Energies of a Trapped Electron — Question 5

Question 5

A mean energy need not equal an allowed measurement outcome

An electron is in the normalized state $\sqrt{3/4}\,u_1+e^{i\alpha}\sqrt{1/4}\,u_3$, where the orthonormal box eigenstates have energies $E_n=n^2E_1$.

1. Find the probabilities of each energy outcome and the mean energy.

2. Find the energy standard deviation and determine whether changing $\alpha$ changes these statistics.

Question 5 - Solution

A mean energy need not equal an allowed measurement outcome

(a) Method and calculation

$$\boxed{P(E_1)=3/4,\quad P(9E_1)=1/4}.$$$$\boxed{\langle E\rangle=(3/4)E_1+(1/4)9E_1=3E_1}.$$

(b) Interpretation and further calculation

$$\langle E^2\rangle=(3/4)E_1^2+(1/4)81E_1^2=21E_1^2.$$$$\boxed{\sigma_E=\sqrt{21-9}\,E_1=2\sqrt3\,E_1},\quad\boxed{\partial_\alpha\langle E\rangle=\partial_\alpha\sigma_E=0}.$$

Physical check

The mean is an ensemble statistic and need not be an eigenvalue. Relative phase can affect other observables even though it cancels from these energy probabilities.