• Homework 1: (Due Thursday, March 5)
  

  
    1) Consider a particle in a box with the potential
  
                  { 0 if |x| < L/2
           V(x) = {
                  { ∞ otherwise
  
       A perturbing potential of the form
  
                             {  V0  if |x| < L/4
                      H(1) = {
                             {  0   otherwise
  
       is also present. Find the correction to the energy of the n'th state to second
       order* in perturbation theory.
  
  
    2) A particle has an unperturbed energy operator with the matrix representation
  
                               1  1
                     H(0)  = E           .
                               1  2
  
  
        If there is also a perturbation with the matrix representation
  
  
                               0  1
                     H(1) = e         ,
                               1  0
  
        (a) find the exact energy values,
  
        (b) find the second order* correction to the energies and discuss their relationship to the exact results.
  
        (c) Take E=1 and e=0.1 . Compare the numrerical values of the exact energies to those obtained from perturbation theory.
  
    * "Find the correction to second order" means find E(1) and E(2).
  
  
  

• Homework 2: (Due Thursday, March 26)
  

  
    1) Consider the Hamiltonian of a charged particle in a three dimensional harmonic oscillator potential:
  
               H(o) = (px2 + py2 + pz2)/2m +k(x2 + y2 + z2)/2
  
       Note that the Schrodinger equation is separable in the x, y, z coordinates. The energy then has the form ħω(nx+ny+nz +3/2) 
       for the state |nx ny nz> where ni is the eigenvalue of the number operator Ni=ai+ai- in the i'th direction:
  
               Ni |nx ny nz> = ni |nx ny nz> 
  
       The states | 1 0 0 > , | 0 1 0 > and | 0 0 1 > are therefore degenerate. Now, assume that a weak magnetic field B in the 
       z-direction is also present so that there is an additional contribution to the energy in the form
  
               H(1) = α BLz ,
  
       where α is a constant (what do you expect it to be?) and Lz=xpy-ypx is the angular momentum operator in the z-direction.
  
       Find first order corrections to the energies of the states | 0 0 0 > , | 1 0 0 > , | 0 1 0 > and | 0 0 1 > .
  

• Homework 3: (Due Thursday, April 2)
  

  
     In your previous homework, you found the effect of a magnetic field on a (spinless) charged particle 
  in the three dimensional harmonic oscillator potential. (Equivalent of the Zeeman effect in Hydrogen.)
  We also discussed the Stark effect for this potential in class. (Remember, no change to first order in the field.)
  
  Now, find the relativistic correction and the spin-orbit effects on the energy for this system. i.e.,
  
   (a) find the first order change in the ground state energy if the kinetic energy is expressed 
       relativistically as K2 = p2c2 + m2c4,
  
   (b) find the first order change in the energy of the states | 0 0 0 > , | 1 0 0 > , | 0 1 0 > and | 0 0 1 > if
       there is an extra energy term of the form H(1) = (α/r3) L⋅S where L and S are the angular momentum
       and spin-1/2 operators respectively. 
  
       [Hint: Consider the set
  
            ψa = | 0 0 1 >
  
            ψb = ( | 0 1 0 > + i | 1 0 0 > ) /√2
  
            ψc = ( | 0 1 0 > - i | 1 0 0 > ) /√2
  
       and the action of the operators Lz and L2 on them. The results should imply that you can use methodology similar to the
       Hydrogen atom. You will have to evaluate the expectation value < α/r3 >r by integrating the wavefunctions.]
  
  

Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.