Two- and Three-Dimensional Electron Traps — Question 5

Question 5

A superposition within a degenerate level can create a stationary diagonal node

In a square well, let $u_{12}=(2/L)\sin(\pi x/L)\sin(2\pi y/L)$ and $u_{21}=(2/L)\sin(2\pi x/L)\sin(\pi y/L)$. They are orthonormal and share energy $5C/L^2$. Consider $\psi=(u_{12}-u_{21})/\sqrt2$.

1. Check normalization and show that the diagonal $x=y$ is a node.

2. Find the time evolution and determine whether the density changes.

Question 5 - Solution

A superposition within a degenerate level can create a stationary diagonal node

(a) Method and calculation

$$\langle\psi|\psi\rangle=\tfrac12(1+1-0-0)=\boxed{1}.$$$$u_{12}(x,x)=u_{21}(x,x)\Rightarrow\boxed{\psi(x,x)=0}.$$

(b) Interpretation and further calculation

$$H\psi=(5C/L^2)\psi\Rightarrow\boxed{\Psi(x,y,t)=e^{-i5Ct/(\hbar L^2)}\psi(x,y)}.$$$$\boxed{\partial_t|\Psi|^2=0}.$$

Physical check

A superposition is not necessarily time-dependent in its density. States within the same degenerate energy share one time phase.