McIntyre 5.2

A particle in an infinite square well potental has an initial state vector

\[\left| \psi (t=0) \right\rangle =A\left( \left| {{\varphi }_{1}} \right\rangle -\left| {{\varphi }_{2}} \right\rangle +i\left| {{\varphi }_{3}} \right\rangle \right)\]

where are the energy eigenstates.

(a) Normalize the state vector

(b) What are the results of a measurement of the energy and with what probablity do they occur?

( c) What is the average value of the energy?

McIntyre 5.5: Expectation values, uncertainties for infinite well

This is a perfect example of where you should use Mathematica to do integrals. The point is not to demonstrate prowess at integrals, but to use a tool to evaluate integrals to show some interesting physics. Note that if an integral evaluates to zero or something quite simple, then there is some physics or symmetry that you should be seeing, and you should seek it out.

Extra practice: Having found $\Delta x$ and $\Delta p$ , also calculate the uncertainty product $\Delta x\Delta p$ for the first two states and demonstrate that $\Delta x\Delta p > \frac{\hbar }{2}$.