Operators and expectation values of observables

In quantum mechanics an operator is a mathematical instruction that acts on a wave function $\psi$ to give another function. Every measurable (observable) quantity such as position, momentum or energy has a corresponding operator. When the operator acts on $\psi$ and returns a constant times $\psi$, that constant is the value we would measure.

Common operators

Writing $\hat{H}\psi = \hat{E}\psi$ gives the Schrödinger equation itself, so the operators are the basis of quantum mechanics.

Eigenvalue equation

If $\hat{A}\psi = a\psi$ with $a$ a constant, then $\psi$ is an eigenfunction of $\hat{A}$ and $a$ is its eigenvalue. A measurement of $A$ on this state always gives exactly $a$. Example: the time-independent Schrödinger equation $\hat{H}\psi = E\psi$ gives the allowed energies $E$.

Expectation value

In general a state is not an eigenfunction, so repeated measurements give different results. The expectation value is the average of many measurements on identical systems. Since $|\psi|^2 dx$ is the probability of finding the particle between $x$ and $x+dx$:

$$\langle A \rangle = \frac{\int_{-\infty}^{\infty} \psi^*\, \hat{A}\, \psi \, dx}{\int_{-\infty}^{\infty} \psi^* \psi \, dx}$$

For a normalised wave function the denominator is 1. Then:

$$\langle x \rangle = \int \psi^* x\, \psi\, dx, \qquad \langle p \rangle = \int \psi^* \left(-i\hbar \frac{\partial \psi}{\partial x}\right) dx, \qquad \langle E \rangle = \int \psi^* \hat{H} \psi\, dx$$

The operator is placed between $\psi^*$ and $\psi$ so that derivatives act only on $\psi$.

Example: particle in an infinite well

For $\psi_n = \sqrt{2/L}\,\sin(n\pi x/L)$ in $0 \le x \le L$:

• $\langle x \rangle = \frac{2}{L}\int_0^L x \sin^2(n\pi x/L)\, dx = \frac{L}{2}$, the centre of the well, as symmetry suggests.
• $\langle p \rangle = 0$, because the particle moves left and right with equal probability.
• $\hat{H}\psi_n = \frac{n^2h^2}{8mL^2}\psi_n$, so $\langle E \rangle = E_n$ exactly: $\psi_n$ is an energy eigenfunction.

Uses

1. They give the possible measured values (eigenvalues), e.g. quantised energy levels of electrons in atoms and solids.
2. They give average values of position, momentum and energy for any state through $\langle A \rangle$.
3. They allow uncertainties, $\Delta x = \sqrt{\langle x^2\rangle - \langle x\rangle^2}$, to be calculated, linking to the uncertainty principle.