The Dirac Delta Function Potential

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle} evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

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Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

A delta potential, eg. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_0\delta(x-a)\!} , is a special case of the finite square well, where the width of the well goes to zero and the depth of the well goes to infinity, while the produce of the height and depth remains constant. For a delta potential, the wavefunction is still continuous across the potential, ie. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=a\!} . However, the first derivative of the wavefunction is discontinuous across the potential.

For a particle subject to an attractive delta potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x) = -V_0\delta(x)\!} the Schrödinger equation is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2}-V_0\delta(x)\psi(x)=E\psi(x)}

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \neq \!0 } the potential term vanishes, and all that is left is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2 \psi(x)}{dx^2} + \frac{2mE}{\hbar^2}\psi(x) = 0}

A bound state(s) may exist when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E < 0 \! } , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x) \! } vanishes at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \pm \infty \!} . The bound state solutions are therefore given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{1}(x) = Ae^{kx} x < 0 \!}

$\psi _{2}(x)=Be^{-kx}x>0\!$

where

The first boundary condition, the continuity of $\psi (x)\!$ at $x=0\!$, yields $A=B\!$.

The second boundary condition, the discontinuity of ${\frac {d\psi (x)}{dx}}\!$ at $x=0\!$, can be obtained by integrating the Schrödinger equation from $-\epsilon \!$ to $\epsilon \!$ and then letting $\epsilon \rightarrow 0\!$

Integrating the whole equation across the potential gives

$\int _{-\epsilon }^{\epsilon }{\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}dx+\int _{-\epsilon }^{+\epsilon }(-V_{0}\delta (x))\psi (x)dx=\int _{-\epsilon }^{\epsilon }E\psi (x)dx$

In the limit $\epsilon \rightarrow 0\!$, we have

${\frac {-\hbar ^{2}}{2m}}\left[{\frac {d\psi _{2}(0)}{dx}}-{\frac {d\psi _{1}(0)}{dx}}\right]+V_{0}\psi (0)=0$

which yields the relation: $2kA={\frac {2mV_{0}}{\hbar ^{2}}}A\!$.

Since we defined $k={\sqrt {\frac {2m|E|}{\hbar ^{2}}}}\!$, we have ${\sqrt {\frac {2m|E|}{\hbar ^{2}}}}={\frac {mV_{0}}{\hbar ^{2}}}\!$. Then, the energy is $E=-{\frac {mV_{0}^{2}}{2\hbar ^{2}}}\!$

Finally, we normalize $\psi (x)\!$:

$\int _{-\infty }^{\infty }|\psi (x)|dx=2|B|^{2}\int _{0}^{\infty }e^{-2kx}dx={\frac {|B|^{2}}{k}}=1$

so,

$B={\sqrt {k}}={\frac {\sqrt {mV_{0}}}{\hbar }}$

Evidently, the delta function well, regardless of its "strength" $V_{0}\!$, has one bound state:

$\psi (x)={\frac {\sqrt {mV_{0}}}{\hbar }}e^{\frac {-mV_{0}|x|}{\hbar ^{2}}}$

Similarly, for a delta potential of the form $V_{0}\delta (x-a)\!$, the discontinuity of the first derivative can be shown as follows:

The Schrödinger equation is

${\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+V_{0}\delta (x-a)\psi (x)=E\psi (x)$

Integrating the whole equation across the potential gives

$\int _{a-\epsilon }^{a+\epsilon }{\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}dx+\int _{a-\epsilon }^{a+\epsilon }V_{0}\delta (x-a)\psi (x)dx=\int _{a-\epsilon }^{a+\epsilon }E\psi (x)dx$

In the limit $\epsilon \rightarrow 0\!$, we have

${\frac {-\hbar ^{2}}{2m}}\left[{\frac {d\psi (a^{+})}{dx}}-{\frac {d\psi (a^{-})}{dx}}\right]+V_{0}\psi (a)=0$

Hence the first derivative of the wave function across a delta potential is discontinuous by an amount:

${\frac {d\psi (a^{+})}{dx}}-{\frac {d\psi (a^{-})}{dx}}={\frac {2mV_{0}}{\hbar ^{2}}}\psi (a)$

External Link

Additional information on the dirac delta function