Practice Problem Set 1

1. Suppose that and are normalized, orthogonal wavefunctions. Let . Show that is normalized only if .
2. Suppose that you want to measure the position of an electron to an accuracy of 1Å. What is then the minimum measurement error in a simultaneous measurement of its speed?
3. Show that Schrödinger's equation is separable for a particle in a two-dimensional universe if the potential energy V if of the form V(x,y) = U(x)+W(y).
4. Spectroscopy involves measuring the energy difference between two energy levels.
   1. Work out a formula for the energy required to change the quantum number n by 1 for a particle in a box during a spectroscopic transition.
   2. If the lowest-energy line of the spectrum of a particle in a box occurs at a frequency of , what would the frequency of the next line in the spectrum be?  
   3. If the particle in question 4b is an electron, what is the length of the box?
5. Use the Hermitian property to show that the expectation value of an observable must be a real number.