Quantum Mechanics Physics 215

Homework 3. Due in Class Wednesday February 9

1. Consider the one-dimensional problem of a particle moving in a delta-function potential:

   

   (a) Solve for the bound state energies and wavefunctions. Consider the cases W > 0 and W< 0 separately. [Hint: Integrate the Schrödinger\ equation between and . Let and note that the derivative of the wave function is discontinuous at x = 0.]

   (b) In the case of E > 0 (where E is the energy), obtain the reflection coefficient R and transmission coefficient T. Write the coefficients in terms of the dimensionless parameter where is the ground state energy obtained in part (a), in the case of W > 0. What is the behavior of T(b) as ?

2. A one-dimensional potential has the following form:

   

   where and b are positive constants.

   (a) Find as a function of b such that there is just one bound state, of about zero binding energy, for a particle of mass M.

   (b) Applying this crude model to the deuteron (a bound state of a proton and a neutron), evaluate in MeV, assuming cm and , (where is the proton mass).

   (c) Why did I set rather than in part (b)?

3. In this problem, we will study the bound states and scattering states of a particle of mass m moving in a potential:

   

   where q is positive. You could attempt to solve the Schrödinger equation directly, but the resulting differential equation is quite complicated, (but see e.g. S. Flügge, Practical Quantum Mechanics, problem 39). Instead, you can go through the following tricks which make the solution of this problem quite simple.

   (a) Define the differential operator

   

   Show that the Hamiltonian for this problem can be written in operator form:

   

   where is the adjoint of A. Prove that the energy eigenvalues must be non-negative.

   (b) Construct a second Hamiltonian given by:

   

   Evaluate explicitly and show that the corresponding potential is independent of x. Solve for the energy eigenfunctions and eigenvalues of . What is the minimum value of the allowed energy eigenvalues?

   (c) Show that if is an eigenfunction of H, then either:

    
   		(i) 		   is an eigenstate of   , or
   
   		(ii) 		   is an eigenstate of H with zero eigenvalue.
   
   

   Similarly show that if is an eigenstate of then is an eigenstate of H. Conclude that H and have the same eigenvalue spectrum, except for one eigenstate of H with zero eigenvalue.

   (d) Let be an energy eigenfunction of H with energy , with k > q. Define . Then takes the form:

   

   Using results of parts (b) and (c), write down the exact energy eigenfunctions of H with k > q. Then determine the reflection and transmission amplitudes R(K) and T(K).

   (e) Check for conservation of probability ( ). Using the expression for T, find all the bound state energies of H.

   (f) Using part (c), show that if is the ground state of H with zero eigenvalue, then . Solve the resulting differential equation (in the coordinate basis) for the ground state wave function of H.

4. Consider a periodic potential in one-dimension which satisfies V(x + L) = V(x).

   (a) Show that the translational operator commutes withe the Hamiltonian:

   

   (b) We may choose the energy eigenstates to be simultaneous eigenstates of the translation operator. Show that the general form of such eigenstates is:

   

   where . That is, the eigenfunctions are plane waves modulated by a function with the periodicity of the potential. (This is called Bloch's theorem).

5. Prove the following identities involving the Pauli matrices:

    
   		(a)		   .
   
   		(b)		    .
   
   		(c)		  ,
   
   

   where is a unit vector.
6. An electron is subject to a uniform, time-independent magnetic field of strength B in the positive z-direction. At time t = 0, the electron is known to be in an eigenstate of with eigenvalue , where is a unit vector, lying in the xz-plane, that makes an angle with the z axis.

   (a) Obtain the probability for finding the electron in the state as a function of time.

   (b) Find the expectation value of as a function of time.

   (c) Check explicitly the extreme cases of and .

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