---
title: Particle in Infinite and Finite Square Wells
module: Wave Mechanics in One Dimension
moduleNumber: 3
lessonNumber: 3
order: 303
summary: >
  The infinite square well is the simplest bound-state problem: two boundary conditions
  quantize the energy into a ladder E_n = n² E_1, and the eigenfunctions are the
  standing waves of a string fixed at both ends. Relaxing the walls to a finite depth
  lets the wave function leak into the classically forbidden region, keeps the number of
  bound states finite, and turns the eigenvalue condition into a transcendental equation
  solved graphically.
topics: [Wave Mechanics in One Dimension]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 6 — The Schrödinger Equation; §6-2 The Infinite Square Well, §6-3 The Finite Square Well"
---

The [time-independent Schrödinger equation](/quantum-mechanics/wave-mechanics-1d/the-schrodinger-equation-in-one-dimension)
becomes a solvable ordinary differential equation once a potential $V(x)$ is fixed.
The most transparent case is a particle confined to a box with perfectly rigid walls,
the **infinite square well**. It admits exact solutions with no difficult mathematics,
reproduces the standing-wave quantization of a string fixed at both ends, and displays
every qualitative feature of a bound quantum system.[^tl-62]

## The infinite square well

A particle is free inside a region of width $L$ and cannot escape it:

$$
V(x) =
\begin{cases}
0, & 0 < x < L, \\
\infty, & x \le 0 \ \text{or}\ x \ge L.
\end{cases}
$$

A macroscopic image is a bead sliding on a frictionless wire between two massive stops;
a physical realization is an electron between charged grids whose repelling fields
become the walls as the voltage is raised. Because the potential is infinite outside,
the wave function must vanish there — the particle is certainly inside — and by
continuity $\psi$ must reach zero at each wall:

$$
\psi(0) = 0, \qquad \psi(L) = 0.
$$

> **Definition (Boundary condition).** A constraint on $\psi$ imposed at the edge of a
> region, here the vanishing of $\psi$ at the infinite walls. Boundary conditions,
> together with the requirement $\psi \to 0$ at infinity, are what select discrete
> energies out of the continuum the free particle enjoys.

### Quantization from standing waves

Inside the well $V = 0$, so the equation is the free-particle equation

$$
\frac{\d^2\psi}{\d x^2} = -k^2\psi, \qquad k^2 = \frac{2mE}{\hbar^2},
$$

with general solution $\psi(x) = A\sin kx + B\cos kx$. The wall at $x = 0$ forces
$\psi(0) = 0$; since $\cos 0 = 1$, the cosine term must drop, $B = 0$. The wall at
$x = L$ then requires

$$
\psi(L) = A\sin kL = 0,
$$

which holds only when $kL$ is an integer multiple of $\pi$:

$$
k_n = \frac{n\pi}{L}, \qquad n = 1, 2, 3, \dots
$$

Written through $\lambda = 2\pi/k$, this is $n(\lambda/2) = L$: an integer number of
half-wavelengths fits the well, exactly the standing-wave condition for a string of
length $L$ clamped at both ends. The value $n = 0$ gives $\psi \equiv 0$, no particle,
and is excluded.

The energy follows from $E = \hbar^2 k^2 / 2m$ with $k = k_n$:

$$
E_n = \frac{\hbar^2 k_n^2}{2m} = n^2\,\frac{\pi^2\hbar^2}{2mL^2} = n^2 E_1,
\qquad
E_1 = \frac{\pi^2\hbar^2}{2mL^2}.
$$

> **Theorem (Energy levels of the infinite well).** A particle of mass $m$ confined
> to a one-dimensional box of width $L$ has quantized energies
> $E_n = n^2 E_1$ with $E_1 = \pi^2\hbar^2/2mL^2$ and $n = 1, 2, 3, \dots$. The
> spacing grows with $n$, and the lowest energy $E_1$ is strictly positive.

The integer $n$ is a **quantum number**: it fixes both the energy and the wave
function.

$$
% caption: Energy levels of the infinite well rise as n squared inside rigid walls;
% a classical particle could sit at any height, but only the marked levels solve the
% wave equation.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % walls
  \draw[black, very thick] (0,0) -- (0,5.4);
  \draw[black, very thick] (5,0) -- (5,5.4);
  \draw[black] (0,0) -- (5,0) node[right, font=\scriptsize, text=black] {$x$};
  \node[font=\scriptsize, text=black] at (2.5,-0.35) {$V = 0$ for $0 < x < L$};
  % levels at n^2 scale, E1 = 0.2
  \foreach \n/\h/\lab in {1/0.2/1, 2/0.8/2, 3/1.8/3, 4/3.2/4, 5/5.0/5} {
    \draw[acc, thick] (0,\h) -- (5,\h);
    \node[acc, font=\scriptsize, anchor=west] at (5.15,\h) {$n = \lab$};
  }
  \node[acc, font=\scriptsize, anchor=east] at (-0.15,0.2) {$E_1$};
  \node[acc, font=\scriptsize, anchor=east] at (-0.15,5.0) {$25\,E_1$};
  \node[align=center, font=\scriptsize, text=black] at (2.5,4.7) {$E_n = n^2 E_1$};
\end{tikzpicture}
$$

### The eigenfunctions

With $B = 0$ and $k = k_n$ the wave functions are sines. The constant $A$ is fixed by
normalization; since $\psi = 0$ outside, only the interval $[0, L]$ contributes:

$$
\int_0^L A^2\sin^2\!\Big(\frac{n\pi x}{L}\Big)\,\d x = A^2\frac{L}{2} = 1
\quad\Longrightarrow\quad
A = \sqrt{\tfrac{2}{L}}.
$$

> **Definition (Eigenfunctions of the infinite well).** The normalized stationary
> states are
> $$
> \psi_n(x) = \sqrt{\tfrac{2}{L}}\,\sin\!\Big(\frac{n\pi x}{L}\Big),
> \qquad 0 \le x \le L,
> $$
> and $\psi_n = 0$ outside. These are identical to the standing-wave shapes of a
> vibrating string, with $n{-}1$ interior nodes.

The lowest state $n = 1$ is the **ground state**; $n = 2, 3, \dots$ are excited states.
Every $\psi_n$ has the same peak amplitude $\sqrt{2/L}$, and every probability density
$P_n = \psi_n^2$ peaks at $2/L$.

$$
% caption: The first three eigenfunctions are half, full, and one-and-a-half sine
% waves pinned to zero at both walls; each higher state adds one interior node.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % three stacked panels, well width 4.4
  \foreach \row/\lab in {0/1, 1/2, 2/3} {
    \begin{scope}[yshift=\row*2.0cm]
      \draw[black, thick] (0,-0.9) -- (0,0.9);
      \draw[black, thick] (4.4,-0.9) -- (4.4,0.9);
      \draw[black] (0,0) -- (4.4,0);
      \node[acc, font=\scriptsize, anchor=west] at (4.6,0) {$n = \lab$};
    \end{scope}
  }
  % n = 1: half sine
  \begin{scope}[yshift=0cm]
    \draw[acc, very thick] (0,0) sin (2.2,0.8) cos (4.4,0);
  \end{scope}
  % n = 2: full sine
  \begin{scope}[yshift=2.0cm]
    \draw[acc, very thick] (0,0) sin (1.1,0.8) cos (2.2,0) sin (3.3,-0.8) cos (4.4,0);
  \end{scope}
  % n = 3
  \begin{scope}[yshift=4.0cm]
    \draw[acc, very thick] (0,0) sin (0.733,0.8) cos (1.467,0) sin (2.2,-0.8) cos (2.933,0) sin (3.667,0.8) cos (4.4,0);
  \end{scope}
\end{tikzpicture}
$$

The zero-point energy $E_1 > 0$ is not an accident of the algebra. A particle pinned
exactly at rest would have definite position and definite (zero) momentum, forbidden by
the [uncertainty principle](/quantum-mechanics/matter-waves/the-uncertainty-principle).
Confining the particle to width $\Delta x \sim L$ forces a momentum spread
$\Delta p \sim \hbar/L$ and thus a minimum kinetic energy of order
$\hbar^2/2mL^2$ — the same size as $E_1$.

## Classical correspondence

Classically a particle in the box moves at constant speed, reflecting off each wall,
and is equally likely to be found anywhere: the position distribution is flat,
$P_C(x) = 1/L$. The quantum ground state looks nothing like this — it is a single hump
peaked at the center with zero probability at the walls. The two pictures reconcile at
large $n$. In state $n$ the density $\psi_n^2 = (2/L)\sin^2(k_n x)$ has $n$ peaks, and
averaged over a small window containing several oscillations $\langle \sin^2\rangle =
\tfrac12$, so

$$
\overline{\psi_n^2} = \frac{2}{L}\cdot\frac{1}{2} = \frac{1}{L} = P_C(x).
$$

This is **Bohr's correspondence principle**: the quantum distribution reproduces the
classical one when $n$ is large. A measurement with finite resolution $\Delta x$
cannot resolve the individual peaks and returns the classical average.[^tl-corr]

$$
% caption: At n equals 10 the probability density has ten peaks; smeared over a
% detector's resolution it flattens to the constant classical distribution shown
% dashed.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,0) -- (0,1.7);
  \draw[black, thick] (7,0) -- (7,1.7);
  \draw[black] (0,0) -- (7,0) node[right, font=\scriptsize, text=black] {$x$};
  % ten sin^2 humps
  \foreach \i in {0,...,9} {
    \draw[acc, very thick] ({\i*0.7},0) sin ({\i*0.7+0.35},1.2) cos ({\i*0.7+0.7},0);
  }
  % classical line
  \draw[dashed, black!70, thick] (0,0.6) -- (7,0.6);
  \node[black!70, font=\scriptsize, anchor=south east] at (7,0.62) {$P_C = \frac{1}{L}$};
  \node[acc, font=\scriptsize, anchor=south west] at (0.1,1.25) {$n = 10$};
\end{tikzpicture}
$$

**A worked spectrum.** Take $L = 0.1\ \text{nm}$, about the size of an atom. Writing
$E_1 = (hc)^2/8mc^2L^2$ with $hc = 1240\ \text{eV}\cdot\text{nm}$ and
$mc^2 = 0.511\ \text{MeV}$,

$$
E_1 = \frac{(1240\ \text{eV}\cdot\text{nm})^2}{8(5.11\times10^5\ \text{eV})(0.1\ \text{nm})^2} = 37.6\ \text{eV},
$$

comparable to the $13.6\ \text{eV}$ binding of hydrogen. The higher levels are
$E_2 = 4E_1 = 150.4\ \text{eV}$ and $E_3 = 9E_1 = 338.4\ \text{eV}$. Photons emitted in
downward transitions carry the level differences:

| Transition | $\Delta E$ | $\lambda = hc/\Delta E$ |
| --- | --- | --- |
| $3 \to 2$ | $188.0\ \text{eV}$ | $6.60\ \text{nm}$ |
| $3 \to 1$ | $300.8\ \text{eV}$ | $4.12\ \text{nm}$ |
| $2 \to 1$ | $112.8\ \text{eV}$ | $11.0\ \text{nm}$ |

These are soft-X-ray wavelengths, the scale expected when an electron is confined to
atomic dimensions.[^tl-ex64]

## The finite square well

Real wells have finite depth. Lowering the walls to a height $V_0$,

$$
V(x) =
\begin{cases}
0, & 0 < x < L, \\
V_0, & \text{otherwise},
\end{cases}
$$

changes the boundary behavior. For a bound state, $E < V_0$. Inside the well $V = 0$ and
the equation is again $\psi'' = -k^2\psi$ with $k^2 = 2mE/\hbar^2$, so $\psi$ oscillates.
Outside the well $E < V(x)$, and the equation becomes

$$
\psi''(x) = \frac{2m}{\hbar^2}(V_0 - E)\,\psi(x) = \alpha^2\,\psi(x),
\qquad
\alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar}.
$$

The exterior solution is a real exponential $\psi \sim e^{-\alpha x}$, decaying into the
barrier. The wave function no longer vanishes at the wall; instead $\psi$ and $\psi'$
must join continuously to the interior sine, and the exterior tail must go to zero at
infinity.

### Origin of energy quantization

The [curvature rule](/quantum-mechanics/wave-mechanics-1d/the-schrodinger-equation-in-one-dimension) does
the selecting. Inside, $\psi'' $ and $\psi$ have opposite signs, so $\psi$ curves toward
the axis and oscillates. Outside, they share a sign, so $\psi$ curves _away_ from the
axis. For a generic energy, the exterior solution that matches the interior at the wall
curves away and diverges to $\pm\infty$ — not normalizable, not physical. Only for
special energies does the interior emerge at the wall with exactly the slope that lets
the exterior decay to zero. Those energies are the bound-state levels.

$$
% caption: Slightly off the allowed energy the exterior solution curls away to plus or
% minus infinity; at the exact eigenvalue the function and its slope reach zero
% together, giving the one normalizable tail.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick, dashed] (2.6,-1.7) -- (2.6,2.0);
  \draw[black] (-0.2,0) -- (5.6,0) node[right, font=\scriptsize, text=black] {$x$};
  \node[font=\scriptsize, text=black, anchor=south] at (1.3,-2.0) {inside: oscillates};
  \node[font=\scriptsize, text=black, anchor=south] at (4.2,-2.0) {outside: decays or diverges};
  % interior half-hump
  \draw[acc, very thick] (0,0) .. controls (0.9,1.4) and (1.7,1.4) .. (2.6,0.6);
  % accepted decaying tail
  \draw[acc, very thick] (2.6,0.6) .. controls (3.4,0.2) and (4.4,0.05) .. (5.4,0.02);
  \node[acc, font=\scriptsize, anchor=west] at (5.0,0.3) {accepted};
  % too-high energy: diverges up
  \draw[black, very thick] (2.6,0.6) .. controls (3.4,0.8) and (4.2,1.4) .. (5.0,2.0);
  \node[black, font=\scriptsize, anchor=west] at (4.9,1.9) {$E$ too high};
  % too-low energy: diverges down
  \draw[black, very thick] (2.6,0.6) .. controls (3.4,0.0) and (4.2,-0.9) .. (5.0,-1.6);
  \node[black, font=\scriptsize, anchor=west] at (4.9,-1.6) {$E$ too low};
\end{tikzpicture}
$$

The finite well differs from the infinite well in three ways.[^tl-63]

- **Penetration.** $\psi$ is nonzero in the regions $x < 0$ and $x > L$, where
  classically $E < V$ and the kinetic energy would be negative. There is a real
  probability of finding the particle in the classically forbidden region, falling off
  as $|\psi|^2 \sim e^{-2\alpha x}$.
- **Fewer, lower levels.** Because the wave function extends into the walls, its
  half-wavelength inside is slightly longer than in the infinite well, so each energy
  is somewhat lower than the corresponding infinite-well level. Only a finite number of
  bound states exist, set by the depth $V_0$; a very shallow well holds just one.
- **Guaranteed ground state.** However shallow, a one-dimensional well always binds at
  least one state.

The penetration does not let a measurement return a negative kinetic energy. Localizing
the particle in the tail region, of extent $\Delta x \sim 1/\alpha$, injects a momentum
uncertainty $\Delta p \sim \hbar\alpha$ and a minimum kinetic energy $\sim (\Delta p)^2
/2m = V_0 - E$, exactly enough to forbid the measurement of a negative value. The same
tail is the seed of
[barrier tunneling](/quantum-mechanics/wave-mechanics-1d/barrier-penetration-and-quantum-tunneling).

$$
% caption: In the finite well the eigenfunctions do not vanish at the walls; they
% decay exponentially into the barriers, so the probability density has tails in the
% classically forbidden region.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % finite well potential outline
  \draw[black, thick] (-1.4,1.6) -- (0,1.6) -- (0,0) -- (4.4,0) -- (4.4,1.6) -- (5.8,1.6);
  \node[black!70, font=\scriptsize] at (-0.9,1.85) {$V_0$};
  \node[black!70, font=\scriptsize] at (2.2,-0.3) {$V = 0$};
  \draw[black] (-1.4,0) -- (5.8,0);
  % ground-state wavefunction with tails, centered
  \draw[acc, very thick]
    (-1.4,0.05) .. controls (-0.7,0.15) and (-0.2,0.5) .. (0,0.75)
    .. controls (1.1,1.7) and (3.3,1.7) .. (4.4,0.75)
    .. controls (4.6,0.5) and (5.1,0.15) .. (5.8,0.05);
  \node[acc, font=\scriptsize, anchor=west] at (4.7,0.55) {tail};
  \node[acc, font=\scriptsize, anchor=east] at (-0.7,0.55) {tail};
\end{tikzpicture}
$$

### The graphical eigenvalue condition

For most finite potentials the matching conditions produce a transcendental equation
with no closed-form solution, but a symmetric well admits a clean graphical reading.
Center the well on the origin with half-width $a$, so it runs from $-a$ to $+a$. The
symmetric potential $V(-x) = V(x)$ splits the solutions by
[parity](/quantum-mechanics/wave-mechanics-1d/operators-expectation-values-and-the-harmonic-oscillator):
the interior is either a cosine (even) or a sine (odd). Matching $\psi$ and $\psi'$ at
$x = a$ gives

$$
\text{even:}\quad \alpha = k\tan(ka),
\qquad
\text{odd:}\quad \alpha = -k\cot(ka).
$$

Introduce the dimensionless variables $u = ka$ and $v = \alpha a$. Their squares sum to
a constant fixed by the well:

$$
u^2 + v^2 = \frac{2mV_0 a^2}{\hbar^2} \equiv R_0^2,
$$

because $k^2 + \alpha^2 = 2mV_0/\hbar^2$. The allowed states are the intersections of
this circle of radius $R_0$ with the curves $v = u\tan u$ (even) and
$v = -u\cot u$ (odd) in the first quadrant. Each intersection is one bound state.

$$
% caption: Bound states of the symmetric finite well are the intersections of the
% quarter circle of radius R0 with the tangent and cotangent branches; a deeper well
% has a larger radius and catches more intersections.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.6,0) node[right, font=\scriptsize, text=black] {$u = ka$};
  \draw[->, black] (0,0) -- (0,3.6) node[above, font=\scriptsize, text=black] {$v$};
  % quarter circle radius R0 = 3.4
  \draw[acc, thick] (3.4,0) arc[start angle=0, end angle=90, radius=3.4];
  \node[acc, font=\scriptsize, anchor=south west] at (2.5,2.5) {$u^2 + v^2 = R_0^2$};
  % even branch: v = u tan u, rising from 0 near u=0, asymptote at pi/2 ~ 1.57
  \draw[black!70, thick] (0,0) .. controls (0.7,0.35) and (1.1,1.1) .. (1.35,3.4);
  \node[black!70, font=\scriptsize] at (0.65,3.1) {even};
  % even branch 2 near u = pi (3.14)
  \draw[black!70, thick] (3.14,0) .. controls (3.5,0.5) and (4.2,2.0) .. (4.5,3.4);
  % odd branch: v = -u cot u, from u=pi/2 rising, asymptote at pi (3.14)
  \draw[black!70, thick, densely dotted] (1.57,0) .. controls (2.1,0.6) and (2.7,1.9) .. (2.95,3.4);
  \node[black!70, font=\scriptsize] at (2.35,2.9) {odd};
  % intersection dots (schematic)
  \fill[acc] (1.28,3.15) circle (0) ;
  \fill[black] (1.2,1.85) circle (2.2pt);
  \fill[black] (2.35,1.55) circle (2.2pt);
  \fill[black] (3.28,0.85) circle (2.2pt);
\end{tikzpicture}
$$

Reading off the geometry: a small radius $R_0 < \pi/2$ meets only the first even branch,
so a shallow well holds exactly one (even, nodeless) ground state. As $V_0$ grows,
$R_0$ increases and the circle sweeps across successive branches, adding an odd state,
then another even state, and so on. The number of bound states is the number of
half-$\pi$ intervals the radius spans,

$$
N = \left\lceil \frac{R_0}{\pi/2} \right\rceil = \left\lceil \frac{a}{\pi}\sqrt{\frac{2mV_0}{\hbar^2}} \right\rceil,
$$

always at least one.[^tl-graph] The same qualitative analysis applies to any well-type
potential: wherever $E > V(x)$ the wave function oscillates, wherever $E < V(x)$ it
curves away, and only discrete energies give a solution that decays at infinity. The
[harmonic oscillator](/quantum-mechanics/wave-mechanics-1d/operators-expectation-values-and-the-harmonic-oscillator)
is the next such potential.

[^tl-62]: **Tipler & Llewellyn**, _Modern Physics_, §6-2 — the infinite square well: setup, boundary conditions, standing-wave quantization, eigenfunctions $\psi_n = \sqrt{2/L}\sin(n\pi x/L)$, and quantum numbers.
[^tl-corr]: **Tipler & Llewellyn**, _Modern Physics_, §6-2 — comparison with the classical flat distribution and the correspondence principle at large $n$ (Figure 6-5, $n = 10$).
[^tl-ex64]: **Tipler & Llewellyn**, _Modern Physics_, §6-2, Example 6-4 — ground-state energy and transition wavelengths for an electron in a $0.1\ \text{nm}$ box.
[^tl-63]: **Tipler & Llewellyn**, _Modern Physics_, §6-3 — the finite square well: penetration into the classically forbidden region, finitely many bound states, energies lower than the infinite well, and the uncertainty-principle resolution of the negative-kinetic-energy puzzle.
[^tl-graph]: **Tipler & Llewellyn**, _Modern Physics_, §6-3, Graphical Solution of the Finite Square Well — the transcendental matching conditions and their graphical solution as circle-versus-tangent intersections.