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Problem Set 10

This document outlines 6 problems from a problem set. The first problem involves evaluating the expectation values of <r> and <1/r> for the n=1 state of a hydrogen atom using results from McQuarrie problem 7-11. The remaining problems reference additional McQuarrie problems to solve, including problems 7-22, 7-24, 7-28, and 7-43. The final problem involves showing that the raising operator L+ satisfies a particular expression and uses properties of Hermitian adjoints to derive and manipulate terms.

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41 views1 page

Problem Set 10

This document outlines 6 problems from a problem set. The first problem involves evaluating the expectation values of <r> and <1/r> for the n=1 state of a hydrogen atom using results from McQuarrie problem 7-11. The remaining problems reference additional McQuarrie problems to solve, including problems 7-22, 7-24, 7-28, and 7-43. The final problem involves showing that the raising operator L+ satisfies a particular expression and uses properties of Hermitian adjoints to derive and manipulate terms.

Uploaded by

kasun1237459
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Problem Set 10

1. For the n = 1 state in an H atom, evaluate <r>, from which you can write an
expression for 1/<r>. Now evaluate <1/r> and compare with the previous result. The
radial integrals you need to evaluate are easy to do if you use the results from McQuarrie
problem 7-11. (This problem is similar to McQuarrie 7-16, but much of the key exercise
here is in doing the evaluations.)

2. McQuarrie 7-22

3. McQuarrie 7-24

4. McQuarrie 7-28

5. McQuarrie 7-43. This problem is not hard if you follow the directions for things to do
at each step. The exercise of doing/being able to do the operations at each stage is useful.

6. The object of this problem is to show that L lm   l  m  l  m  1 lm  1 . This


is analogous to showing that L  lm   l  m  l  m  1 lm  1 . We already know
that L lm  c  lm  1 (i.e. raising operation in m quantum number. We’re putting in
an  multiplier as well to go with the c factor). Now we need to find the value for c .
The basic procedure follows the same kind of steps as were done for a †n  cnn 1
(Chapter 5 Powerpoint slides) for the harmonic oscillator. The key property that is
needed is that L and L are Hermitian adjoints of one another. Using that property, it
follows that lm L L lm  lm L Llm  L lm Llm . Continue the analysis to
2 2
show/argue that lm L L lm  c  2 lm  1 lm  1  c  2 . Once this is established,
express L L in terms of L2 and Lz operators to evaluate the lm L L lm matrix
element. Manipulate the pieces you get and show that they can be expressed in the form
 l  m  l  m  1 2 .

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