| course |
course_year |
question_number |
tags |
title |
year |
Quantum Mechanics |
IB |
47 |
IB |
2007 |
Quantum Mechanics |
|
3.II.16B |
2007 |
A quantum system has a complete set of orthonormal eigenstates, $\psi_{n}(x)$, with nondegenerate energy eigenvalues, $E_{n}$, where $n=1,2,3 \ldots$ Write down the wave-function, $\Psi(x, t), t \geqslant 0$ in terms of the eigenstates.
A linear operator acts on the system such that
$$\begin{aligned}
&A \psi_{1}=2 \psi_{1}-\psi_{2} \\
&A \psi_{2}=2 \psi_{2}-\psi_{1} \\
&A \psi_{n}=0, n \geqslant 3
\end{aligned}$$
Find the eigenvalues of $A$ and obtain a complete set of normalised eigenfunctions, $\phi_{n}$, of $A$ in terms of the $\psi_{n}$.
At time $t=0$ a measurement is made and it is found that the observable corresponding to $A$ has value 3. After time $t, A$ is measured again. What is the probability that the value is found to be 1 ?