# 3.5: The Energy of a Particle in a Box is Quantized

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The *particle in the box *model system is the simplest non-trivial application of the Schrödinger equation, but one which illustrates many of the fundamental concepts of quantum mechanics. For a particle moving in one dimension (again along the x- axis), the Schrödinger equation can be written

\[-\dfrac{\hbar^2}{2m}\psi {}''(x)+ V (x)\psi (x) = E \psi (x)\label{3.5.1}\]

Assume that the particle can move freely between two endpoints \(x = 0\) and \(x = L\), but cannot penetrate past either end. This is equivalent to a potential energy dependent on *x* with

\[V(x)=\begin{cases}

0 & 0\leq x\leq L \\

\infty & x< 0 \; and\; x> L \end{cases}\label{3.5.2}\]

This potential is represented in Figure \(\ref{3.5.1}\). The infinite potential energy constitutes an impenetrable barrier since the particle would have an infinite potential energy if found there, which is clearly impossible.

The particle is thus bound to a *"potential well*" since the particle cannot penetrate beyond \(x = 0\) or \(x = L\)

\[\psi (x)=0\; \; \; for \; \; x<0\; \; and\; \; x>L\label{3.5.3}\]

By the requirement that the wavefunction be continuous, it must be true as well that

\[\psi (0)=0\; \; \; and\; \; \; \psi (L)=0\label{3.5.4}\]

which constitutes a pair of boundary conditions on the wavefunction within the box. Inside the box, \(V(x) = 0\), so the Schrödinger equation reduces to the free-particle form:

\[-\dfrac{\hbar^2}{2m}\psi{}''(x)=E\psi (x) \label{3.5.5}\]

with \( 0\leq x\leq L\).

We again have the differential equation

\[\psi {}''(x) +k^2\psi (x)=0 \label{3.5.6}\]

with

\[k^2=2mE/\hbar^2 \label{3.5.6a}\]

The general solution can be written

\[\psi (x)=A\: \sin\; kx\,+\, B\: \cos\; kx\label{3.5.7}\]

where \(A\) and \(B\) are constants to be determined by the boundary conditions in Equation \(\ref{3.5.4}\). By the first condition, we find

\[\psi (0)=A\, \sin\, 0\, +\, B\, \cos\, 0\, =\, B\,= 0\label{3.5.8}\]

The second boundary condition at \(x = L\) then implies

\[\psi (a)=A\, \sin\, kL\,=\, 0\label{3.5.9}\]

It is assumed that \(A \neq 0\), for otherwise \(\psi(x)\) would be zero everywhere and the particle would disappear (i.e., the trivial solution). The condition that \(\sin kx = 0\) implies that

\[kL\, =\, n\pi \label{3.5.10}\]

where \(n\) is a integer, positive, negative or zero. The case \(n = 0\) must be excluded, for then \(k = 0\) and again \(\psi(x)\) would vanish everywhere. Eliminating \(k\) between Equation \(\ref{3.5.6}\) and \(\ref{3.5.10}\), we obtain

\[E_{n}=\dfrac{\hbar^2\pi^2}{2mL^2}\, n^2=\dfrac{h^2}{8mL^2}n^2 \label{3.5.11}\]

with \(n=1,2,,3...\).

These are the only values of the energy which allows solutions of the Schrödinger Equation \(\ref{3.5.5}\) consistent with the boundary conditions in Equation \(\ref{3.5.4}\). The integer \(n\), called a *quantum number*, is appended as a subscript on \(E\) to label the allowed energy levels. Negative values of \(n\) add nothing new because the energies in Equation \(\ref{3.5.11}\) depends on \(n^2\).

Figure \(\PageIndex{2}\) shows part of the energy-level diagram for the particle in a box. The occurrence of discrete or quantized energy levels is characteristic of a bound system, that is, one confined to a finite region in space. For the free particle, the absence of confinement allowed an energy continuum. Note that, in both cases, the number of energy levels is infinite-denumerably infinite for the particle in a box, but nondenumerably infinite for the free particle.

The particle in a box assumes its lowest possible energy when \(n = 1\), namely

\[E_{1}=\dfrac{h^2}{8mL^2}\label{3.5.12}\]

The state of lowest energy for a quantum system is termed its *ground state*.

The particle-in-a-box eigenfunctions are given by Equation \(\ref{3.5.14}\), with \(B = 0\) and \(k = n\pi/L=a\), in accordance with Equation \(\ref{3.5.10}\)

\[\psi _{n}(x)=A\, \sin\dfrac{n\pi x}{L} \label{3.5.14} \]

with

\[n=1,2,3...\label{3.5.14a}\]

These, like the energies, can be labeled by the quantum number \(n\). The constant \(A\), thus far arbitrary, can be adjusted so that \(\psi _{n}(x)\) is normalized. The normalization condition is, in this case,

\[\int_{0}^{a}\begin{bmatrix}\psi_{n}(x) \end{bmatrix}^2\,dx=1\label{3.5.15}\]

the integration running over the domain of the particle \(0\leq x\leq a\). Substituting Equation \(\ref{3.5.14}\) into Equation \(\ref{3.5.15}\),

\[\begin{align} A^2\: \int_{0}^{L}\, \sin^2\, \dfrac{n\pi x}{L}dx &=A^2\dfrac{L}{n\pi}\int_{0}^{n\pi}\sin^2\, \theta \,d\theta \\[4pt] &=A^2\dfrac{L}{2}=1\label{3.5.16} \end{align}\]

We have made the substitution \(\theta=n\pi x/L\) and used the fact that the average value of \(\sin^2 \theta\) over an integral number of half wavelengths equals 1/2 (alternatively, one could refer to standard integral tables). From Equation \(\ref{3.5.16}\), we can identify the general normalization constant

\[A = \sqrt{ \dfrac{2}{L}}\]

for all values of \(n\). Finally we can write the normalized eigenfunctions:

\[\psi _{n}(x)=\sqrt{\dfrac{2}{L}} \sin \dfrac{n\pi x}{L} \label{3.5.17} \]

with

\[n=1,2,3...\label{3.5.17b}\]

The first few eigenfunctions and the corresponding probability distributions are plotted in Figure \(\PageIndex{3}\). There is a close analogy between the states of this quantum system and the modes of vibration of a violin string. The patterns of standing waves on the string are, in fact, identical in form with the wavefunctions in Equation \(\ref{3.5.17}\).

It is generally true in quantum systems (not just for particles in boxes) that the number of nodes in a wavefunction increases with the energy of the quantum state.

## Orthonormality

Another important property of the eigenfunctions in Equation \(\ref{3.5.17}\) applies to the integral over a product of two *different* eigenfunctions. It is easy to see from Figure \(\PageIndex{5}\) that the integral

\[\int_{0}^{a}\psi _{2}(x)\psi _{1}(x)dx=0 \label{3.5.18}\]

To prove this result in general, use the trigonometric identity

\[\sin\,\alpha \: \sin\, \beta =\dfrac{1}{2}\begin{bmatrix}\cos(\alpha -\beta )-\cos(\alpha +\beta )\end{bmatrix} \label{trig}\]

to show that

\[\int_{0}^{L}\psi _{m}(x)\psi _{n}(x)dx=0\: \: \: if\: \: \: m \neq n\label{3.5.19}\]

This property is called *orthogonality*. We will show in the next Chapter, that this is a general result from quantum-mechanical eigenfunctions. The normalization (Equation \(\ref{3.5.18}\)) together with the orthogonality (Equation \(\ref{3.5.19}\)) can be combined into a single relationship

\[\int_{0}^{L}\psi _{m}(x)\psi _{n}(x)dx=\delta _{mn}\label{3.5.20}\]

In terms of the* Kronecker delta*

\[\delta _{mn}=\begin{cases}

1 & \text{if} \,\, m=n \\[4pt]

0 & \text{if} \,\, m\neq n \end{cases}\label{3.5.21}\]

A set of functions \(\begin{Bmatrix}\psi_{n}\end{Bmatrix}\) which obeys Equation \(\ref{3.5.20}\) is called *orthonormal*.

## Contributors

Seymour Blinder (Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor)