Particle in a box

The particle in a box or infinite square well problem is one of the simplest non-trivial solutions to Schrödinger's wave equation. As such it is often encountered in introductory quantum mechanics material as a demonstration of the quantization of energy. In its simplest form the problem is one-dimensional, and involves a single particle living in an infinite potential well.

The 1D square well

Setting up the problem

The potential can be given by

$V(x)=0\,\!$ for $0\leq x\leq L$ and $V(x)=\infty$ otherwise.

That is, in a region of length $L$ (the box or well) the potential is zero; everywhere else it is infinite. An immediate consequence of this is that the particle must be located somewhere between $x=0$ and $x=L$, since if it were anywhere else it would have to have infinite energy.

Since $V(x)$ does not depend on time we can use the time-independent version of the Schrödinger equation,

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x)=E\psi (x)\ ,$

where $\hbar$ is Planck's constant divided by 2π, and $m$ and $E$ are the mass and energy of the particle, respectively. This is nothing but the eigenvalue problem, and our task is to determine the energy eigenvalue(s) $E$ and eigenstate(s) $\psi$ that solve it. The equation is a second-order linear ODE with constant coefficients, and the general solution can be written as

$\psi (x)=a\cos \left({\sqrt {\frac {2mE}{\hbar ^{2}}}}x\right)+b\sin \left({\sqrt {\frac {2mE}{\hbar ^{2}}}}x\right)\ ,$

where $a$ and $b$ are complex constants to be determined from conditions on the system. The first condition we can use is that the wavefunction must be continuous. We know that the particle cannot exist at positions $x<0$, which tells us that the wavefunction must be identically zero there (in order for the probability of finding the particle there to also be zero). Continuity then implies that at $x=0$ the wavefunction is also zero, so we have

$0=\psi (0)=a\cos(0)+b\sin(0)=a\ ,$

meaning that the wavefunction can now be written

$\psi (x)=b\sin \left({\sqrt {\frac {2mE}{\hbar ^{2}}}}x\right)\ .$

Energy quantization

Using the same continuity argument at $x=L$ tells us that

$0=\psi (L)=b\sin \left({\sqrt {\frac {2mE}{\hbar ^{2}}}}L\right)\ .$

One solution to this is of course $b=0$. However, this would mean that the wavefunction vanishes everywhere -- implying that there is no particle! The other way to satisfy this equality is to have the sine term vanish, which will happen if

${\sqrt {\frac {2mE}{\hbar ^{2}}}}L=2n\pi \ ,$

where $n$ is any integer. We can solve this for the particle's energy to find

$E_{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}\ ,$

where we have now labelled the energy by the integer $n$. We have just derived energy quantization! Without the potential well, i.e. if the particle was free, its energy would be allowed to take on any real number. Once inside the box though, only a specific discrete set of energy eigenvalues is permitted.

Substituting $E_{n}$ back into $\psi$, we now have an infinite number of possible wavefunctions for the particle,

$\psi _{n}(x)=b\sin \left({\frac {n\pi x}{L}}\right)\ .$

The final step is to determine the value of $b$. This requires another condition that we can impose upon the problem.

Normalization

The probability of finding the particle somewhere must be unity. Since the wavefunction vanishes everywhere outside $[0,L]$, this means that the normalization condition is

$1=\int _{0}^{L}\psi ^{*}(x)\psi (x)\,\mathrm {d} x=|b|^{2}\int _{0}^{L}\sin ^{2}\left({\frac {n\pi x}{L}}\right)\,\mathrm {d} x=|b|^{2}{\frac {L}{2}}\ .$

Solving this for $b$ tells us $b=e^{i\varphi }{\sqrt {2/L}}$, where $\varphi$ is some overall quantum phase (a real number). Because overall phases do not affect measurable results we are free to choose any value we want for $\varphi$ without affecting the physics. For simplicity then, we set $\varphi =0$.

The solutions

A plot of $\psi _{n}(x)$ for $n=1$ (darkest) to $n=7$ (lightest).

We have now solved the particle-in-a-box problem. According to the Schrödinger wave equation the particle confined to the (infinite) box can only take on certain energy eigenvalues, namely

$E_{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}\ ,$

and the wavefunction for the particle when it is in the $n$th energy eigenstate is

$\psi _{n}(x)={\sqrt {\frac {2}{L}}}\sin \left({\frac {n\pi x}{L}}\right)$

inside the box, and $\psi (x)=0$ outside it.

Properties / Discussion / Comments (under construction)

The $\psi _{n}$ are complete.

$E=0$ is not allowed because $n=0$ implies wavefunction vanishes everywhere; no particle. So lowest energy state is above zero.

Energy levels are degenerate; $\psi _{-n}=-\psi _{n}$ but $E_{-n}=E_{n}$.

As $L\to \infty$ energy spacing goes to zero; recover the free space result.