Physics 471 Problem Set 10 Fall 2003
36. The normalized eigenstates of the harmonic oscillator, |n, can be constructed from the
ground state |0using the raising operator a†as
|n=(a†)n
√n!|0,E
n=¯hω (n+1
2).
Determine the effect of the application of aand a†to |n. Hint: Remember that that a|n
is an eigenstate with the energy lowered by ¯hω and a†|nis an eigenstate with the energy
raised by ¯hω.
37. Using the representation
a=mω
2¯hx+¯h
2mω
d
dx
a†=mω
2¯hx−¯h
2mω
d
dx ,
express xin terms of aand a†and calculate m|x|nusing the properties of the raising
and lowering operators. (No integrals are necessary!)
38. Find the momentum space wave function for a particle of mass mmoving in the potential
V(x)=Fx by integrating the momentum space version of the Schr¨odinger equation.
Assume E>0.
39. Griffiths Problem 4.1 (a) and (c).
