6.676ex;" alt="{\displaystyle {\sqrt {T}}={\frac {2{\sqrt {k_{1}k_{2}}}}{k_{1}+k_{2}}}}" loading="lazy"></span></dd>
<dd><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 {\sqrt {R}}={\frac {k_{1}-k_{2}}{k_{1}+k_{2}}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>R</mi>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {R}}={\frac {k_{1}-k_{2}}{k_{1}+k_{2}}}.}</annotation>
</semantics>
</math></span><img src="./455ebd533d947d8bd616cd5bc34d35ec94eb0039.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.653ex; height:5.843ex;" alt="{\displaystyle {\sqrt {R}}={\frac {k_{1}-k_{2}}{k_{1}+k_{2}}}.}" loading="lazy"></span></dd></dl>
<p>The model is symmetric with respect to a <a href="Parity_transformation" class="mw-redirect" title="Parity transformation">parity transformation</a> and at the same time interchange <i>k</i><sub>1</sub> and <i>k</i><sub>2</sub>. For incidence from the right we have therefore the amplitudes for transmission and reflection
</p>
<dl><dd><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 {\sqrt {T'}}={\sqrt {T}}={\frac {2{\sqrt {k_{1}k_{2}}}}{k_{1}+k_{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mi>T</mi>
<mo>′</mo>
</msup>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>T</mi>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</msqrt>
</mrow>
</mrow>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {T'}}={\sqrt {T}}={\frac {2{\sqrt {k_{1}k_{2}}}}{k_{1}+k_{2}}}}</annotation>
</semantics>
</math></span><img src="./b8345563be7ad6783cad2ce9c418d6ce76d63b4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.351ex; height:6.676ex;" alt="{\displaystyle {\sqrt {T'}}={\sqrt {T}}={\frac {2{\sqrt {k_{1}k_{2}}}}{k_{1}+k_{2}}}}" loading="lazy"></span></dd>
<dd><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 {\sqrt {R'}}=-{\sqrt {R}}={\frac {k_{2}-k_{1}}{k_{1}+k_{2}}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mi>R</mi>
<mo>′</mo>
</msup>
</msqrt>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>R</mi>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mrow>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {R'}}=-{\sqrt {R}}={\frac {k_{2}-k_{1}}{k_{1}+k_{2}}}.}</annotation>
</semantics>
</math></span><img src="./64e1aa144f8fd7e98e027c409e17c02175bf94de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.332ex; height:5.843ex;" alt="{\displaystyle {\sqrt {R'}}=-{\sqrt {R}}={\frac {k_{2}-k_{1}}{k_{1}+k_{2}}}.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Analysis_of_the_expressions">Analysis of the expressions</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Energy_less_than_step_height_(E_<_V0)">Energy less than step height (<i>E</i> < <i>V</i><sub>0</sub>)</h3></div>
<p>For energies <i>E</i> < <i>V</i><sub>0</sub>, the wave function to the right of the step is exponentially decaying over a distance <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 1/(k_{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1/(k_{2})}</annotation>
</semantics>
</math></span><img src="./2b81e3973e9ddbd32edf59da54244e0b7fc4e9a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.4ex; height:2.843ex;" alt="{\displaystyle 1/(k_{2})}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Energy_greater_than_step_height_(E_>_V0)">Energy greater than step height (<i>E</i> > <i>V</i><sub>0</sub>)</h3></div>
<p>In this energy range the transmission and reflection coefficient differ from the classical case. They are the same for incidence from the left and right:
</p>
<dl><dd><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 T=|T'|={\frac {4k_{1}k_{2}}{(k_{1}+k_{2})^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>T</mi>
<mo>′</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>4</mn>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T=|T'|={\frac {4k_{1}k_{2}}{(k_{1}+k_{2})^{2}}}}</annotation>
</semantics>
</math></span><img src="./e4c8866d73e5c35b97996bf09a13f718d0f41d52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.603ex; height:6.176ex;" alt="{\displaystyle T=|T'|={\frac {4k_{1}k_{2}}{(k_{1}+k_{2})^{2}}}}" loading="lazy"></span></dd>
<dd><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 R=|R'|=1-T={\frac {(k_{1}-k_{2})^{2}}{(k_{1}+k_{2})^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>R</mi>
<mo>′</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle R=|R'|=1-T={\frac {(k_{1}-k_{2})^{2}}{(k_{1}+k_{2})^{2}}}}</annotation>
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</math></span><img src="./d9cc25cd9d3ab24eaf46aab9534936427059ca74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.512ex; height:6.676ex;" alt="{\displaystyle R=|R'|=1-T={\frac {(k_{1}-k_{2})^{2}}{(k_{1}+k_{2})^{2}}}}" loading="lazy"></span></dd></dl>
<p>In the limit of large energies <i>E</i> ≫ <i>V</i><sub>0</sub>, we have <i>k</i><sub>1</sub> ≈ <i>k</i><sub>2</sub> and the classical result <i>T</i> = 1, <i>R</i> = 0 is recovered.
</p><p>Thus there is a finite probability for a particle with an energy larger than the step height to be reflected.
</p>
<div class="mw-heading mw-heading2"><h2 id="Negative_steps">Negative steps</h2></div>
<ul><li>In the case of a large positive <i>E</i>, and a small positive step, then <i>T</i> is almost 1.</li>
<li>But, in the case of a small positive <i>E</i> and a large negative <i>V</i>, then <b>R</b> is almost 1.</li></ul>
<p>In other words, a quantum particle reflects off a large potential drop (just as it does off a large potential step). This makes sense in terms of impedance mismatches, but it seems classically counter-intuitive...
</p>
<div class="mw-heading mw-heading2"><h2 id="Classical_limit">Classical limit</h2></div>
<p>The result obtained for R depends only on the ratio <i>E</i>/<i>V</i><sub>0</sub>. This seems superficially to violate the <a href="Correspondence_principle" title="Correspondence principle">correspondence principle</a>, since we obtain a finite probability of reflection regardless of the value of the Planck constant or the mass of the particle. For example, we seem to predict that when a marble rolls to the edge of a table, there can be a large probability that it is reflected back rather than falling off. Consistency with classical mechanics is restored by eliminating the unphysical assumption that the step potential is discontinuous. When the step function is replaced with a ramp that spans some finite distance <i>w</i>, the probability of reflection approaches zero in the limit <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 wk\to \infty }">
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<annotation encoding="application/x-tex">{\displaystyle wk\to \infty }</annotation>
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</math></span><img src="./3a71367c4e069c8f04e35c02535a9721e4c3bd64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.813ex; height:2.176ex;" alt="{\displaystyle wk\to \infty }" loading="lazy"></span>, where <i>k</i> is the wavenumber of the particle.<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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Relativistic_calculation">Relativistic calculation</h2></div>
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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Klein_paradox" title="Klein paradox">Klein paradox</a></div>
<p>The relativistic calculation of a free particle colliding with a step potential can be obtained using <a href="Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">relativistic quantum mechanics</a>. For the case of 1/2 fermions, like <a href="Electron" title="Electron">electrons</a> and <a href="Neutrino" title="Neutrino">neutrinos</a>, the solutions of the <a href="Dirac_equation" title="Dirac equation">Dirac equation</a> for high energy barriers produce transmission and reflection coefficients that are not bounded. This phenomenon is known as the <a href="Klein_paradox" title="Klein paradox">Klein paradox</a>. The apparent paradox disappears in the context of <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<p>The Heaviside step potential mainly serves as an exercise in introductory quantum mechanics, as the solution requires understanding of a variety of quantum mechanical concepts: wavefunction normalization, continuity, incident/reflection/transmission amplitudes, and probabilities.
</p><p>A similar problem to the one considered appears in the physics of normal-metal <a href="Superconductor" class="mw-redirect" title="Superconductor">superconductor</a> interfaces. <a href="Quasiparticle" title="Quasiparticle">Quasiparticles</a> are <a href="Scattering" title="Scattering">scattered</a> at the <a href="Superconductivity" title="Superconductivity">pair potential</a> which in the simplest model may be assumed to have a step-like shape. The solution of the Bogoliubov-de Gennes equation resembles that of the discussed Heaviside-step potential. In the superconductor normal-metal case this gives rise to <a href="Andreev_reflection" title="Andreev reflection">Andreev reflection</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Rectangular_potential_barrier" title="Rectangular potential barrier">Rectangular potential barrier</a></li>
<li><a href="Finite_potential_well" title="Finite potential well">Finite potential well</a></li>
<li><a href="Infinite_potential_well" class="mw-redirect" title="Infinite potential well">Infinite potential well</a></li>
<li><a href="Delta_potential_barrier" class="mw-redirect" title="Delta potential barrier">Delta potential barrier</a></li>
<li><a href="Finite_potential_barrier" class="mw-redirect" title="Finite potential barrier">Finite potential barrier</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> The <a href="Transmission_coefficient" title="Transmission coefficient">transmission coefficient</a> is defined as the ratio of the transmitted <a href="Probability_current" title="Probability current">probability current</a> to the incoming probability current. However, the quantities directly involved in this potential step problem are called <a href="S-matrix" title="S-matrix">scattering amplitudes</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 S_{ij}}">
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<annotation encoding="application/x-tex">{\displaystyle S_{ij}}</annotation>
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</math></span><img src="./7207b997c68f09a399b687baba2430393420dbba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.902ex; height:2.843ex;" alt="{\displaystyle S_{ij}}" loading="lazy"></span>. They are related to the transmission and reflection coefficients <a href="S-matrix#Transmission_coefficient_and_reflection_coefficient" title="S-matrix">here</a>. We can see in <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=bX-k26w-tsU">this YouTube video</a> that the most general expression for <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 A_{\leftarrow }}">
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<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A_{\leftarrow }}</annotation>
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</math></span><img src="./65181f227fc2fb0d7fca8f166fcb760234f9bf94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle A_{\leftarrow }}" loading="lazy"></span> 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 r={\sqrt {R}}}">
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</math></span><img src="./49a9b13263cfcdf061e760c6c72bdfa99e1ab724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.847ex; height:3.009ex;" alt="{\displaystyle r={\sqrt {R}}}" loading="lazy"></span>, and for <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 B_{\rightarrow }}">
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</math></span><img src="./7bf028c5ef708d69e46108be81185b8b9caf681a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.639ex; height:2.509ex;" alt="{\displaystyle B_{\rightarrow }}" loading="lazy"></span> we have the ratio of k-vectors and possibly different masses on their respective sides: <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="{\textstyle t{\sqrt {m_{2}k_{1}/m_{1}k_{2}}}={\sqrt {Tm_{2}k_{1}/m_{1}k_{2}}}}">
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<annotation encoding="application/x-tex">{\textstyle t{\sqrt {m_{2}k_{1}/m_{1}k_{2}}}={\sqrt {Tm_{2}k_{1}/m_{1}k_{2}}}}</annotation>
</semantics>
</math></span><img src="./3d0c1b0d051783326bcb6ab391ad0a5e95bf8e7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.987ex; height:3.343ex;" alt="{\textstyle t{\sqrt {m_{2}k_{1}/m_{1}k_{2}}}={\sqrt {Tm_{2}k_{1}/m_{1}k_{2}}}}" loading="lazy"></span>. The masses come from the definition of the probability current and the k-vectors from the derivatives of the wavefunctions. </span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFBranson1979" class="citation journal cs1">Branson, D. (1979). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://www.deepdyve.com/lp/american-association-of-physics-teachers/correspondence-principle-and-scattering-from-potential-steps-tKM85ATfDZ/1">"The correspondence principle and scattering from potential steps"</a></span>. <i>American Journal of Physics</i>. <b>47</b> (12): <span class="nowrap">1101–</span>1102. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979AmJPh..47.1101B">1979AmJPh..47.1101B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.11582">10.1119/1.11582</a>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2></div>
<ul><li><i>Quantum Mechanics Demystified</i>, D. McMahon, Mc Graw Hill (USA), 2006, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-145546 9</bdi></li>
<li><i>Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition)</i>, R. Eisberg, R. Resnick, John Wiley & Sons, 1985, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-87373-0</bdi></li>
<li><i>Quantum Mechanics</i>, 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></li>
<li><i>Elementary Quantum Mechanics</i>, N.F. Mott, Wykeham Science, Wykeham Press (Taylor & Francis Group), 1972, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-85109-270-5</bdi></li>
<li><i>Stationary States</i>, A. Holden, College Physics Monographs (USA), Oxford University Press, 1971, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-851121-3</bdi></li>
<li><i>Quantum mechanics</i>, E. Zaarur, Y. Peleg, R. Pnini, Schaum's Outlines, Mc Graw Hill (USA), 1998, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>007-0540187</bdi></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li><i>The New Quantum Universe</i>, T.Hey, P.Walters, Cambridge University Press, 2009, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-56457-1</bdi>.</li>
<li><i>Quantum Field Theory</i>, D. McMahon, Mc Graw Hill (USA), 2008, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-154382-8</bdi></li>
<li><i>Quantum mechanics</i>, E. Zaarur, Y. Peleg, R. Pnini, Schaum's Easy Outlines Crash Course, Mc Graw Hill (USA), 2006, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>007-145533-7</bdi> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-007-145533-6</bdi></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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