Operators and expected energy in a well

Significance of operators

In quantum mechanics every measurable quantity (observable) is represented by an operator, a mathematical instruction that acts on the wave function $\psi$.

Why they matter:

• They give a rule for getting physical information from $\psi$; $\psi$ alone is not measurable.
• Eigenvalue equation: if $\hat{A}\psi = a\psi$, a measurement of $A$ always gives the exact value $a$. The Schrödinger equation itself is $\hat{H}\psi = E\psi$, so allowed energies are eigenvalues of $\hat{H}$.
• Expectation value: when $\psi$ is not an eigenfunction, operators give the average result of many measurements: $\langle A\rangle = \int\psi^*\hat{A}\psi\,dx$.
• Non-commuting operators (e.g. $\hat{x}$ and $\hat{p}$) lead to the uncertainty principle.

Expected energy of a particle in a well (0 to L)

For an infinite well of width $L$ ($V = 0$ inside), the normalized wave function is

$$\psi_n(x,t) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)e^{-jE_nt/\hbar}$$

The expected (average) energy is

$$\langle E\rangle = \int_0^L \psi^*\,\hat{E}\,\psi\,dx = \int_0^L \psi^*\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\right)\psi\,dx$$

Using $\dfrac{\partial^2\psi}{\partial x^2} = -\left(\dfrac{n\pi}{L}\right)^2\psi$:

$$\begin{aligned} \langle E\rangle &= \frac{\hbar^2}{2m}\left(\frac{n\pi}{L}\right)^2\int_0^L\psi^*\psi\,dx \\ &= \frac{\hbar^2}{2m}\left(\frac{n\pi}{L}\right)^2\cdot\frac{2}{L}\int_0^L\sin^2\frac{n\pi x}{L}\,dx \\ &= \frac{\hbar^2n^2\pi^2}{2mL^2}\cdot\frac{2}{L}\cdot\frac{L}{2} = \frac{n^2h^2}{8mL^2} \end{aligned}$$

(The same result follows from $\hat{E} = j\hbar\,\partial/\partial t$, since $j\hbar\,\partial\psi/\partial t = E_n\psi$.)

Result: $\langle E\rangle = \dfrac{n^2h^2}{8mL^2}$. Because $\psi_n$ is an eigenfunction of the energy operator, every measurement gives this same value; there is no spread in energy.