Problem 1:

(a)  Explicitly find the expectation value of the potential energy for the first excited energy level of the one-dimensional harmonic oscillator and compare it to the total energy of this level.

(2)  Find the most general expression for the first excited state of the two-dimensional isotropic harmonic oscillator in terms of the eigenstates {|n>} of the 1-d oscillator.

(3)  Find the expectation values of x² and y² using the state vector from part (b). What is the expectation value of V(x,y)=(1/2)mw²(x²+y²)?

Problem 2:

Find a linear combination of |+> and |-> kets that minimizes the uncertainty product (DS_x)²(DS_y)².  Verify explicitly, that for the product you found the uncertainty relation for S_x and S_y is not violated.

Problem 3:

Consider the state space E=E₁ÄE₂ of two non-identical spin 1/2 particles spanned by the basis vectors {|++>, |+->, |-+>, |-->}. Find the common eigenvectors of S² and S_y.  Express these eigenvectors in terms of the basis vectors {|++>, |+->, |-+>, |-->}.

Problem 4:

The wave function y(r) of a spinless particle is y(r)=Nx²exp(-r²/b²), where b is a real constant and N is a normalization constant.

(a)  If L² is measured, what results can be obtained and with what probabilities?

(b)  If L_z is measured, what results can be obtained and with what probabilities?

(c)  Is y(r) an eigenfunction of L² or L_z?