Energies of a Trapped Electron — Question 7

Question 7

A constant potential offset shifts levels without changing transition frequencies

A box has energies $E_n=V_b+n^2A$, where $A=\pi^2\hbar^2/(2mL^2)$. The interior potential is raised uniformly by a constant $\Delta V$ without changing L or the shape of the well. Ignore any external reference detector.

1. Find the new energy levels and the shift of a transition $n\to r$.

2. Find the extra time-dependent phase acquired after a duration t and its effect on position density.

Question 7 - Solution

A constant potential offset shifts levels without changing transition frequencies

(a) Method and calculation

$$\boxed{E_n\prime=E_n+\Delta V}.$$$$\boxed{E_n\prime-E_r\prime=E_n-E_r=(n^2-r^2)A}.$$

(b) Interpretation and further calculation

$$\boxed{\Psi\prime(x,t)=e^{-i\Delta Vt/\hbar}\Psi(x,t)}.$$$$\boxed{|\Psi\prime(x,t)|^2=|\Psi(x,t)|^2}.$$

Physical check

The common offset changes the chosen energy reference. Relative phases between levels and internal transition frequencies remain unchanged.