Quantum Mechanics Physics 215
Homework 1. Due in Class Friday January 14.
(b) 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 .
Verify that:
(b) Let . Show that:
(c) Interpret the result of part (b) in terms of Dirac bras and kets.
(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?
[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.
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.
(a) Show that it can be written in the form .
(b) Evaluate , expressing your answer in terms of I (the identity), A and .