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Quantum Mechanics Physics 215

Homework 1. Due in Class Friday January 14.

  1. (a) Let tex2html_wrap_inline70 be the eigenvalues of the matrix tex2html_wrap_inline72 . Prove that:

    displaymath74

    displaymath76

    (b) Show that:

    displaymath78

  2. Show that:

    displaymath80

    where the tex2html_wrap_inline82 are the zeros of f(x) and tex2html_wrap_inline86 is d f/dx evaluated at tex2html_wrap_inline90 . (Assume that tex2html_wrap_inline92 for all n.) Use this result to obtain simplified expressions for tex2html_wrap_inline96 and tex2html_wrap_inline98 .

  3. The tex2html_wrap_inline100 -function may be defined as:

    displaymath102

    Verify that:

    displaymath104

    (b) Let tex2html_wrap_inline106 . Show that:

    displaymath108

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

  4. Define the operators X and K such that tex2html_wrap_inline114 and tex2html_wrap_inline116 . 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:

    displaymath126

    (b) Evaluate the classical Poisson bracket tex2html_wrap_inline128 , 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 tex2html_wrap_inline134 be an eigenstate of X with eigenvalue x. Prove that tex2html_wrap_inline140 is an eigenstate of X. What is the corresponding eigenvalue?

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

    displaymath148

    [Hint: Define tex2html_wrap_inline150 , determine tex2html_wrap_inline152 , solve the resulting equation and then set tex2html_wrap_inline154 .]

    (b) For reasons you discovered in problem 4(c), tex2html_wrap_inline156 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 tex2html_wrap_inline162 changes under translation.

  6. The Legendre polynomials are defined by

    displaymath164

    Consider the space of square integrable functions defined on a region tex2html_wrap_inline166 . This space is denoted by tex2html_wrap_inline168 . Note that the set tex2html_wrap_inline170 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 tex2html_wrap_inline172 antisymmetric matrix tex2html_wrap_inline174 .

    (a) Show that it can be written in the form tex2html_wrap_inline176 .

    (b) Evaluate tex2html_wrap_inline178 , expressing your answer in terms of I (the identity), A and tex2html_wrap_inline184 .




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Peter Young
Mon Dec 20 12:35:51 PST 1999