Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 3 / 20D (Quantum Mechanics)

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 $H_3$.

In the ordered eigenbasis, $H_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, a Pauli matrix. The coefficient equations are $i\hslash {\dot{a}}_0=a_1$ and $i\hslash {\dot{a}}_1=a_0$, with $a_0(0)=1$, $a_1(0)=0$. Differentiating the first gives ${\ddot{a}}_0=-a_0/{\hslash }^2$, and its initial derivative vanishes. ThusThe population of ${\chi }_1$ is ${\sin }^2(t/\hslash )$. To be an $H_3$ eigenstate of eigenvalue $+1$, the coefficient of ${\chi }_0$ must vanish. Its first positive zero isThe global phase does not change its eigenstate or measurement probabilities.

Here $H_2=\left(\begin{array}{cc}0& i\\ -i& 0\end{array}\right)$ and $H_2^2=I$. Its time evolution for an additional interval $\tau$ isAt $\tau =T=\pi \hslash /2$, $U_2(T)=-i{H}_2$. Acting on the state from the previous part givesIt differs from the initial state only by a global phase. For every observable or projection the phase cancels between bra and ket, so the final state is observationally indistinguishable from the original isolated state. A relative phase between two components or between coherently controlled alternatives would be observable; this overall phase of the entire state is not.