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3220_PS13 - Physics 3220 Quantum Mechanics 1 Fall 2008...

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Physics 3220 – Quantum Mechanics 1 – Fall 2008 Problem Set #13 Due Wednesday, December 10 at 2pm Problem 13.1 : Surveys! (20 points) Please take the following surveys. You will not be graded for accuracy for these surveys, you get credit just for participating. a) http://www.colorado.edu/sei/surveys/Fall08/Clicker Phys3220 fa08-post.html b) http://www.colorado.edu/physics/EducationIssues/baily/SurveyFa08/MPASFall08Post 3220.htm Problem 13.2 : Projection operators. (20 points) Consider first a Hilbert space spanned by a basis of orthonormal states {| n i} labeled by one discrete quantum number, which we will call n . The states | n i are the eigenstates of some operator ˆ Q , and n labels the various eigenvalues, ˆ Q | n i = q n | n i . (For example, ˆ Q could be the Hamiltonian, and n could label the allowed energies.) For each n , we define the projection operator onto the state | n i as ˆ P n ≡ | n ih n | . (1) Thus there is a different ˆ P n for each state | n i . a) Demonstrate that ˆ P n is Hermitian, and that ˆ P 2 n = ˆ P n . b) What is the result of acting ˆ P n on an arbitrary state | ψ i = m c m | m i ? Explain why the name “projection operator” is justified. If there are N distinct values of n , all operators will be N × N matrices; what does ˆ P n look like as such a matrix? c) In general ˆ P n | ψ i is not normalized; show that the state ˆ P n | ψ i / q h ψ | ˆ P n | ψ i is properly normalized. d) How are the number h ψ | ˆ P n | ψ i and the state ˆ P n | ψ i / q h ψ | ˆ P n | ψ i related to the result of making a measurement of Q ? Relate them to the postulates of quantum mechanics. Now consider a system where the Hilbert space is labeled by more than one quantum number: the hydrogen atom, with states | n ‘ m i labeled by n , and m . The projection operator associated to a given value of n now has a sum over all values of the other quantum numbers: ˆ P n = n - 1 X =0 X m = - | n ‘ m ih n ‘ m | . (2) 1
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In the following consider the hydrogen atom wavefunction | ψ i = 1 2 | 2 1 0 i + 2 | 2 0 0 i + | 1 0 0 i . (3) e) Consider a measurement of energy. Which values of n might be observed? Write down
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