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<title>Delta potential</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>

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

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<p>In <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> the <b>delta potential</b> is a <a href="Potential_well" title="Potential well">potential well</a> mathematically described by the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> - a <a href="Generalized_function" title="Generalized function">generalized function</a>. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.

</p><p>The delta potential well is a <a href="Limiting_case_(mathematics)" title="Limiting case (mathematics)">limiting case</a> of the <a href="Finite_potential_well" title="Finite potential well">finite potential well</a>, which is obtained if one maintains the product of the width of the well and the potential constant while decreasing the well's width and increasing the potential.

</p><p>This article, for simplicity, only considers a one-dimensional potential well, but analysis could be expanded to more dimensions.

</p>

<div class="mw-heading mw-heading2"><h2 id="Single_delta_potential">Single delta potential</h2></div>

<p>The time-independent <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> for the <a href="Wave_function" title="Wave function">wave function</a> <span class="texhtml"><i>ψ</i>(<i>x</i>)</span> of a particle in one dimension in a <a href="Scalar_potential" title="Scalar potential">potential</a> <span class="texhtml"><i>V</i>(<i>x</i>)</span> is

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+V(x)\psi (x)=E\psi (x),}</annotation>

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where <span class="texhtml mvar" style="font-style:italic;">ħ</span> is the reduced <a href="Planck_constant" title="Planck constant">Planck constant</a>, and <span class="texhtml mvar" style="font-style:italic;">E</span> is the <a href="Energy" title="Energy">energy</a> of the particle.

</p><p>The delta potential is the potential

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<annotation encoding="application/x-tex">{\displaystyle V(x)=\lambda \delta (x),}</annotation>

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where <span class="texhtml"><i>δ</i>(<i>x</i>)</span> is the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>.

</p><p>It is called a <i>delta potential well</i> if <span class="texhtml mvar" style="font-style:italic;">λ</span> is negative, and a <i>delta potential barrier</i> if <span class="texhtml mvar" style="font-style:italic;">λ</span> is positive. The delta has been defined to occur at the origin for simplicity; a shift in the delta function's argument does not change any of the following results.

</p>

<div class="mw-heading mw-heading3"><h3 id="Solving_the_Schrödinger_equation">Solving the Schrödinger equation</h3></div>

<p>Source:<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 potential splits the space in two parts (<span class="texhtml"><i>x</i> < 0</span> and <span class="texhtml"><i>x</i> > 0</span>). In each of these parts the potential is zero, and the Schrödinger equation reduces to

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}=-{\frac {2mE}{\hbar ^{2}}}\psi ;}</annotation>

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this is a <a href="Linear_differential_equation" title="Linear differential equation">linear differential equation</a> with <a href="Constant_coefficients" class="mw-redirect" title="Constant coefficients">constant coefficients</a>, whose solutions are <a href="Linear_combination" title="Linear combination">linear combinations</a> of <span class="texhtml"><i>e<sup>ikx</sup></i></span> and <span class="texhtml"><i>e</i><sup>−<i>ikx</i></sup></span>, where the <a href="Wave_number" class="mw-redirect" title="Wave number">wave number</a> <span class="texhtml mvar" style="font-style:italic;">k</span> is related to the energy by

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<annotation encoding="application/x-tex">{\displaystyle k={\frac {\sqrt {2mE}}{\hbar }}.}</annotation>

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</p><p>In general, due to the presence of the delta potential in the origin, the coefficients of the solution need not be the same in both half-spaces:

<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 (x)={\begin{cases}\psi _{\text{L}}(x)=A_{\text{r}}e^{ikx}+A_{\text{l}}e^{-ikx},&{\text{ if }}x<0,\\\psi _{\text{R}}(x)=B_{\text{r}}e^{ikx}+B_{\text{l}}e^{-ikx},&{\text{ if }}x>0,\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)={\begin{cases}\psi _{\text{L}}(x)=A_{\text{r}}e^{ikx}+A_{\text{l}}e^{-ikx},&{\text{ if }}x<0,\\\psi _{\text{R}}(x)=B_{\text{r}}e^{ikx}+B_{\text{l}}e^{-ikx},&{\text{ if }}x>0,\end{cases}}}</annotation>

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where, in the case of positive energies (real <span class="texhtml mvar" style="font-style:italic;">k</span>), <span class="texhtml"><i>e<sup>ikx</sup></i></span> represents a wave traveling to the right, and <span class="texhtml"><i>e</i><sup>−<i>ikx</i></sup></span> one traveling to the left.

</p><p>One obtains a relation between the coefficients by imposing that the wavefunction be continuous at the origin:

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<annotation encoding="application/x-tex">{\displaystyle \psi (0)=\psi _{L}(0)=\psi _{R}(0)=A_{r}+A_{l}=B_{r}+B_{l},}</annotation>

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</p><p>A second relation can be found by studying the derivative of the wavefunction. Normally, we could also impose differentiability at the origin, but this is not possible because of the delta potential. However, if we integrate the Schrödinger equation around <span class="texhtml"><i>x</i> = 0</span>, over an interval <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 [-\varepsilon ,\varepsilon ]}">

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<annotation encoding="application/x-tex">{\displaystyle [-\varepsilon ,\varepsilon ]}</annotation>

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</math></span><img src="./b0cdf7c5d5d8f4048ab95ea906ff07e95837d0d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.303ex; height:2.843ex;" alt="{\displaystyle [-\varepsilon ,\varepsilon ]}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}\int _{-\varepsilon }^{+\varepsilon }\psi ''(x)\,dx+\int _{-\varepsilon }^{+\varepsilon }V(x)\psi (x)\,dx=E\int _{-\varepsilon }^{+\varepsilon }\psi (x)\,dx.}</annotation>

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</p><p>In the limit 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 \varepsilon \to 0}">

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

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}[\psi _{R}'(0)-\psi _{L}'(0)]+\lambda \psi (0),}</annotation>

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because

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\varepsilon }^{+\varepsilon }\psi ''(x)\,dx=[\psi '(+\varepsilon )-\psi '(-\varepsilon )].}</annotation>

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Substituting the definition of <span class="texhtml mvar" style="font-style:italic;">ψ</span> into this expression yields

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}ik(-A_{r}+A_{l}+B_{r}-B_{l})+\lambda (A_{r}+A_{l})=0.}</annotation>

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</p><p>The boundary conditions thus give the following restrictions on the coefficients

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}A_{r}+A_{l}-B_{r}-B_{l}&=0,\\-A_{r}+A_{l}+B_{r}-B_{l}&={\frac {2m\lambda }{ik\hbar ^{2}}}(A_{r}+A_{l}).\end{cases}}}</annotation>

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</p>

<div class="mw-heading mw-heading3"><h3 id="Bound_state_(E_<_0)">Bound state (<i>E</i> < 0)</h3></div>

<p>In any one-dimensional attractive potential there will be a <a href="Bound_state" title="Bound state">bound state</a>. To find its energy, note that for <span class="texhtml"><i>E</i> < 0</span>, <span class="texhtml"><i>k</i> = <i>i</i><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2<i>m</i>|<i>E</i>|</span></span>/<i>ħ</i> = <i>iκ</i></span> is imaginary, and the wave functions which were oscillating for positive energies in the calculation above are now exponentially increasing or decreasing functions of <i>x</i> (see above). Requiring that the wave functions do not diverge at infinity eliminates half of the terms: <span class="texhtml"><i>A</i><sub>r</sub> = <i>B</i><sub>l</sub> = 0</span>. The wave function is then

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<annotation encoding="application/x-tex">{\displaystyle \psi (x)={\begin{cases}\psi _{\text{L}}(x)=A_{\text{l}}e^{\kappa x},&{\text{ if }}x\leq 0,\\\psi _{\text{R}}(x)=B_{\text{r}}e^{-\kappa x},&{\text{ if }}x\geq 0.\end{cases}}}</annotation>

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</p><p>From the boundary conditions and normalization conditions, it follows that

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}A_{\text{l}}=B_{\text{r}}={\sqrt {\kappa }},\\\kappa =-{\frac {m\lambda }{\hbar ^{2}}},\end{cases}}}</annotation>

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from which it follows that <span class="texhtml mvar" style="font-style:italic;">λ</span> must be negative, that is, the bound state only exists for the well, and not for the barrier. The <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of this wave function is a <a href="Cauchy_distribution#Characteristic_function" title="Cauchy distribution">Lorentzian function</a>.

</p><p>The energy of the bound state is then

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<annotation encoding="application/x-tex">{\displaystyle E=-{\frac {\hbar ^{2}\kappa ^{2}}{2m}}=-{\frac {m\lambda ^{2}}{2\hbar ^{2}}}.}</annotation>

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</p>

<div class="mw-heading mw-heading3"><h3 id="Scattering_(E_>_0)">Scattering (<i>E</i> > 0)</h3></div>

<p>For positive energies, the particle is free to move in either half-space: <span class="texhtml"><i>x</i> < 0</span> or <span class="texhtml"><i>x</i> > 0</span>. It may be scattered at the delta-function potential.

</p><p>The quantum case can be studied in the following situation: a particle incident on the barrier from the left side <span class="texhtml">(<i>A</i><sub>r</sub>)</span>. It may be reflected <span class="texhtml">(<i>A</i><sub>l</sub>)</span> or transmitted <span class="texhtml">(<i>B</i><sub>r</sub>)</span>.

To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations <span class="texhtml"><i>A</i><sub>r</sub> = 1</span> (incoming particle), <span class="texhtml"><i>A</i><sub>l</sub> = <i>r</i></span> (reflection), <span class="texhtml"><i>B</i><sub>l</sub> = 0</span> (no incoming particle from the right) and <span class="texhtml"><i>B</i><sub>r</sub> = <i>t</i></span> (transmission), and solve for <span class="texhtml mvar" style="font-style:italic;">r</span> and <span class="texhtml mvar" style="font-style:italic;">t</span> even though we do not have any equations in <span class="texhtml mvar" style="font-style:italic;">t</span>.

The result is

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<annotation encoding="application/x-tex">{\displaystyle t={\cfrac {1}{1-{\cfrac {m\lambda }{i\hbar ^{2}k}}}},\quad r={\cfrac {1}{{\cfrac {i\hbar ^{2}k}{m\lambda }}-1}}.}</annotation>

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</p><p>Due to the mirror <a href="Symmetry" title="Symmetry">symmetry</a> of the model, the amplitudes for incidence from the right are the same as those from the left. The result is that there is a non-zero probability

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<annotation encoding="application/x-tex">{\displaystyle R=|r|^{2}={\cfrac {1}{1+{\cfrac {\hbar ^{4}k^{2}}{m^{2}\lambda ^{2}}}}}={\cfrac {1}{1+{\cfrac {2\hbar ^{2}E}{m\lambda ^{2}}}}}}</annotation>

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for the particle to be reflected. This does not depend on the sign of <span class="texhtml mvar" style="font-style:italic;">λ</span>, that is, a barrier has the same probability of reflecting the particle as a well. This is a significant difference from <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, where the reflection probability would be 1 for the barrier (the particle simply bounces back), and 0 for the well (the particle passes through the well undisturbed).

</p><p>The probability for transmission is

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<annotation encoding="application/x-tex">{\displaystyle T=|t|^{2}=1-R={\cfrac {1}{1+{\cfrac {m^{2}\lambda ^{2}}{\hbar ^{4}k^{2}}}}}={\cfrac {1}{1+{\cfrac {m\lambda ^{2}}{2\hbar ^{2}E}}}}.}</annotation>

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</p>

<div class="mw-heading mw-heading3"><h3 id="Remarks_and_application">Remarks and application</h3></div>

<p>The calculation presented above may at first seem unrealistic and hardly useful. However, it has proved to be a suitable model for a variety of real-life systems.

</p><p>One such example regards the interfaces between two <a href="Electrical_conductivity" class="mw-redirect" title="Electrical conductivity">conducting</a> materials. In the bulk of the materials, the motion of the electrons is quasi-free and can be described by the <a href="Kinetic_term" title="Kinetic term">kinetic term</a> in the above Hamiltonian with an <a href="Effective_mass_(solid-state_physics)" title="Effective mass (solid-state physics)">effective mass</a> <span class="texhtml mvar" style="font-style:italic;">m</span>. Often, the surfaces of such materials are covered with oxide layers or are not ideal for other reasons. This thin, non-conducting layer may then be modeled by a local delta-function potential as above. Electrons may then tunnel from one material to the other giving rise to a current.

</p><p>The operation of a <a href="Scanning_tunneling_microscope" title="Scanning tunneling microscope">scanning tunneling microscope</a> (STM) relies on this tunneling effect. In that case, the barrier is due to the air between the tip of the STM and the underlying object. The strength of the barrier is related to the separation being stronger the further apart the two are. For a more general model of this situation, see <a href="Finite_potential_barrier_(QM)" class="mw-redirect" title="Finite potential barrier (QM)">Finite potential barrier (QM)</a>. The delta function potential barrier is the limiting case of the model considered there for very high and narrow barriers.

</p><p>The above model is one-dimensional while the space around us is three-dimensional. So, in fact, one should solve the Schrödinger equation in three dimensions. On the other hand, many systems only change along one coordinate direction and are translationally invariant along the others. The Schrödinger equation may then be reduced to the case considered here by an Ansatz for the wave function of the type <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,y,z)=\psi (x)\phi (y,z)\,\!}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi (x,y,z)=\psi (x)\phi (y,z)\,\!}</annotation>

</semantics>

</math></span><img src="./fa3f70c1ea8173b84fec61e97c53fbe8ac93108f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:23.869ex; height:2.843ex;" alt="{\displaystyle \Psi (x,y,z)=\psi (x)\phi (y,z)\,\!}" loading="lazy"></span>.

</p><p>Alternatively, it is possible to generalize the delta function to exist on the surface of some domain <i>D</i> (see <a href="Laplacian_of_the_indicator" title="Laplacian of the indicator">Laplacian of the indicator</a>).<sup id="cite_ref-Lange_2012_2-0" class="reference"><a href="#cite_note-Lange_2012-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>The delta function model is actually a one-dimensional version of the <a href="Hydrogen_atom" title="Hydrogen atom">Hydrogen atom</a> according to the <i>dimensional scaling</i> method developed by the group of <a href="Dudley_R._Herschbach" title="Dudley R. Herschbach">Dudley R. Herschbach</a><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>

The delta function model becomes particularly useful with the <i>double-well</i> Dirac Delta function model which represents a one-dimensional version of the <a href="Hydrogen_molecule_ion" class="mw-redirect" title="Hydrogen molecule ion">Hydrogen molecule ion</a>, as shown in the following section.

</p>

<div class="mw-heading mw-heading2"><h2 id="Double_delta_potential">Double delta potential</h2></div>

<p>The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation:

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+V(x)\psi (x)=E\psi (x),}</annotation>

</semantics>

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

where the potential is now

<mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(x)=-q\left[\delta \left(x+{\frac {R}{2}}\right)+\lambda \delta \left(x-{\frac {R}{2}}\right)\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 0<R<\infty }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./fda38a174244607174938a4b715851d66fed1ea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.447ex; height:2.176ex;" alt="{\displaystyle 0<R<\infty }" loading="lazy"></span> is the "internuclear" distance with Dirac delta-function (negative) peaks located at <span class="texhtml"><i>x</i> = ±<i>R</i>/2</span> (shown in brown in the diagram). Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use <a href="Atomic_units" title="Atomic units">atomic units</a> and set <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 \hbar =m=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \hbar =m=1}</annotation>

</semantics>

</math></span><img src="./29d75fe7f363953c54924c4f6e404fd82a7d29e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.706ex; height:2.176ex;" alt="{\displaystyle \hbar =m=1}" loading="lazy"></span>. 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 0<\lambda <1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<\lambda <1}</annotation>

</semantics>

</math></span><img src="./1e4999282806a988bca6204e12ad076984f8cea4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.877ex; height:2.176ex;" alt="{\displaystyle 0<\lambda <1}" loading="lazy"></span> is a formally adjustable parameter. From the single-well case, we can infer the "<a href="Ansatz" title="Ansatz">ansatz</a>" for the solution to be

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (x)=Ae^{-d\left|x+{\frac {R}{2}}\right|}+Be^{-d\left|x-{\frac {R}{2}}\right|}.}</annotation>

</semantics>

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

Matching of the wavefunction at the Dirac delta-function peaks yields the determinant

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mtext>where </mtext>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}q-d&qe^{-dR}\\q\lambda e^{-dR}&q\lambda -d\end{vmatrix}}=0,\quad {\text{where }}E=-{\frac {d^{2}}{2}}.}</annotation>

</semantics>

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

Thus, <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 d}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> is found to be governed by the <i>pseudo-quadratic</i> equation

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

<msub>

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</msub>

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</msup>

</mrow>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{\pm }(\lambda )={\frac {1}{2}}q(\lambda +1)\pm {\frac {1}{2}}\left\{q^{2}(1+\lambda )^{2}-4\lambda q^{2}\left[1-e^{-2d_{\pm }(\lambda )R}\right]\right\}^{1/2},}</annotation>

</semantics>

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

which has two solutions <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 d=d_{\pm }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=d_{\pm }}</annotation>

</semantics>

</math></span><img src="./eb99c7dc89ac4837aeda8fecd1bc316ed72acdb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.034ex; height:2.509ex;" alt="{\displaystyle d=d_{\pm }}" loading="lazy"></span>. For the case of equal charges (symmetric homonuclear case), <span class="texhtml"><i>λ</i> = 1</span>, and the pseudo-quadratic reduces to

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{\pm }=q\left[1\pm e^{-d_{\pm }R}\right].}</annotation>

</semantics>

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

The "+" case corresponds to a wave function symmetric about the midpoint (shown in red in the diagram), where <span class="texhtml"><i>A</i> = <i>B</i></span>, and is called <i><a href="Molecular_term_symbol" title="Molecular term symbol">gerade</a></i>. Correspondingly, the "−" case is the wave function that is anti-symmetric about the midpoint, where <span class="texhtml"><i>A</i> = −<i>B</i></span>, and is called <i>ungerade</i> (shown in green in the diagram). They represent an approximation of the two lowest discrete energy states of the three-dimensional <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 {\ce {H2^+}}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\ce {H2^+}}}</annotation>

</semantics>

</math></span><img src="./b4bd681c1d347b9f0767eb35b39cd55cffe45819.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.254ex; height:3.176ex;" alt="{\displaystyle {\ce {H2^+}}}" loading="lazy"></span> and are useful in its analysis. Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by<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>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{\pm }=q+W(\pm qRe^{-qR})/R,}</annotation>

</semantics>

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

where <i>W</i> is the standard <a href="Lambert_W_function" title="Lambert W function">Lambert <i>W</i> function</a>. Note that the lowest energy corresponds to the symmetric solution <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 d_{+}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./38d541b460e2682b9f68c293273b3a60d41b3bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.509ex;" alt="{\displaystyle d_{+}}" loading="lazy"></span>. In the case of <i>unequal</i> charges, and for that matter the three-dimensional molecular problem, the solutions are given by a <i>generalization</i> of the Lambert <i>W</i> function (see <a href="Lambert_W_function#Generalizations" title="Lambert W function">Lambert W function § Generalizations</a>).

</p><p>One of the most interesting cases is when <i>qR</i> ≤ 1, which results in <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 d_{-}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./451b4631b3b2570ec52dee410c13231890c873e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.981ex; height:2.509ex;" alt="{\displaystyle d_{-}=0}" loading="lazy"></span>. Thus, one has a non-trivial bound state solution with <span class="texhtml"><i>E</i> = 0</span>. For these specific parameters, there are many interesting properties that occur, one of which is the unusual effect that the <a href="Transmission_coefficient" title="Transmission coefficient">transmission coefficient</a> is unity at zero energy.<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>

</p>

<ul><li><a href="Free_particle" title="Free particle">Free particle</a></li>

<li><a href="Particle_in_a_box" title="Particle in a box">Particle in a box</a></li>

<li><a href="Finite_potential_well" title="Finite potential well">Finite potential well</a></li>

<li><a href="Particle_in_a_ring" title="Particle in a ring">Particle in a ring</a></li>

<li><a href="Particle_in_a_spherically_symmetric_potential" title="Particle in a spherically symmetric potential">Particle in a spherically symmetric potential</a></li>

<li><a href="Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">Quantum harmonic oscillator</a></li>

<li><a href="Hydrogen_atom" title="Hydrogen atom">Hydrogen atom</a> or <a href="Hydrogen-like_atom" title="Hydrogen-like atom">hydrogen-like atom</a></li>

<li><a href="Ring_wave_guide" class="mw-redirect" title="Ring wave guide">Ring wave guide</a></li>

<li><a href="Particle_in_a_one-dimensional_lattice_(periodic_potential)" class="mw-redirect" title="Particle in a one-dimensional lattice (periodic potential)">Particle in a one-dimensional lattice (periodic potential)</a></li>

<li><a href="Hydrogen_molecular_ion" class="mw-redirect" title="Hydrogen molecular ion">Hydrogen molecular ion</a></li>

<li><a href="Holstein%E2%80%93Herring_method" title="Holstein–Herring method">Holstein–Herring method</a></li>

<li><a href="Laplacian_of_the_indicator" title="Laplacian of the indicator">Laplacian of the indicator</a></li>

<li><a href="List_of_quantum-mechanical_systems_with_analytical_solutions" title="List of quantum-mechanical systems with analytical solutions">List of quantum-mechanical systems with analytical solutions</a></li></ul>

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

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

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

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://physics.stackexchange.com/questions/92240/wave-function-with-a-delta-potential">"quantum mechanics - Wave function with a delta potential"</a>. <i>Physics Stack Exchange</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-03-29</span></span>.</cite></span>

</li>

<li id="cite_note-Lange_2012-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lange_2012_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFLange2012" class="citation cs2">Lange, Rutger-Jan (2012), "Potential theory, path integrals and the Laplacian of the indicator", <i>Journal of High Energy Physics</i>, <b>2012</b> (11): <span class="nowrap">1–</span>49, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1302.0864">1302.0864</a></span>, <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/2012JHEP...11..032L">2012JHEP...11..032L</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FJHEP11%282012%29032">10.1007/JHEP11(2012)032</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:56188533">56188533</a></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"><a href="Dudley_R._Herschbach" title="Dudley R. Herschbach">D.R. Herschbach</a>, J.S. Avery, and O. Goscinski (eds.), <i>Dimensional Scaling in Chemical Physics</i>, Springer, (1992). <a rel="nofollow" class="external autonumber" href="https://www.amazon.com/Dimensional-Scaling-Chemical-Physics-Herschbach/dp/0792320360">[1]</a></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">T. C. Scott, J. F. Babb, <a href="Alexander_Dalgarno" title="Alexander Dalgarno">A. Dalgarno</a> and John D. Morgan III, <a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1993JChPh..99.2841S">"The Calculation of Exchange Forces: General Results and Specific Models"</a>, <a href="Journal_of_Chemical_Physics" class="mw-redirect" title="Journal of Chemical Physics">J. Chem. Phys.</a>, 99, pp. 2841–2854, (1993).</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 id="CITEREFvan_DijkKiers1992" class="citation journal cs1">van Dijk, W.; Kiers, K. A. (1992). "Time delay in simple one-dimensional systems". <i>American Journal of Physics</i>. <b>60</b> (6). American Association of Physics Teachers (AAPT): <span class="nowrap">520–</span>527. <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/1992AmJPh..60..520V">1992AmJPh..60..520V</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%2F1.16866">10.1119/1.16866</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9505">0002-9505</a>.</cite></span>

</li>

</ol></div>

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

<ul><li><cite id="CITEREFGriffiths,_David_J.2005" class="citation book cs1">Griffiths, David J. (2005). <i>Introduction to Quantum Mechanics</i> (2nd ed.). Prentice Hall. pp. <span class="nowrap">68–</span>78. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-111892-8</bdi>.</cite></li>

<li>For the 3-dimensional case look for the "delta shell potential"; further see K. Gottfried (1966), <i>Quantum Mechanics Volume I: Fundamentals</i>, ch. III, sec. 15.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="noviewer" typeof="mw:File"></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Delta_potential" class="extiw external" title="commons:Category:Delta potential">Delta potential</a> at Wikimedia Commons</li></ul></div><div><div class="zim-footer">

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