Delta potential

In quantum mechanics, the delta potential, also known as the delta function potential, is a simplified model for a one-dimensional attractive potential defined by V(x) = -\alpha \delta(x), where \delta(x) is the Dirac delta function and \alpha > 0 is a constant with units of energy times length, representing an infinitely narrow and deep potential well localized at x = 0.^[1]^[2] This potential is strictly speaking a distribution rather than a conventional function, often arising as a limit of finite-width potentials, and serves as a pedagogical tool for exactly solvable problems involving bound states and scattering.^[1]^[3] For bound states with energy E < 0, the time-independent Schrödinger equation yields a single solution, characterized by the wavefunction \psi(x) = \sqrt{\kappa} e^{-\kappa |x|}, where \kappa = m \alpha / \hbar^2 and the binding energy is E = -m \alpha^2 / (2 \hbar^2), with m the particle mass and \hbar the reduced Planck's constant.^[1]^[2] The wavefunction is continuous at x = 0, but its derivative exhibits a discontinuity \Delta (d\psi/dx) = -(2m \alpha / \hbar^2) \psi(0), where \Delta (d\psi/dx) = [d\psi/dx](0+) - [d\psi/dx](0-), reflecting the singular nature of the potential.^[2]^[3]^[4] No excited bound states exist, as higher-energy solutions fail to satisfy normalization and boundary conditions.^[2] For scattering states with E > 0, the potential supports both reflection and transmission, with the transmission coefficient T = k^2 / (k^2 + k_0^2) and reflection coefficient R = k_0^2 / (k^2 + k_0^2), where k = \sqrt{2mE}/\hbar is the wave number and k_0 = m \alpha / \hbar^2.^[1] This model can represent physical scenarios like electron interactions with localized impurities, such as a heavy neutron, and extends to repulsive cases (\alpha < 0) with no bound states but similar scattering behavior.^[3] The delta potential's exact solvability makes it a cornerstone for understanding quantum tunneling, resonance, and the node theorem in introductory quantum mechanics.^[1]^[2]

Fundamentals

Definition

The delta potential in quantum mechanics is a simplified model for a localized attractive interaction, mathematically expressed as
V(x) = -\alpha \, \delta(x),
where \alpha > 0 is the strength parameter and \delta(x) is the Dirac delta function.^[2]^[5] This form represents an idealized point-like potential well that is zero everywhere except at x = 0, where it becomes infinitely deep and narrow. The Dirac delta function \delta(x) possesses the defining property that it is zero for all x \neq 0 and integrates to unity over the entire real line: \int_{-\infty}^{\infty} \delta(x) \, dx = 1.^[5] This sifting property allows \delta(x) to model an impulsive interaction concentrated at a single point, serving as an idealized representation of short-range forces in one-dimensional quantum systems. For a wave function \psi(x) satisfying the time-independent Schrödinger equation with this potential, the boundary conditions at x = 0 require continuity of the wave function itself, \psi(0^+) = \psi(0^-), while the first derivative exhibits a discontinuity:
\psi'(0^+) - \psi'(0^-) = -\frac{2m \alpha}{\hbar^2} \psi(0),
where m is the particle mass and \hbar is the reduced Planck's constant.^[5] These conditions arise from integrating the Schrödinger equation across the singularity at x = 0. The parameter \alpha has dimensions of energy multiplied by length, ensuring that V(x) carries the appropriate units of energy given the inverse-length dimension of \delta(x). This delta potential often approximates more realistic finite-range interactions in scenarios where the range is negligible compared to other length scales.

Physical Significance

The delta potential models short-range interactions in quantum mechanics by idealizing them as infinitely narrow and strong forces, capturing essential physics when the interaction range is negligible compared to other system scales. This approximation is particularly valuable in atomic physics for representing contact interactions in few-body systems like ultracold atoms or molecular clusters, where it simplifies the description of s-wave scattering without losing key low-energy behaviors. In solid-state physics, it serves as a building block for periodic structures, approximating electron-lattice interactions in crystalline materials.^[6]^[7] Historically, the delta potential gained prominence through its use in the Kronig-Penney model, introduced in 1931 to explain energy band structures in solids by treating the periodic potential as a series of delta functions. This approach provided an exactly solvable framework for understanding electron motion in crystals, influencing early developments in solid-state theory. In quantum field theory, delta potentials act as toy models for point-like interactions, analogous to contact terms in effective field theories, highlighting renormalization needs similar to those in particle physics.^[8] One key advantage of the delta potential is its exact solvability via the time-independent Schrödinger equation, allowing precise calculations of wavefunctions and energies that reveal quantum phenomena approximated in finite-range potentials, such as the emergence of bound states in attractive cases despite the idealized point-like nature. This exactness aids pedagogical insights into tunneling, scattering, and localization effects central to quantum behavior. However, as an idealization, its zero width introduces ultraviolet divergences in perturbative expansions or higher-dimensional treatments, requiring regularization techniques like cutoff procedures or renormalization to ensure physical consistency.^[9]^[10]

Single Delta Potential

Time-Independent Schrödinger Equation

The time-independent Schrödinger equation for a particle of mass m in one dimension interacting with an attractive delta potential V(x) = -\alpha \delta(x), where \alpha > 0 sets the interaction strength, reads

-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} - \alpha \delta(x) \psi(x) = E \psi(x),

with \psi(x) the energy eigenfunction and E the energy eigenvalue.^[11] For x \neq 0, the delta function vanishes, so the equation simplifies to the free-particle form, whose general solutions consist of piecewise exponential functions for E < 0 (bound states) or plane waves for E > 0 (scattering states). The overall strategy is to solve this free equation independently in the left (x < 0) and right (x > 0) regions, then enforce matching conditions at x = 0 to ensure a valid global solution.^[2] Explicitly, for scattering states with E > 0, the wave functions are

\psi(x) = \begin{cases} A e^{i k x} + B e^{-i k x} & x < 0, \\ C e^{i k x} + D e^{-i k x} & x > 0, \end{cases}

where k = \sqrt{2 m E}/\hbar is the wave number, and A, B, C, D are complex coefficients to be fixed by the boundary conditions.^[11] The matching conditions arise from integrating the Schrödinger equation over the infinitesimal interval [-\epsilon, \epsilon] around x = 0 and letting \epsilon \to 0^+. The integral of the second-derivative term yields the jump in the first derivative across the origin, while the delta function integrates to -\alpha \psi(0); all other terms vanish in the limit. This enforces continuity of the wave function at x = 0, \psi(0^+) = \psi(0^-), together with a discontinuous derivative,

\left. \frac{d\psi}{dx} \right|_{0^+} - \left. \frac{d\psi}{dx} \right|_{0^-} = -\frac{2 m \alpha}{\hbar^2} \psi(0).

^[11] The potential's even symmetry, V(x) = V(-x), implies that the Hamiltonian is parity-invariant, so eigenfunctions can always be classified by definite parity: even solutions satisfy \psi(x) = \psi(-x) and odd ones satisfy \psi(x) = -\psi(-x). Exploiting this reduces the problem to solving in one half-line with appropriate boundary conditions at the origin.^[2]

Bound States

For the attractive delta potential V(x) = -\alpha \delta(x) with \alpha > 0, there exists exactly one bound state corresponding to a negative energy eigenvalue given by

E = -\frac{m \alpha^2}{2 \hbar^2},

where [m](/page/M) is the particle mass and \hbar is the reduced Planck's constant.^[12]^[13] The corresponding normalized wave function for this bound state is

\psi(x) = \sqrt{\kappa} \, e^{-\kappa |x|},

where \kappa = \frac{m \alpha}{\hbar^2}. This form satisfies the time-independent Schrödinger equation for E < 0 away from x = 0 and the appropriate matching conditions at the delta function location.^[12]^[13] The normalization of the wave function can be verified by evaluating the integral

\int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 2 \kappa \int_{0}^{\infty} e^{-2 \kappa x} \, dx = 2 \kappa \left[ -\frac{e^{-2 \kappa x}}{2 \kappa} \right]_{0}^{\infty} = 1,

confirming that the probability density integrates to unity across the real line.^[12]^[13] Due to the even symmetry of |\psi(x)|^2, the expectation value of position is \langle x \rangle = 0. The expectation value of x^2 is

\langle x^2 \rangle = 2 \kappa \int_{0}^{\infty} x^2 e^{-2 \kappa x} \, dx = \frac{1}{2 \kappa^2},

providing a measure of the spatial spread of the bound state.^[12]^[13] In the repulsive case where \alpha < 0, no bound states exist because the parameter \kappa becomes negative, leading to non-physical exponentially growing solutions rather than decaying ones.^[12]^[13]

Scattering States

For scattering states of the single delta potential V(x) = -\alpha \delta(x), with \alpha > 0, the energy E > 0 corresponds to continuum solutions where a particle incident on the potential can be transmitted or reflected. The wave function takes the form of plane waves asymptotically far from the potential, with the incident wave approaching from the left: \psi(x) \to e^{ikx} + r e^{-ikx} as x \to -\infty, and \psi(x) \to t e^{ikx} as x \to +\infty, where k = \sqrt{2mE}/\hbar, r is the reflection amplitude, and t is the transmission amplitude.^[13] The reflection and transmission amplitudes are derived from the boundary conditions at x = 0: continuity of the wave function and a discontinuity in its derivative given by \psi'(0^+) - \psi'(0^-) = -(2m \alpha / \hbar^2) \psi(0). This yields r = -\frac{\gamma}{2ik + \gamma} and t = 1 + r = \frac{2ik}{2ik + \gamma}, where \gamma = 2m \alpha / \hbar^2.^[13] The transmission probability is T = |t|^2 = \frac{1}{1 + (m \alpha / \hbar^2 k)^2}, which approaches 1 for large k (high energy) and exhibits enhanced transmission relative to naive expectations at low k due to the potential's sharpness. The reflection probability is R = |r|^2 = 1 - T.^[13] In one dimension, the scattering can also be characterized by a phase shift \delta(k) = -\arctan(\gamma / 2k), which encodes the potential's effect on the outgoing wave phase.^[14] In the low-energy limit (k \to 0), T \approx (2k / \gamma)^2, indicating near-total reflection, and the s-wave scattering length is a = \hbar^2 / (m \alpha), providing a measure of the potential's effective range.^[13]

Applications and Limitations

The single delta potential serves as a simplified model for shallow impurities in semiconductors, where dopant atoms create localized attractive potentials that can be approximated as delta functions due to their narrow spatial extent. In delta-doped layers, such as those formed by phosphorus in silicon, this model captures the formation of quasi-two-dimensional electron gases and shallow sub-bands, aiding in the analysis of high-mobility transport properties.^[15]^[16] This potential also arises as an approximation to more realistic finite square wells, obtained in the limit where the well width approaches zero while the product of width and depth remains fixed, preserving the bound state energy and illustrating the transition from extended to point-like interactions.^[2] In quantum mechanics education, the delta potential exemplifies exact solvability for both bound and scattering states, contrasting with the numerical methods required for finite-range potentials and highlighting key concepts like discontinuity in the wave function derivative.^[17] Despite these applications, the model exhibits limitations, particularly at high energies where the transmission coefficient approaches unity, yet the point-like nature leads to non-physical behaviors such as kinked wave functions with power-law tails in momentum space (∝ 1/p²), rendering expectation values of p⁴ and higher powers ill-defined.^[17] In quantum field theory contexts, the ultraviolet divergences necessitate a momentum cutoff for renormalization.^[18] Extensions address these issues through regularization techniques, such as replacing the delta with separable potentials that introduce a finite range while preserving low-energy physics, or embedding in lattice models like Dirac lattices to handle zero-range interactions via self-adjoint extensions.^[19]^[20]

Multiple Delta Potentials

Double Delta Potential

The double delta potential consists of two attractive Dirac delta functions separated by a distance a, defined as

V(x) = -\alpha \left[ \delta\left(x + \frac{a}{2}\right) + \delta\left(x - \frac{a}{2}\right) \right],

where \alpha > 0 is the strength parameter. This configuration models a simple "diatomic molecule" in one dimension, allowing exact solutions for both bound and scattering states due to the piecewise free-particle nature of the Schrödinger equation away from the deltas. The system exhibits interaction effects between the two sites, leading to level splitting compared to isolated single deltas.^[21] For bound states (E < 0), the time-independent Schrödinger equation yields exponentially decaying solutions outside the interval [-a/2, a/2] and hyperbolic functions inside. The wave functions possess definite parity due to the symmetric potential. The even-parity mode has the form \psi_\text{even}(x) = A \cosh(\kappa x) for |x| < a/2 (with \kappa = \sqrt{-2mE}/\hbar > 0) and \psi_\text{even}(x) = B e^{-\kappa |x|} for |x| > a/2. The odd-parity mode is \psi_\text{odd}(x) = A \sinh(\kappa x) for |x| < a/2 and \psi_\text{odd}(x) = C \operatorname{sgn}(x) e^{-\kappa |x|} for |x| > a/2. Matching continuity of the wave function and the derivative jump condition \psi'(x_0^+) - \psi'(x_0^-) = -(2m\alpha/\hbar^2) \psi(x_0) at each delta (x_0 = \pm a/2) yields transcendental equations for the energies. For the even mode,

\tanh\left(\kappa \frac{a}{2}\right) = \frac{\beta}{\kappa} - 1,

where \beta = 2m\alpha / \hbar^2. The odd mode satisfies

\tanh\left(\kappa \frac{a}{2}\right) = 1 - \frac{\beta}{\kappa},

with solutions existing only if \beta a > 2. Thus, there is always one even bound state, but the odd state appears only for sufficiently strong \alpha or large a, resulting in up to two bound states total. As a \to 0, the potential reduces to a single delta with effective strength $2\alpha, and the bound energy approaches E = -m (2\alpha)^2 / (2 \hbar^2).^[21]^[22] For scattering states (E > 0), plane waves describe the motion in the regions x < -a/2, -a/2 < x < a/2, and x > a/2, with wave number k = \sqrt{2mE}/\hbar. The even-parity scattering solution between the deltas takes the form \psi_\text{even}(x) = A \cos(kx) for |x| < a/2, decaying to incoming/outgoing waves outside modulated by reflection. The transfer matrix method efficiently connects the coefficients across the deltas, accounting for the derivative jumps. The transmission coefficient is

T(k) = \frac{1}{1 + \frac{\beta^2}{4k^2} \sin^2(ka)},

where resonances occur at T=1 when \sin(ka) = 0 (i.e., ka = n\pi), corresponding to hard-wall-like states inside the separation. These perfect transmission points manifest the Ramsauer-Townsend effect, where low-energy scattering minima arise from constructive interference, tunable by \alpha and a. Asymmetric configurations (different strengths) introduce additional phase shifts but retain similar structure.^[21]

Kronig-Penney Model

The Kronig-Penney model describes the motion of electrons in a one-dimensional periodic lattice potential, serving as a foundational example for understanding energy band formation in crystalline solids. The potential is given by V(x) = \sum_{n=-\infty}^{\infty} -\alpha \delta(x - na), where a is the lattice constant, \alpha > 0 is the strength of each attractive delta function potential, and the sum extends over all integer sites n. This idealized model simplifies the complex periodic potentials in real crystals while capturing essential features like allowed energy bands and forbidden band gaps.^[23] Due to the translational symmetry of the lattice, solutions to the time-independent Schrödinger equation obey Bloch's theorem, taking the form \psi(x) = e^{i q x} u(x), where q is the Bloch wave vector and u(x) is a periodic function with period a, i.e., u(x + a) = u(x). For scattering states with energy E = \frac{\hbar^2 k^2}{2m} > 0, where k = \sqrt{2mE}/\hbar, the dispersion relation that determines the allowed energies is

\cos(q a) = \cos(k a) + \frac{m \alpha}{\hbar^2 k} \sin(k a).

This transcendental equation relates the energy E (via k) to the Bloch wave vector q, which is confined to the first Brillouin zone -\pi/a \leq q \leq \pi/a.^[23] Allowed energy bands correspond to values of E for which |\cos(q a)| \leq 1, permitting real q for propagating states; regions where |\cos(q a)| > 1 form forbidden band gaps, preventing electron propagation at those energies. Band gaps are particularly prominent at the Brillouin zone edges (q = \pm \pi/a), where the right-hand side of the dispersion relation exceeds unity or falls below minus unity, reflecting Bragg-like scattering from the periodic lattice. In the tight-binding limit of weak \alpha, the model yields an effective mass approximation near band minima, with parabolic dispersion E(q) \approx E_0 + \frac{\hbar^2 (q - q_0)^2}{2 m^*}, where m^* incorporates lattice effects.^[23] The model was originally introduced in 1931 by R. de L. Kronig and W. G. Penney to describe nearly free electrons in metals, using finite-range potentials but inspiring the delta-function simplification for analytical tractability.^[8]