The finite square well: bound states, tunneling, and resonant scattering

Intuition Beginner

The particle in a box (unit 12.04.01) has infinitely high walls. No real system has infinitely high walls. The finite square well replaces the infinite walls with walls of finite height $V_0$. The particle can still be trapped, but something new happens: the wave function does not drop to zero at the walls. Instead it leaks into the region outside, decaying exponentially into the "forbidden" zone.

A particle trapped inside a finite well has discrete energy levels, just like the infinite box. Two key differences arise. First, the number of bound states is finite, not infinite. A shallow well may support only one or two. Second, the wave function extends beyond the well boundaries, forming an evanescent tail that decays over a characteristic distance called the penetration depth.

What about particles with energy above the well? Classically they pass straight over without hesitation. Quantum mechanics says otherwise: there is always some probability of reflection, and at certain energies the transmission reaches 100%. These are resonant tunneling energies, where the well becomes perfectly transparent.

The finite square well is the simplest model that shows both bound states (trapped inside, discrete energies) and scattering states (passing over, continuous energies). Every real quantum-confined system — a quantum well in a semiconductor, an electron trapped by an impurity atom, a neutron in a nuclear potential — is described by a finite, not infinite, potential.

The matching conditions at the walls are the central technical tool. At each boundary, the wave function and its first derivative must be continuous. These two conditions, applied at both walls, determine the allowed energies. For bound states the result is a transcendental equation — an equation that cannot be solved in closed form but whose solutions can be found graphically or numerically.

A remarkable property of the one-dimensional finite square well: it always has at least one bound state, no matter how shallow or narrow the well. In three dimensions this is not true — a well that is too shallow supports no bound states at all.

Visual Beginner

The diagram shows the well as a rectangular depression in the potential. Inside, the bound-state wave functions are sinusoidal — the same shapes as the particle in a box. Outside, they decay exponentially. The shallower the well, the longer the evanescent tail and the fewer bound states fit. The scattering state at the top has a shorter wavelength inside the well (higher kinetic energy) and a longer wavelength outside (lower kinetic energy), with amplitude changes at each boundary.

Worked example Beginner

An electron ($m=9.109\times 10^{-31}$ kg) is trapped in a finite square well of width $2a=1$ nm and depth $V_0=10$ eV.

How many bound states? Define $z_0=a\sqrt{2m{V}_0}/\hslash$. With $a=5\times 10^{-10}$ m and $V_0=10\times 1.602\times 10^{-19}$ J:

$$z_0=\frac{5\times 10^{-10}}{1.055\times 10^{-34}}\sqrt{2\times 9.109\times 10^{-31}\times 1.602\times 10^{-18}}\approx \frac{5\times 10^{-10}}{1.055\times 10^{-34}}\times 5.40\times 10^{-24}\approx \mathrm{25.6.}$$

The number of bound states is $N=\lfloor 2z_0/\pi \rfloor +1=\lfloor 51.2/\pi \rfloor +1=\lfloor 16.3\rfloor +1=17$.

Penetration depth for the ground state. The ground state energy is close to the infinite-well value $E_1\approx {\pi }^2{\hslash }^2/(2m(2a{)}^2)\approx 0.376$ eV. The penetration depth is

$$\delta =\frac{\hslash }{\sqrt{2m(V_0-E)}}\approx \frac{1.055\times 10^{-34}}{\sqrt{2\times 9.109\times 10^{-31}\times (10-0.376)\times 1.602\times 10^{-19}}}\approx 0.019\text{nm}.$$

The wave function extends about 0.02 nm beyond each wall — small compared to the 1 nm well width, but nonzero. For a shallower well, the penetration would be much larger.

Check your understanding Beginner

Formal definition Intermediate+

Consider a potential that is zero inside a region of width $2a$ and takes a constant positive value $V_0$ outside:

$$V(x)=\{\begin{array}{ll}0& \text{if }|x|<a,\\ V_0& \text{if }|x|\ge a.\end{array}$$

The potential is symmetric about $x=0$, so the Hamiltonian commutes with the parity operator $P\psi (x)=\psi (-x)$. Energy eigenstates can be chosen with definite parity: even ($\psi (-x)=\psi (x)$) or odd ($\psi (-x)=-\psi (x)$).

Bound states ($0<E<V_0$)

For energies below $V_0$, the solutions are confined. Inside the well the Schrödinger equation is the free-particle equation with wave number $k=\sqrt{2mE}/\hslash$. Outside the well it is exponentially decaying with decay constant $\kappa =\sqrt{2m(V_0-E)}/\hslash$.

Even-parity solutions:

$$\psi (x)=\{\begin{array}{ll}A\cos (kx)& |x|<a,\\ B{e}^{-\kappa |x|}& |x|\ge a.\end{array}$$

Odd-parity solutions:

$$\psi (x)=\{\begin{array}{ll}A\sin (kx)& |x|<a,\\ B\text{sgn}(x)e^{-\kappa |x|}& |x|\ge a.\end{array}$$

Continuity of $\psi$ and ${\psi }^{\prime }$ at $x=a$ gives the transcendental equations:

$$\text{Even:}k\tan (ka)=\kappa ,\text{Odd:}-k\cot (ka)=\kappa .$$

These are conventionally expressed in dimensionless form. Define $z=ka$ and $z_0=a\sqrt{2m{V}_0}/\hslash$. Then $\kappa a=\sqrt{z_0^2-z^2}$ and the equations become:

$$\text{Even:}z\tan z=\sqrt{z_0^2-z^2},\text{Odd:}-z\cot z=\sqrt{z_0^2-z^2}.$$

The parameter $z_0$ encodes the well's strength. The number of bound states is $N=\lfloor 2z_0/\pi \rfloor +1$, always at least one.

Scattering states ($E>V_0$)

For energies above $V_0$, the wave function is oscillatory everywhere. A particle incident from the left with wave number $k^{\prime }=\sqrt{2m(E-V_0)}/\hslash$ outside and $k=\sqrt{2mE}/\hslash$ inside has the stationary state:

$$\psi (x)=\{\begin{array}{ll}{e}^{i{k}^{\prime }x}+r{e}^{-i{k}^{\prime }x}& x<-a,\\ C{e}^{ikx}+D{e}^{-ikx}& |x|\le a,\\ t{e}^{i{k}^{\prime }x}& x>a,\end{array}$$

where $r$ is the reflection amplitude and $t$ is the transmission amplitude. The transmission coefficient is $T=(k^{\prime }/k)|t{|}^2$ and the reflection coefficient is $R=|r{|}^2$, with $R+T=1$.

Matching $\psi$ and ${\psi }^{\prime }$ at $x=\pm a$ and solving the resulting $4\times 4$ linear system yields:

$$T=\frac{1}{1+\frac{V_0^2}{4E(E-V_0)}{\sin }^2(2ka)}.$$

Transmission is perfect ($T=1$) when $\sin (2ka)=0$, i.e., $2ka=n\pi$ for integer $n$. These are the resonant tunneling conditions. At resonance the well width $2a$ contains an integer number of half-wavelengths of the interior oscillation, and the reflected waves from the two boundaries cancel by destructive interference.

Domain considerations

The Hamiltonian $H=p^2/(2m)+V(x)$ on $L^2(\mathbb{R})$ is self-adjoint on its natural domain. For bound states ($E<V_0$), the domain requires $\psi \in H^2(\mathbb{R})$ and $\psi \in L^2(\mathbb{R})$, which enforces the exponential decay at infinity. For scattering states, the domain is the locally $H^2$ functions with the asymptotic form of incoming plus reflected waves. The self-adjointness of $H$ with a piecewise-constant potential follows from the general theory of Schrödinger operators with bounded potentials.

Counterexamples to common slips

• The bound-state energies are lower than the corresponding infinite-well values, not higher. The finite wall effectively widens the well, and wider confinement means lower energy.
• The number of bound states depends on the product $a\sqrt{V_0}$, not on $a$ or $V_0$ separately. Doubling $a$ and halving $V_0$ preserves the number of bound states.
• Transmission resonance ($T=1$) does not mean the particle passes through without interacting with the well. At resonance the probability density $|\psi (x){|}^2$ is enhanced inside the well — the particle spends more time there, but all of it eventually transmits.
• The statement "every 1D well has at least one bound state" applies only to potentials that decay at infinity. A potential that does not vanish at infinity can bind zero states.

Key theorem with proof Intermediate+

Theorem (Bound states of the finite square well). The finite square well of half-width $a$ and depth $V_0$ supports a finite number of bound states. The number of bound states is $N=\lfloor 2z_0/\pi \rfloor +1$ where $z_0=a\sqrt{2m{V}_0}/\hslash$. At least one bound state always exists. The bound-state energies are the solutions of the transcendental equations $z\tan z=\sqrt{z_0^2-z^2}$ (even parity) and $-z\cot z=\sqrt{z_0^2-z^2}$ (odd parity), with $z\in (0,z_0)$.

Proof.

Step 1: Even-parity transcendental equation. For even-parity states, $\psi (x)=A\cos (kx)$ inside the well and $\psi (x)=B{e}^{-\kappa |x|}$ outside. Continuity of $\psi$ at $x=a$: $A\cos (ka)=B{e}^{-\kappa a}$. Continuity of ${\psi }^{\prime }$ at $x=a$: $-Ak\sin (ka)=-B\kappa e^{-\kappa a}$. Dividing:

$$k\tan (ka)=\kappa .$$

In dimensionless variables $z=ka$, $\kappa a=\sqrt{z_0^2-z^2}$, this becomes $z\tan z=\sqrt{z_0^2-z^2}$.

Step 2: Existence of at least one even solution. The left-hand side $f(z)=z\tan z$ starts at $f(0)=0$ with $f^{\prime }(0)=1$, and increases monotonically to $+\infty$ as $z\to \pi /2^{-}$. The right-hand side $g(z)=\sqrt{z_0^2-z^2}$ starts at $g(0)=z_0>0$ and decreases monotonically to $g(z_0)=0$. Since $f$ diverges upward while $g$ remains finite, $f$ must cross $g$ at least once in $(0,\min (\pi /2,z_0))$. At least one even bound state exists for any $z_0>0$.

Step 3: Counting bound states. The even-parity equation has a solution in each interval $(0,\pi /2)$, $(\pi ,3\pi /2)$, $(2\pi ,5\pi /2)$, ... and the odd-parity equation $-z\cot z=\sqrt{z_0^2-z^2}$ has a solution in each interval $(\pi /2,\pi )$, $(3\pi /2,2\pi )$, ... as long as $z_0$ exceeds the left endpoint of the interval. The total count of intervals that fit in $(0,z_0)$ is $\lfloor 2z_0/\pi \rfloor$, and adding the guaranteed even ground state gives $N=\lfloor 2z_0/\pi \rfloor +1$.

Step 4: Finiteness. Since $z_0$ is finite, only finitely many intervals fit, and the number of bound states is finite. $\square$

Corollary (Infinite-well limit). As $V_0\to \infty$, $z_0\to \infty$, the bound-state energies converge to the infinite-square-well values $E_n=n^2{\pi }^2{\hslash }^2/(2m(2a{)}^2)$, and the wave functions converge to the sinusoidal eigenstates of the infinite well.

Bridge. The finite square well is the first exactly solvable potential that exhibits both discrete and continuous spectra. The same matching-condition machinery — continuity of $\psi$ and ${\psi }^{\prime }$ at each discontinuity of $V$ — appears in every multilayer quantum structure: semiconductor heterostructures, tunnel junctions, and superlattices. The transcendental eigenvalue equations build toward the WKB quantisation condition (unit 12.07.04), which approximates the bound states of an arbitrary smooth potential by matching oscillatory and exponential solutions across the turning points. The scattering analysis builds toward the full scattering theory of unit 12.08.01, where the transmission and reflection coefficients of the 1D well become the $S$-matrix elements of a 1D scattering problem.

Exercises Intermediate+

Lean formalization Intermediate+

The finite square well presents a formalization challenge distinct from the particle in a box: the eigenvalue condition is a transcendental equation, not a closed-form expression. The key objects are:

• The potential $V(x)$ as a piecewise-constant function on $\mathbb{R}$.
• The Schrödinger operator $H=-{\hslash }^2/(2m)d^2/d{x}^2+V(x)$ as a self-adjoint operator on $L^2(\mathbb{R})$.
• The transcendental eigenvalue equations $z\tan z=\sqrt{z_0^2-z^2}$ and $-z\cot z=\sqrt{z_0^2-z^2}$.
• The existence and count of solutions as a function of $z_0$.

Mathlib has the basic analysis (tangent, cotangent, square root, the intermediate value theorem) needed to formalize the graphical existence proof. It does not have: the self-adjointness of $H$ with a piecewise-constant potential; the spectral theorem applied to prove the discrete spectrum; or the matching-condition derivation as a rigorous theorem about ODE boundary-value problems. A formalization would proceed by proving existence of solutions to the transcendental equations by IVT, then constructing the eigenfunctions and verifying the eigenvalue equation. This unit ships with lean_status: none.

Advanced results Master

The Ramsauer-Townsend effect

The condition $T=1$ at specific energies has a celebrated experimental manifestation. In 1921, Ramsauer and Townsend independently observed that the scattering cross section of electrons on noble-gas atoms drops nearly to zero at certain electron energies (around 0.7 eV for argon). The atom acts as a finite potential well for the incident electron, and at the resonant energy the well is transparent. This was one of the earliest experimental confirmations of wave mechanics and cannot be explained by classical scattering theory, which predicts a monotonically varying cross section.

The Ramsauer-Townsend effect is essentially one-dimensional: the $s$-wave ($\ell =0$) partial wave dominates at low energies, and the radial equation for $\ell =0$ is a 1D Schrödinger equation. The condition for the minimum in the cross section is exactly the condition $2ka=n\pi$ derived above, where $a$ is now the effective radius of the atomic potential well.

Quantum wells and semiconductor heterostructures

The finite square well is the standard model for quantum wells in semiconductor physics. A thin layer of a narrow-bandgap semiconductor (e.g., GaAs) sandwiched between wider-bandgap materials (e.g., AlGaAs) creates a potential well for both electrons and holes. The conduction-band offset $V_0$ (typically 0.1--0.3 eV) and the well width $2a$ (typically 5--20 nm) determine the number of confined sub-bands and their energies.

For an electron with effective mass $m^{\ast }\approx 0.067m_e$ in a GaAs well of width 10 nm and depth 0.25 eV:

$$z_0=\frac{5\times 10^{-9}}{1.055\times 10^{-34}}\sqrt{2\times 0.067\times 9.109\times 10^{-31}\times 0.25\times 1.602\times 10^{-19}}\approx \mathrm{3.5.}$$

This gives $N=\lfloor 7/\pi \rfloor +1=3$ bound sub-bands. The inter-sub-band transition energies are in the far-infrared (terahertz) range and are used in quantum cascade lasers.

Tunneling through a rectangular barrier

The finite square well with $V_0<0$ (a barrier instead of a well) gives the simplest model of quantum tunneling through a barrier of height $V_0$ and width $2a$. For $E<V_0$:

$$T=\frac{1}{1+\frac{V_0^2}{4E(V_0-E)}{\sinh }^2(2\kappa a)},$$

where $\kappa =\sqrt{2m(V_0-E)}/\hslash$. For thick barriers ($\kappa a\gg 1$), the $\sinh$ grows exponentially and $T\approx 16E(V_0-E)/V_0^2\cdot e^{-4\kappa a}$. This exponential dependence on barrier width and height is the hallmark of quantum tunneling. It explains alpha decay (tunneling through the nuclear Coulomb barrier), scanning tunneling microscopy (electrons tunneling from a tip through vacuum into a surface), and Josephson junctions (Cooper pairs tunneling through an insulating barrier).

The double-barrier structure and resonant tunneling diode

Two barriers separated by a well form a double-barrier structure. At energies matching the quasi-bound states of the central well, the transmission probability peaks sharply — a resonance. The resonance width $\Gamma$ is related to the barrier penetration: $\Gamma \sim \hslash /\tau$ where $\tau$ is the lifetime of the quasi-bound state.

The resonant tunneling diode exploits this effect. The current-voltage characteristic shows negative differential resistance: as the applied voltage increases past a resonance, the current drops. This enables high-frequency oscillators (up to THz) and is the fastest electronic device known.

Analytic properties of the $S$-matrix

The scattering matrix for the finite square well, viewed as a function of complex energy $E$, is a meromorphic function whose poles in the lower half of the complex energy plane correspond to resonances (quasi-bound states with finite lifetime). The poles on the negative real axis correspond to true bound states. The analytic continuation from $E>V_0$ (scattering region) to $E<0$ (bound-state region) connects the two parts of the spectrum through a single meromorphic function — a preview of the analytic $S$-matrix theory developed in unit 12.08.01.

The transfer-matrix method

For a general piecewise-constant potential with $n$ regions, the matching conditions at each interface can be encoded in a $2\times 2$ transfer matrix $M_j$ that relates the amplitude vector $(A,B{)}^T$ (coefficients of $e^{ikx}$ and $e^{-ikx}$) on one side of an interface to that on the other side. The total transfer matrix is the product $M=M_n\cdots M_2{M}_1$.

For a single finite well of half-width $a$, the transfer matrix is

$$M=\left(\begin{array}{cc}\cos (2ka)+\frac{i}{2}\left(\frac{k}{\kappa }+\frac{\kappa }{k}\right)\sin (2ka)& \frac{i}{2}\left(\frac{k}{\kappa }-\frac{\kappa }{k}\right)\sin (2ka)\\ -\frac{i}{2}\left(\frac{k}{\kappa }-\frac{\kappa }{k}\right)\sin (2ka)& \cos (2ka)-\frac{i}{2}\left(\frac{k}{\kappa }+\frac{\kappa }{k}\right)\sin (2ka)\end{array}\right).$$

The transmission coefficient is $T=1/|M_{11}{|}^2$. For a periodic lattice of wells (a superlattice), the product of transfer matrices gives rise to band structure: allowed energy bands where $|\mathrm{tr}M|\le 2$ alternate with forbidden gaps where $|\mathrm{tr}M|>2$. This is the Kronig-Penney model of band theory.

Synthesis. The finite square well is the minimal model that captures the three fundamental phenomena of quantum mechanics in a potential: discrete bound states, tunneling into classically forbidden regions, and resonant scattering above the barrier. The same matching-condition formalism handles all three regimes. The central insight is that the continuity of $\psi$ and ${\psi }^{\prime }$ at potential discontinuities, combined with the oscillatory or exponential character of the solutions in each region, is sufficient to determine the complete spectral and scattering properties. This is the pattern that generalizes to multilayer structures, to the WKB approximation for smooth potentials, and to the full scattering theory of the $S$-matrix.

Connections Master

The finite square well connects to every area of physics that involves quantum confinement. In semiconductor physics (unit 11.05.01), quantum wells are finite square wells engineered by band-offset engineering in heterostructures. The sub-band energies and wave function penetration into the barriers determine the optical absorption edge, the gain spectrum of quantum-well lasers, and the design of high-electron-mobility transistors.

In nuclear physics, the neutron-nucleus potential is modelled as a finite square well (the Woods-Saxon potential is a smoothed version). The number of bound states determines which nuclei are stable, and the tunneling calculation explains alpha decay lifetimes.

In scanning tunneling microscopy (STM), the tip-surface gap is a rectangular barrier. The exponential sensitivity of the tunneling current to barrier width ($I\propto e^{-2\kappa d}$) gives STM its atomic-scale resolution.

The transmission/reflection analysis is the 1D prototype of the $S$-matrix (unit 12.08.01). The $S$-matrix for the finite well is a $2\times 2$ unitary matrix relating incoming and outgoing amplitudes, and its analytic structure (bound-state poles, resonance peaks) encodes the complete physics of the interaction.

The transfer-matrix formalism developed for the finite well generalizes directly to superlattices and photonic crystals. A periodic array of finite wells produces energy bands and gaps — the Kronig-Penney model. This band structure is the one-dimensional analogue of the electronic band structure of crystals (unit 11.05.01) and the photonic band structure of dielectric superlattices. The exponential decay in the gap regions of a superlattice is mathematically identical to the evanescent decay outside a single well: both arise from imaginary wave vectors in classically forbidden energy ranges.

Historical notes Master

The finite square well was first analysed in the early days of wave mechanics (1926--1927). The Ramsauer-Townsend effect (1921) was a key experimental motivation: the dramatic dip in electron scattering cross sections at specific energies could not be explained by classical mechanics and was one of the first qualitative successes of the Schrödinger equation when applied to scattering.

The concept of quantum tunneling emerged from the work of Hund (1927) on molecular double-well potentials, and was quickly applied by Gamow, Gurney, and Condon (1928) to explain alpha decay. Gamow modelled the nuclear Coulomb barrier as a finite barrier and showed that the exponential tunneling probability gave the observed Geiger-Nuttall relation between decay lifetime and alpha-particle energy.

The semiconductor quantum well was proposed by Esaki and Tsu at IBM in 1970 and experimentally realized by Dingle, Wiegmann, and Henry at Bell Labs in 1974. The development of molecular beam epitaxy (MBE) made it possible to grow quantum wells with atomic-level precision, launching the field of bandgap engineering. The 2000 Nobel Prize in Physics (Alferov, Kroemer) was awarded for semiconductor heterostructures that exploit finite-well physics.

Philosophically, the finite square well forces a confrontation with the non-classical nature of quantum mechanics. The wave function extends beyond the classical turning points: there is a nonzero probability of finding the particle in a region where its kinetic energy would be negative. This is not a mathematical artifact — tunneling currents are measured in STM, alpha particles escape nuclei, and quantum cascade lasers operate because of it. The finite well is the simplest system where the failure of classical mechanics is not just a correction but a qualitative change in behaviour.

Bibliography Master

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