• Homework 1: (Due Tuesday, October 23)   

    A particle of mass m is in the ground state of a one dimensional quantum well which
    extends from x=-a/2 to x=a/2 with wavefunction w(x).
  
    At time t=0, the size of the well is suddenly tripled (i.e. a → 3a), the well now extending
    from x=-3a/2 to x=3a/2. Due to the fast expansion of the well, the wavefunction
    at t=0 is still w(x), finite in the range from x=-a/2 to x=a/2: Ψ(x,t=0)=w(x).
  
    Find the energies and corresponding probabilities for measuring them after the expansion.
    What is the numerical probability that the a measurement of the energy of the particle
    after the expansion will be the same as that before the expansion.
   
  

• Homework 2: (Due Thursday, November 29)   

      1. A particle is in a superposition of two states of a harmonic oscillator potential:
  
           Ψ(t=0) = A ( uo(x) + u2(x) )
  
          Where A is the normalization constant. Find the uncertainy of position x as a function of time:
  
            σx(t) = ?
  
       2. A particle with mass m is moving in a one dimensional potential
  
            V(x) = α ( δ(x) + δ(x-a) )
  
          where α is a constant. If the particle is incident from x = -∞ with energy E,
          find the transmission coefficient as a function of E and plot it for various values of α. 
          (Hint: For obtaining computer solutions and plots, it is useful to work with scaled
          quantities which are unitless. For example, you could divide the Schrodinger equation by
          ħ2/(2ma2) - which has units of energy, to obtain unitless variables.)
        
  

• Homework 3: (Due Tuesday, December 11)   

      1. Given that
  
           <x|f> = A (a-|x|)/a  for |x| < a and zero otherwise
  
           <x|p> = exp(ipx/ħ)/√(2πħ)
  
           <p|g> = δ(p - po)
  
         find <f|g>. The quantities a, A, and po are constants.
  
      2. Consider the 2x2 matrix A
  
               [ 1  -i ]
           A = [       ] 
               [ i   1 ]
  
         and the vector V
  
               [ 1 ]
           V = [   ] .
               [ 0 ]
  
         (a) Find the eigenvalues and eigenvectors of A.
  
         (b) Expand V in terms of the eigenvectors of A.
  
         (c) Find exp(iA) V.
  
         (d) Find exp(iA). [Note that you can do part (c) without knowing exp(iA).]
  
  
  

• Homework 4: (Due Thursday, December 20)   

  
    The time independent Schrodinger Equation in three dimensions is
  
        (Px2 + Py2 + Pz2)/2m ψ(x,y,z) + V(x,y,z) ψ(x,y,z) = E ψ(x,y,z)
  
    where Pk = ħ/i ∂/∂xk.
  
    Consider now a cubic box whose one corner is at the origin, and three edges
    extend a distance a in the x, y, and z directions. The potential is zero inside
    the box and infinite outside.
  
    (a) Use separation of variables to show that ψ may be written as 
  
            ψ(x,y,z) = X(x) Y(y) Z(z)
  
        and that the eigenstates of the energy operator are
  
            |nx ny nz > ↔ uE = (2/a)3/2 sin(nxπx/a) sin(nyπy/a) sin(nzπz/a) .
  
    (b) Use the state |Ψ> = ( |1 2 1> + i |2 1 1>)/√2 to find the
        expectation value of the z-component of the angular momentum operator
        Lz = xPy - yPx:
  
            <Lz>Ψ = <Ψ|Lz|Ψ>
  
  

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