Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 3 / 20D / a / Solution

The Time-dependent Schrödinger equation is $i\hslash {\partial }_t\Psi =H\Psi$. The normalized eigenstates of the Hermitian Hamiltonian $H_3$ with distinct eigenvalues are orthogonal, so they form an orthonormal basis of this two-state system. Substitute the expansion of $\Psi$ and compare the two coefficients:Hence $a_0(t)=a_0(0)e^{it/\hslash }$ and $a_1(t)=a_1(0)e^{-it/\hslash }$. For the normalized test state $\varphi =\alpha {\chi }_0+\beta {\chi }_1$, the Born rule givesThe complex conjugates are essential; relative phases can affect this probability even though the energy-state populations remain constant under $H_3$.