Quantum Mechanics Physics 215

Homework 2. Due in Class Friday January 28

1. (a) Consider a quantum mechanical ensemble characterized by a density matrix . Suppose that the system is governed by a Hermitian Hamiltonian H (which may be time dependent). Show that the time evolution of (in the Schrödinger picture) is given by

   

   (b) Let be the time evolution operator. Find a general expression for in terms of and U.

   (c) Prove that is time independent. Hence show that a pure state cannot evolve into a mixed state.

2. In a one-dimensional problem, consider a particle of potential energy V(x) = - f x, where f > 0.

   (a) Write Ehrenfest's theorem for the mean values of the position x and momentum p. Integrate these equations and compare with the classical motion.

   (b) Show that the root-mean-square deviation does not vary with time.

   (c) Write the Schrödinger equation in the p-representation. Deduce from it a relation between

   

   Solve the equation thus obtained and give a physical interpretation.

   (d) Write the Schrödinger equation in the x-representation. What are the energy eigenfunctions? (This will take a little research on your part into the area of special functions).

   (e) Is the energy spectrum continuous or discrete? What is the behavior of the energy eigenfunctions as ?

   (f) If the potential is replaced by V(x) = f|x|, how would your answer to part (e) change?

3. Consider a particle in three dimensions whose Hamiltonian is given by

   

   By calculating , obtain

   

   This becomes the quantum mechanical virial theorem,

   

   provided . Under what conditions does this happen?

4. In this problem you are asked to derive the Feynman-Hellmann Theorem.

   (a) Suppose that the Hamiltonian depends on a real parameter . That is . Show that

   

   [Hint: Evaluate .]

   (b) Consider the one-dimensional problem with the Hamiltonian

   

   where V is independent of the mass m. Suppose that this Hamiltonian possesses a particular energy eigenstate with energy eigenvalue E. Describe the behavior of E as m decreases.

5. Let be the nth energy eigenstate of the one-dimensional harmonic oscillator. Let X and P be the position and momentum operators respectively.

   (a) Compute , , , and . Calculate for arbitrary n.

   (b) Check that the Virial Theorem holds for the expectation values of the kinetic and the potential energy taken with respect to an energy eigenstate.

6. Consider the operator:

   

   (a) Let be an eigenvector of a with eigenvalue . This is called a coherent state. Compute . Using this result show that . Why does a lack of orthogonality not violate any of our quantum mechanics postulates?

   (b) Consider the operator a in the context of the one-dimensional harmonic oscillator. Compute , where is the nth energy eigenstate. (Assume that is normalized to unity.) Given a coherent state , find the most probable value of n (and corresponding energy E).

   (c) Prove that the normalized coherent state can be written as:

   

   (d) Prove that the coherent state is a state of minimum uncertainty, i.e. .

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