Quantum Mechanics Physics 215

Homework 1. Due in Class Friday January 14.

1. (a) Let be the eigenvalues of the matrix . Prove that:

   

   

   (b) Show that:

   

2. Show that:

   

   where the are the zeros of f(x) and is d f/dx evaluated at . (Assume that for all n.) Use this result to obtain simplified expressions for and .

3. The -function may be defined as:

   

   Verify that:

   

   (b) Let . Show that:

   

   (c) Interpret the result of part (b) in terms of Dirac bras and kets.

4. Define the operators X and K such that and . Then [X, K] = i I, where I is the identity operator in an infinite dimensional space.

   (a) Show that if the functions F and G can be expressed as power series in their arguments, then:

   

   (b) Evaluate the classical Poisson bracket , where x and p are the coordinate and linear momentum in one dimension, and compare your result to the one obtained in part (a) above.

   (c) Let be an eigenstate of X with eigenvalue x. Prove that is an eigenstate of X. What is the corresponding eigenvalue?

5. (a) Show that if A commutes with [A,B] then:

   

   [Hint: Define , determine , solve the resulting equation and then set .]

   (b) For reasons you discovered in problem 4(c), is called the translation operator. Using the results of (a) and the definition of X and K given in Problem 4, demonstrate how the expectation value changes under translation.

6. The Legendre polynomials are defined by

   

   Consider the space of square integrable functions defined on a region . This space is denoted by . Note that the set form a basis which is not orthogonal. Construct the first three members of an orthogonal basis using the Gram-Schmidt procedure and show that these states are proportional to the corresponding Legendre polynomials.

7. Consider an arbitrary antisymmetric matrix .

   (a) Show that it can be written in the form .

   (b) Evaluate , expressing your answer in terms of I (the identity), A and .

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