Schrödinger's Equation and the Hydrogen Atom — Question 9

Question 9

Degenerate hydrogen states can form a stationary state with a nonzero mean position

Treat hydrogen with a fixed nucleus, a nonrelativistic Coulomb potential and no spin, fine structure or external field. Write the Bohr length as a. Let $|s\rangle=|2s\rangle$ and $|p\rangle=|2p,m_\ell=0\rangle$ be orthonormal, with $\langle s|z|s\rangle=\langle p|z|p\rangle=0$ and real $\langle s|z|p\rangle=-3a$. Initially $|\psi\rangle=(|s\rangle+|p\rangle)/\sqrt2$.

1. Find the mean z coordinate and determine whether it evolves in the ideal degenerate Coulomb model.

2. Now suppose a supplied diagonal perturbation gives $E_p-E_s=\delta>0$ without changing the eigenfunctions. Find $\langle z\rangle(t)$ and its period.

Question 9 - Solution

Degenerate hydrogen states can form a stationary state with a nonzero mean position

(a) Method and calculation

$$\boxed{\langle z\rangle=\tfrac12(0+0-3a-3a)=-3a}.$$$$E_s=E_p\Rightarrow|\psi(t)\rangle=e^{-iE_st/\hbar}|\psi(0)\rangle\Rightarrow\boxed{\langle z\rangle(t)=-3a}.$$

(b) Interpretation and further calculation

$$E_p-E_s=\delta:\quad\boxed{\langle z\rangle(t)=-3a\cos(\delta t/\hbar)}.$$$$\boxed{T=2\pi\hbar/\delta=h/\delta}.$$

Physical check

The coherent state is not an eigenstate of parity. Equal energy permits a stationary asymmetric density; a splitting introduces a relative phase and oscillating dipole expectation.