Schrödinger's Equation — Question 9

Question 9

An energy superposition has an evolving density even though mean energy is constant

Two real orthonormal eigenfunctions $u_1,u_2$ of the same time-independent Hamiltonian have energies $E_1,E_2$, with $E_2>E_1$. A normalized state is $\Psi=(u_1e^{-iE_1t/\hbar}+u_2e^{-iE_2t/\hbar})/\sqrt2$.

1. Derive the density and its beat period at points where $u_1u_2\ne0$.

2. Find the mean energy and explain why total probability remains constant.

Question 9 - Solution

An energy superposition has an evolving density even though mean energy is constant

(a) Method and calculation

$$\boxed{\rho=\tfrac12(u_1^2+u_2^2)+u_1u_2\cos[(E_2-E_1)t/\hbar]}.$$$$\boxed{T_{\mathrm{beat}}=\frac{2\pi\hbar}{E_2-E_1}=\frac h{E_2-E_1}}.$$

(b) Interpretation and further calculation

$$\boxed{\langle H\rangle=\tfrac12(E_1+E_2)}.$$$$\int u_1u_2\,dx=0\quad\Rightarrow\quad\boxed{\int\rho\,dx=\tfrac12(1+1)=1}.$$

Physical check

Local probability oscillates through interference between different energies. Orthonormality cancels the integrated cross term, preserving total probability.