(Solutions to very similar problems can be found in my lecture notes, starting on P. 13)

Physics 4410 - Review Exercise: A particle is in a harmonic oscillator potential well. Its normalized wave function at t=0 is given by . (Remember, the ground state of the HO has n=0!)

1) What is c? (Careful!)

 

 

 

 

 

2) What is the energy of this particle? (Is this question a good/well defined one? If not, *make* it well defined)

 

 

 

 

 

 

 

3) What is ?

 

 

 

4a) On the back of this sheet, sketch the function. (First excited state of the HO)

b) What is the parity of the starting wave function,?

 

 

 

 

 

 

 

5) An operator O has eigenvalues with corresponding eigenvectors . Write a formal expression (involving an integral) telling the probability that a measurement of O on our state would yield the result, say, .

(What's the answer, numerically, if O happends to be the Hamiltonian operator?)

 

 

 

 

 

 

 

 

 

 

6) How would you write out the starting wave function in Dirac notation? Repeat the last question using Dirac notation. (There's some arbitrariness of notation here, pick what you think would be most reasonable)