Central Forces 2021
In this problem, you will carry out calculations on the following normalized
abstract quantum state on a ring:
\begin{equation}
\left| \Psi\right\rangle
= \sqrt\frac{ 1}{4} \left(\left| 1\right\rangle
+ \sqrt{2}\left| 2\right\rangle
+\left| 3\right\rangle\right)
\end{equation}
-
You carry out a measurement to determine the energy of the particle at time
\(t=0\). Calculate the probability that you measure the energy to be \(\frac{4
\hbar^2}{2 I}\).
-
You carry out a measurement to determine the \(z\)-component of the angular
momentum of the particle at time \(t=0\). Calculate the probability that you
measure the \(z\)-component of the angular momentum to be \(3 \hbar\).
-
You carry out a measurement on the location of the particle at time, \(t=0\).
Calculate the probability that the particle can be found in the region
\(0<\phi< \frac{\pi}{2}\).
-
You carry out a measurement to determine the energy of the particle at time \(t
= \frac{2 I}{\hbar} \frac{\pi}{4}\). Calculate the probability that you measure
the energy to be \(\frac{4 \hbar^2}{2 I}\).
-
You carry out a measurement to determine the \(z\)-component of the angular
momentum of the particle at time \(t = \frac{2 I}{\hbar}\frac{\pi}{4}\).
Calculate the probability that you measure the \(z\)-component of the angular
momentum to be \(3 \hbar\).
You carry out a measurement on the location of the particle at time
\(t = \frac{2 I}{\hbar}\frac{\pi}{4}\).
Calculate the probability that the particle can be found in the region
\(0<\phi< \frac{\pi}{2}\).
-
Write a short paragraph explaining what representation/basis you used for each
of the calculations above and why you chose to use that
representation/basis.
-
In the calculations above, you should have found some of the quantities to be
time dependent and others to be time independent. Briefly explain why this is
so. That is, for a time dependent state like \(\left| \Psi(t)\right\rangle\)
explain what makes some observables time dependent and others time
independent.