HW9 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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Physics 3220 – Quantum Mechanics 1 – Fall 2008 Problem Set #9 Due Wednesday, Oct 29 at 2pm 9. 1 Commutators [15 pts] Prove the following commutator identities: A) [A+B,C] = [A,C] + [B,C] B) [AB, C] = A[B,C] + [A,C]B C) df [f (x),p] i dx = h , where x and p are the position and momentum operators (in 1D) Next, given our usual definition of the Hamiltonian operator, H = p 2 2 m + V ( x ) , D) show that [ H , p ] = i h V / x E) What is [H, x]? (Quick check: does this result make sense when inserted into the right side Griffiths’ eqn 3.71?) F) In 2 dimensions, p x = - i h / x , p y = - i h / y . What is [ p x , p y ] ? (Note: Why do we care about commutators? They tell us whether observables are “compatible”, they are the basis for generalized uncertainty principles, and commutators with the Hamiltonian H teach us about time dependence of expectation values. So these rather formal looking relations turn out to have lots of practical use in quantum mechanics!) 9. 2 Quantum measurements [ 20 pts] An operator ˆ A (representing observable A) has two normalized eigenstates ψ 1 and ψ 2 , with eigenvalues a 1 and a 2 . Operator B (representing observable B) has two normalized eigenstates φ 1 and φ 2 , with eigenvalues b 1 and b 2 . Suppose these eigenstates are related by the following: ψ 1 = 2 3 f 1 + 5 3 f 2 , y 2 = - 5 3 f 1 + 2 3 f 2 , A) Show us that (assuming φ 1 and φ 2 are properly normalized), ψ 1 and ψ 2 are normalized too. B) Let’s start in some unspecified random state, and then observable A is measured. Further, assume that you DO in fact measure the particular value a 1 - what is the state of the system immediately after this measurement? C) Immediately after the measurement of A (which, recall, happened to yield a
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HW9 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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