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Unformatted text preview: 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 } labeled by one discrete quantum number, which we will call n . The states  n are the eigenstates of some operator ˆ Q , and n labels the various eigenvalues, ˆ Q  n = q n  n . (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 as ˆ P n ≡  n n  . (1) Thus there is a different ˆ P n for each state  n . 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 = ∑ m c m  m ? 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 is not normalized; show that the state ˆ P n  ψ / „ ψ  ˆ P n  ψ is properly normalized. d) How are the number ψ  ˆ P n  ψ and the state ˆ P n  ψ / „ ψ  ˆ P n  ψ 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 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 = ∞ „ =0 „ m =  n m n m  . (2) 1 In the following consider the hydrogen atom wavefunction  ψ = 1 2 „  2 1 0 + √ 2...
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 Fall '08
 STEVEPOLLOCK
 mechanics, 3%, Hilbert space, 0 m

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