The delta-function potential: exactly solvable bound state and scattering

Intuition Beginner

A delta-function potential is a square well squeezed to zero width while keeping its area (strength) constant. Think of it as an infinitely narrow, infinitely deep spike at a single point. Despite its simplicity, it captures real physics: a short-range impurity in a wire, a thin barrier between two materials, or a point-like defect in a crystal.

For an attractive delta well (a negative spike), the particle can be trapped in a bound state. Because the well has zero width, it supports exactly one bound state — never more. The wave function forms a symmetric peak centred on the spike, decaying exponentially in both directions. The tighter the spike (stronger the attraction), the deeper the bound state and the more localised the wave function.

For a particle with enough energy to pass over the spike, the delta potential acts as a partial barrier. The particle can be reflected or transmitted, just like light hitting a thin film. The probabilities depend on the strength of the spike and the particle's energy, but unlike the finite square well, there are no resonances — the transmission is a smooth, monotonic function of energy.

The key mathematical feature is what happens to the wave function at the spike. The wave function itself is continuous, but its slope jumps. The size of the jump is proportional to the strength of the delta potential. This discontinuity in the derivative replaces the two matching conditions of the finite square well with a single condition, making the algebra much cleaner.

Visual Beginner

The diagram shows three things. First, the potential: a single downward spike at the origin, representing $-\alpha \delta (x)$ with $\alpha >0$. Second, the bound-state wave function: a tent-like peak that is continuous at $x=0$ but has a visible change in slope — the left half rises with slope $+\kappa A$ and the right half falls with slope $-\kappa A$. Third, a scattering state: an incoming wave from the left, a partially reflected wave, and a partially transmitted wave continuing to the right.

Worked example Beginner

An electron ($m=9.109\times 10^{-31}$ kg) sits in a delta-function potential $V(x)=-\alpha \delta (x)$ with $\alpha =10\text{eV}\cdot \text{nm}$.

Bound-state energy. The single bound state has energy

$$E=-\frac{m{\alpha }^2}{2{\hslash }^2}.$$

Converting $\alpha =10\times 1.602\times 10^{-19}\times 10^{-9}=1.602\times 10^{-27}$ J$\cdot$m:

$$E=-\frac{9.109\times 10^{-31}\times (1.602\times 10^{-27}{)}^2}{2\times (1.055\times 10^{-34}{)}^2}=-\frac{9.109\times 10^{-31}\times 2.566\times 10^{-54}}{2\times 1.113\times 10^{-68}}=-\frac{2.338\times 10^{-84}}{2.226\times 10^{-68}}\approx -1.05\times 10^{-16}\text{J}\approx -0.656\text{eV}.$$

Penetration depth. The decay constant is $\kappa =m\alpha /{\hslash }^2$:

$$\kappa =\frac{9.109\times 10^{-31}\times 1.602\times 10^{-27}}{(1.055\times 10^{-34}{)}^2}=\frac{1.459\times 10^{-57}}{1.113\times 10^{-68}}\approx 1.31\times 10^{11}{\text{m}}^{-1}.$$

The penetration depth is $1/\kappa \approx 7.6\times 10^{-12}$ m = 0.0076 nm. The bound state is extremely localised around the spike.

Transmission at 1 eV. For a scattering state with $E=1$ eV, $k=\sqrt{2mE}/\hslash \approx 5.12\times 10^9{\text{m}}^{-1}$. The dimensionless parameter is $\beta =m\alpha /({\hslash }^{2}k)\approx 25.6$. The transmission coefficient is

$$T=\frac{1}{1+{\beta }^2}=\frac{1}{1+655}\approx \mathrm{0.0015.}$$

Only about 0.15% of the particle flux passes through — the spike is almost opaque at this energy.

Check your understanding Beginner

Formal definition Intermediate+

Consider the potential

$$V(x)=-\alpha \delta (x),\alpha >0,$$

where $\delta (x)$ is the Dirac delta distribution. The time-independent Schrödinger equation is

$$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}-\alpha \delta (x)\psi (x)=E\psi (x).$$

Because the delta function is supported at a single point, the Schrödinger equation reduces to the free-particle equation for $x\ne 0$. The effect of the delta function appears entirely through the matching condition at the origin.

Derivation of the matching condition

Integrate the Schrödinger equation from $-ϵ$ to $+ϵ$ and take $ϵ\to 0^{+}$:

$$-\frac{{\hslash }^2}{2m}{\int }_{-ϵ}^{+ϵ}{\psi }^{\prime \prime }(x)dx-\alpha {\int }_{-ϵ}^{+ϵ}\delta (x)\psi (x)dx=E{\int }_{-ϵ}^{+ϵ}\psi (x)dx.$$

The right side vanishes as $ϵ\to 0$ (assuming $\psi$ is bounded). The second term on the left gives $-\alpha \psi (0)$ by the sifting property. The first term gives $-\frac{{\hslash }^2}{2m}[{\psi }^{\prime }(ϵ)-{\psi }^{\prime }(-ϵ)]$. Taking $ϵ\to 0$:

$${\psi }^{\prime }(0^{+})-{\psi }^{\prime }(0^{-})=-\frac{2m\alpha }{{\hslash }^2}\psi (0).$$

This is the central result: the derivative of $\psi$ has a jump discontinuity at $x=0$ proportional to the value of $\psi$ at the origin and to the strength $\alpha$. The wave function itself is continuous: $\psi (0^{+})=\psi (0^{-})=\psi (0)$.

Bound state ($E<0$)

For $E<0$, define $\kappa =\sqrt{-2mE}/\hslash >0$. The free-particle solutions away from the origin are exponentials:

$$\psi (x)=\{\begin{array}{ll}A{e}^{\kappa x}& x<0,\\ A{e}^{-\kappa x}& x>0,\end{array}$$

where continuity at $x=0$ forces the same coefficient $A$ on both sides and the exponential decay at $\pm \infty$ has eliminated the growing solutions. The derivative jump condition gives:

$$-A\kappa -A\kappa =-\frac{2m\alpha }{{\hslash }^2}A⟹2\kappa =\frac{2m\alpha }{{\hslash }^2}.$$

So $\kappa =m\alpha /{\hslash }^2$ and the bound-state energy is

$$E=-\frac{{\hslash }^2{\kappa }^2}{2m}=-\frac{m{\alpha }^2}{2{\hslash }^2}.$$

Normalisation requires ${\int }_{-\infty }^{\infty }|\psi {|}^{2}dx=|A{|}^2/\kappa =1$, giving $A=\sqrt{\kappa }=\sqrt{m\alpha /{\hslash }^2}$. There is exactly one bound state.

Scattering states ($E>0$)

For $E>0$, define $k=\sqrt{2mE}/\hslash$. A particle incident from the left has the stationary scattering state:

$$\psi (x)=\{\begin{array}{ll}{e}^{ikx}+r{e}^{-ikx}& x<0,\\ t{e}^{ikx}& x>0,\end{array}$$

where $r$ is the reflection amplitude and $t$ is the transmission amplitude. Continuity at $x=0$: $1+r=t$. The derivative jump: $ikt-ik(1-r)=-(2m\alpha /{\hslash }^2)t$. Solving these two equations:

$$t=\frac{1}{1+i\beta },r=\frac{-i\beta }{1+i\beta },\beta =\frac{m\alpha }{{\hslash }^{2}k}.$$

The transmission and reflection coefficients are:

$$T=|t{|}^2=\frac{1}{1+{\beta }^2},R=|r{|}^2=\frac{{\beta }^2}{1+{\beta }^2},R+T=1.$$

At high energies ($k\to \infty$, $\beta \to 0$), $T\to 1$ and the spike becomes transparent. At low energies ($k\to 0$, $\beta \to \infty$), $T\to 0$ and the spike is nearly opaque.

Domain considerations

The Hamiltonian $H=-{\hslash }^2/(2m)d^2/d{x}^2-\alpha \delta (x)$ is defined on a domain of functions in $H^2(\mathbb{R}\setminus \{0\})\cap H^1(\mathbb{R})$ satisfying the jump condition ${\psi }^{\prime }(0^{+})-{\psi }^{\prime }(0^{-})=-(2m\alpha /{\hslash }^2)\psi (0)$. This domain makes $H$ self-adjoint. The delta-function potential is a special case of a point interaction — the most general self-adjoint extension of the free Hamiltonian restricted to $\mathbb{R}\setminus \{0\}$ involves four parameters (the Berezin-Friedrichs family), of which the delta interaction is the physically most important one-parameter subfamily.

Counterexamples to common slips

• The bound-state energy is proportional to ${\alpha }^2$, not $\alpha$. Doubling the well strength quadruples the binding energy.
• The derivative discontinuity is $-2m\alpha /{\hslash }^2$ times $\psi (0)$, not ${\psi }^{\prime }(0)$. Students sometimes confuse which quantity is continuous and which jumps.
• The transmission coefficient $T=1/(1+{\beta }^2)$ is the same for the repulsive barrier $+\alpha \delta (x)$ and the attractive well $-\alpha \delta (x)$. The sign of $\alpha$ does not affect scattering probabilities, only whether a bound state exists.
• There is no resonance ($T<1$ for all finite energies). Unlike the finite square well, the single-point potential has no width to set up constructive interference.

Key theorem with proof Intermediate+

Theorem (Spectrum of the delta-function potential). The Hamiltonian $H=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}-\alpha \delta (x)$ with $\alpha >0$ on $L^2(\mathbb{R})$ has exactly one bound state with energy $E_0=-m{\alpha }^2/(2{\hslash }^2)$ and a continuous spectrum $[0,\infty )$. The bound-state eigenfunction is ${\psi }_0(x)=\sqrt{m\alpha /{\hslash }^2}\exp (-m\alpha |x|/{\hslash }^2)$. For scattering states of energy $E={\hslash }^2{k}^2/(2m)$, the transmission and reflection coefficients are $T=1/(1+{\beta }^2)$ and $R={\beta }^2/(1+{\beta }^2)$ where $\beta =m\alpha /({\hslash }^{2}k)$.

Proof.

Step 1: Bound state. For $E<0$, set $\kappa =\sqrt{-2mE}/\hslash$. The normalisable solutions of the free Schrödinger equation are $\psi =A{e}^{\kappa x}$ for $x<0$ and $\psi =B{e}^{-\kappa x}$ for $x>0$. Continuity at $x=0$ gives $A=B$. The jump condition ${\psi }^{\prime }(0^{+})-{\psi }^{\prime }(0^{-})=-(2m\alpha /{\hslash }^2)\psi (0)$ gives $-B\kappa -A\kappa =-(2m\alpha /{\hslash }^2)A$, hence $2\kappa =2m\alpha /{\hslash }^2$. This is a single equation for $\kappa$, yielding exactly one solution $\kappa =m\alpha /{\hslash }^2$ and hence one bound-state energy $E_0=-{\hslash }^2{\kappa }^2/(2m)=-m{\alpha }^2/(2{\hslash }^2)$.

Step 2: No second bound state. Any additional bound state would need a different value of $\kappa$, but the jump condition fixes $\kappa$ uniquely. There is no free parameter left — the delta well has exactly one bound state.

Step 3: Scattering states. For $E>0$ the wave function is oscillatory with $k=\sqrt{2mE}/\hslash$. The scattering ansatz $e^{ikx}+r{e}^{-ikx}$ (left) and $t{e}^{ikx}$ (right) with continuity $1+r=t$ and jump condition $ikt-ik(1-r)=-(2m\alpha /{\hslash }^2)t$ yields $t=1/(1+i\beta )$ and $r=-i\beta /(1+i\beta )$. Computing $T=|t{|}^2=1/(1+{\beta }^2)$ and $R=|r{|}^2={\beta }^2/(1+{\beta }^2)$, with $R+T=1$.

Step 4: Self-adjointness. The domain of $H$ is the set of all $\psi \in L^2(\mathbb{R})$ such that $\psi \in H^2(\mathbb{R}\setminus \{0\})$, $\psi \in H^1(\mathbb{R})$ (continuity at $x=0$), and the jump condition holds. Integration by parts on both half-lines with the jump condition reproduces the boundary terms, establishing $⟨\varphi ,H\psi ⟩=⟨H\varphi ,\psi ⟩$ for all $\psi ,\varphi$ in the domain. $\square$

Corollary (Limit of the finite square well). The delta-function potential is the limit of a finite square well of depth $V_0$ and half-width $a$ as $a\to 0$ with $2a{V}_0=\alpha$ held constant. In this limit the finite well's ground state energy converges to $-m{\alpha }^2/(2{\hslash }^2)$ and all excited bound states are pushed into the continuum and lost.

Bridge. The delta-function potential is the exactly solvable entry point for point-interaction scattering theory. The single matching condition (derivative jump) replaces the four matching conditions of the finite square well, producing closed-form results for both bound states and scattering. The same point-interaction framework extends to multiple delta potentials (double-delta, periodic delta arrays) and connects directly to the Kronig-Penney model of band theory.

Exercises Intermediate+

Lean formalization Intermediate+

The delta-function potential raises interesting formalization challenges because the potential is a distribution, not a function. The key objects are:

• The Dirac delta $\delta (x)$ as a distribution (Mathlib has this).
• The Schrödinger equation with distributional potential, interpreted through the matching condition on $H^2(\mathbb{R}\setminus \{0\})\cap H^1(\mathbb{R})$.
• The unique bound state ${\psi }_0(x)=\sqrt{\kappa }{e}^{-\kappa |x|}$ and its energy.
• The scattering amplitudes $t=1/(1+i\beta )$ and $r=-i\beta /(1+i\beta )$.

Mathlib has the exponential function, complex arithmetic, and distribution theory. It does not have: the self-adjointness of the point-interaction Hamiltonian; the identification of the spectrum as one discrete eigenvalue plus $[0,\infty )$; or the scattering theory with explicit computation of $T$ and $R$. A formalization would proceed by defining the domain with the jump condition, proving self-adjointness by integration by parts on each half-line, then computing the spectrum directly. This unit ships with lean_status: none.

The double delta-function potential and molecular binding Master

Even and odd bound states

Two delta-function wells at $x=\pm a$,

$$V(x)=-\alpha \delta (x+a)-\alpha \delta (x-a),$$

provide the simplest model of a diatomic molecule: two short-range attractive centres separated by a distance $2a$. The system has the parity symmetry $\psi (-x)=\pm \psi (x)$, so bound states come in even and odd varieties.

The even-parity transcendental equation (Exercise 4) is

$${\kappa }_{\text{even}}=\frac{m\alpha }{{\hslash }^2}(1+e^{-2{\kappa }_{\text{even}}a}),$$

and the odd-parity equation (Exercise 7) is

$${\kappa }_{\text{odd}}=\frac{m\alpha }{{\hslash }^2}(1-e^{-2{\kappa }_{\text{odd}}a}).$$

Since $1+e^{-2\kappa a}>1-e^{-2\kappa a}$ for any $\kappa a>0$, the even state always has a larger $\kappa$ and hence a more negative (deeper) energy. The even state is the bonding orbital and the odd state is the antibonding orbital.

For $a\to \infty$ (widely separated wells), both $\kappa$ values approach $m\alpha /{\hslash }^2$ — each well binds independently and the two states become degenerate. The splitting $\Delta E=E_{\text{odd}}-E_{\text{even}}>0$ is the tunnelling splitting, caused by the wave function leaking from one well to the other. For large $a$:

$$\Delta E\approx \frac{2m{\alpha }^2}{{\hslash }^2}{e}^{-2m\alpha a/{\hslash }^2}.$$

The exponential dependence on separation is characteristic of tunnelling.

For $a\to 0$, the even state has $\kappa \to 2m\alpha /{\hslash }^2$ (a single delta of strength $2\alpha$), while the odd state disappears: it requires $a>{\hslash }^2/(2m\alpha )$ to exist. This is the exact analogue of the finite square well losing bound states as the width shrinks.

The Kronig-Penney model

A periodic array of delta functions,

$$V(x)=-\alpha \sum _{n=-\infty }^{\infty }\delta (x-na),$$

is the Kronig-Penney model — the simplest model that produces energy bands in a periodic potential. By Bloch's theorem, the eigenstates satisfy $\psi (x+a)=e^{iqa}\psi (x)$ where $q$ is the Bloch wave number.

In a single period $[0,a]$, the wave function is $\psi (x)=A{e}^{ikx}+B{e}^{-ikx}$ for $0<x<a$ (where $E={\hslash }^2{k}^2/(2m)$). Applying Bloch periodicity and the derivative jump at $x=0$ (equivalently at $x=a$ by periodicity) yields the condition

$$\cos (qa)=\cos (ka)+\frac{m\alpha }{{\hslash }^{2}k}\sin (ka).$$

Define $\beta =m\alpha /({\hslash }^{2}k)$. The right-hand side $f(k)=\cos (ka)+\beta \sin (ka)$ is not bounded by $\pm 1$ in general. Allowed energy bands correspond to values of $k$ where $|f(k)|\le 1$ (so a real $q$ exists), and band gaps correspond to $|f(k)|>1$ (no real $q$, exponential decay — the state cannot propagate).

The band structure is most transparent in the limit $\beta \ll 1$ (weak delta potentials). Near the band edges $ka=n\pi$, the function $f(k)$ exceeds $\pm 1$ by an amount proportional to $\beta$, opening narrow gaps. Near $ka=0$, $f(k)\approx 1$ and the lowest band spans $0\le ka\lesssim \pi -\beta$.

In the limit $\alpha \to 0$, $f(k)\to \cos (ka)$, and all band gaps close, recovering the free-particle spectrum $E={\hslash }^2{k}^2/(2m)$. In the limit $\alpha \to \infty$, the delta functions become impenetrable barriers and the spectrum converges to the particle-in-a-box levels $E_n=n^2{\pi }^2{\hslash }^2/(2m{a}^2)$ on each period.

The Kronig-Penney model is the starting point for the band theory of solids. The periodic delta functions represent the ionic potentials in a one-dimensional crystal. The band gaps determine whether the material is a conductor, semiconductor, or insulator: if the Fermi level falls in a band gap, the material is an insulator; if it falls in an allowed band, the material is a conductor.

Point interactions and self-adjoint extensions Master

The Berezin-Friedrichs classification

The free Hamiltonian $H_0=-{\hslash }^2/(2m)d^2/d{x}^2$ restricted to $\mathbb{R}\setminus \{0\}$ (functions that vanish at the origin) is symmetric but not self-adjoint. By von Neumann's theory of self-adjoint extensions, all self-adjoint extensions are parametrised by a $2\times 2$ unitary matrix $U$ relating the boundary values. The most general point interaction is defined by the boundary conditions

$$\left(\begin{array}{c}\psi (0^{+})\\ {\psi }^{\prime }(0^{+})\end{array}\right)=U\left(\begin{array}{c}\psi (0^{-})\\ -{\psi }^{\prime }(0^{-})\end{array}\right),$$

where $U$ is a $2\times 2$ unitary matrix. The four-parameter family of self-adjoint extensions includes:

1. Free particle ($U=I$, identity): $\psi (0^{+})=\psi (0^{-})$ and ${\psi }^{\prime }(0^{+})=-(-{\psi }^{\prime }(0^{-}))={\psi }^{\prime }(0^{-})$. Full continuity.
2. Delta interaction ($U$ with one parameter $\alpha$): $\psi$ continuous, ${\psi }^{\prime }$ has a jump proportional to $\psi (0)$.
3. Delta-prime interaction: ${\psi }^{\prime }$ continuous, $\psi$ has a jump proportional to ${\psi }^{\prime }(0)$.
4. Separated half-lines: $U=-I$, completely decoupling the left and right half-lines.

The delta-function potential is the most physically important member of this family. Its single parameter $\alpha$ controls the strength of the interaction and interpolates between the free particle ($\alpha =0$) and the decoupled half-lines ($\alpha \to \infty$).

Renormalisation and the delta potential in higher dimensions

In two and three dimensions, a delta-function potential is problematic: the Schrödinger equation with $V(\mathbf{r})=-\alpha {\delta }^{(d)}(\mathbf{r})$ in $d\ge 2$ dimensions has no well-defined self-adjoint extension for a fixed coupling $\alpha$. The issue is that the Green's function $G(\mathbf{r};E)$ diverges at $\mathbf{r}=0$ in $d\ge 2$, and the naive matching condition fails.

The resolution is renormalisation: one introduces a cutoff (e.g., a finite well of width $ϵ$), computes the bound-state energy or scattering length as a function of $ϵ$ and the bare coupling ${\alpha }_0(ϵ)$, and then takes $ϵ\to 0$ while holding a physical observable (such as the scattering length $a_s$) fixed. The bare coupling ${\alpha }_0$ must be adjusted ("renormalised") as $ϵ\to 0$ to keep $a_s$ constant.

In two dimensions, the bound-state energy scales as $E\propto e^{-4\pi {\hslash }^2/(m{\alpha }_0)}$ — a non-perturbative result that cannot be expanded in powers of ${\alpha }_0$. In three dimensions, the analogous result is the Bethe-Peierls boundary condition: $\psi (r)\propto (1/r-1/a_s)$ as $r\to 0$, where $a_s$ is the $s$-wave scattering length. This is the starting point for the effective-range expansion in nuclear physics and for the description of Feshbach resonances in ultracold atomic gases.

The one-dimensional delta potential avoids all of these complications — the Green's function is finite at the origin in 1D, and the matching condition is well-defined without any renormalisation. This is why the 1D delta potential is exactly solvable while its higher-dimensional counterparts require regularisation.

Connections Master

The delta-function potential connects to several areas of physics and mathematics.

In solid-state physics, the Kronig-Penney model is the zeroth-order model for electronic band structure. The periodic delta potentials represent the ionic cores in a crystal lattice, and the band gaps predict whether a material conducts. The nearly-free electron model and the tight-binding model are refinements that replace the delta functions with more realistic atomic potentials.

In nuclear physics, short-range nuclear forces are modelled by delta-function (contact) interactions. The effective-range expansion for nucleon-nucleon scattering starts from a delta-function $s$-wave interaction, with corrections from the effective range. The Bethe-Peierls condition is used extensively in few-body nuclear physics.

In ultracold atomic physics, the $s$-wave scattering length $a_s$ of two atoms in a dilute gas is the effective coupling constant of a delta-function interaction. The Feshbach resonance — where $a_s$ can be tuned from $-\infty$ to $+\infty$ by adjusting a magnetic field — is a direct experimental realisation of varying the effective $\alpha$ in a delta potential. The unitarity limit ($|a_s|\to \infty$) produces universal thermodynamics independent of the microscopic potential.

In mathematical physics, point interactions are a well-studied class of exactly solvable models. The four-parameter family of self-adjoint extensions of the Laplacian on $\mathbb{R}\setminus \{0\}$ (the Berezin-Friedrichs family) provides a complete classification. These models are used to study spectral theory, resonance phenomena, and the behaviour of quantum systems with singular potentials.

In quantum field theory, contact interactions of the form $g{\delta }^{(3)}(\mathbf{x})$ appear in the Lagrangian density of scalar field theories. The renormalisation of these interactions — the need to absorb divergences into redefined coupling constants — is the simplest instance of the renormalisation procedure that pervades QFT. The 1D delta potential is the non-relativistic, one-dimensional toy model for this process.

Historical notes Master

The delta-function potential was first analysed systematically by Kronig and Penney (1931) in their model of electrons in a one-dimensional crystal lattice. Their paper introduced the periodic delta array as a tractable model for band structure, replacing the realistic (but analytically intractable) periodic Coulomb potentials of the ionic lattice with delta functions.

The single delta-function bound state and scattering problem appears as a textbook exercise in many quantum mechanics texts, with the most thorough treatment given by Griffiths (Ch. 2.5). The mathematical theory of point interactions was developed by Berezin and Friedrichs in the 1960s and 1970s, providing a rigorous foundation for the delta-function potential as a self-adjoint extension of the free Hamiltonian.

The connection to renormalisation in QFT is discussed in detail by Jackiw (1991) in his analysis of delta-function potentials in two and three dimensions. The 1D case is straightforwardly renormalisable (no divergences), while the 2D and 3D cases require the full machinery of renormalisation to define. This makes the 1D delta potential a pedagogically valuable bridge between quantum mechanics and quantum field theory.

The double delta-function potential as a model for molecular binding was introduced by quantum chemists in the 1950s as the simplest illustration of the bonding-antibonding orbital splitting. The tunnelling splitting $\Delta E\propto e^{-2\kappa a}$ is the same mechanism that underlies the hydrogen molecular ion $H_2^{+}$ and the ammonia inversion doubling.

Bibliography Master

• Griffiths, D. J. and Schroeter, D. F. Introduction to Quantum Mechanics, 3rd ed. Cambridge University Press, 2018. Ch. 2.5.
• Shankar, R. Principles of Quantum Mechanics, 2nd ed. Plenum, 2012. Ch. 5.
• Cohen-Tannoudji, C., Diu, B. and Laloe, F. Quantum Mechanics, Vol. 1. Wiley, 1991. Complement B-I.
• Kronig, R. de L. and Penney, W. G. "Quantum Mechanics of Electrons in Crystal Lattices." Proc. R. Soc. Lond. A 130, 499-513 (1931).
• Albeverio, S., Gesztesy, F., Hoegh-Krohn, R. and Holden, H. Solvable Models in Quantum Mechanics, 2nd ed. AMS Chelsea, 2005.
• Jackiw, R. "Delta-function potentials in two-dimensional and three-dimensional quantum mechanics." In M. A. B. Bég Memorial Volume (1991).
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