HW14 - the expectation value of L z ? d) Assume that we...

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University of Colorado, Department of Physics PHYS3220, Fall 09, HW#14 due Wed, Dec 9, 2PM at start of class 1. Survey (Total: 20 pts) Please take the following survey under http://www.colorado.edu/sei/surveys/Fall09/Clicker Phys3220 fa09-post.html Note: Cutting and pasting the link may not work, so type it or go to the link at the web page of the course. 2. Superposition of orbital angular momentum eigenstates (Total: 20 pts) Consider a system which is initially in the state Ψ( θ,φ ) = 1 5 Y 1 , - 1 ( θ,φ ) + r 3 5 Y 1 , 0 ( θ,φ ) + AY 1 , 1 ( θ,φ ) where A is a real number. a) Find A such that the state is normalized. Is your answer unique? Explain. b) Find < Ψ | ˆ L + | Ψ > . (Hint: We have shown in class, that ˆ L ± Y l,m Y l,m ± 1 . For the problem you need the proportionality factor. This is ˆ L ± Y l,m = C ± l,m Y l,m ± 1 with C ± l,m = ~ p l ( l + 1) - m ( m ± 1).) c) If ˆ L z is measured what values will one obtain and with what probabilities? What is
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Unformatted text preview: the expectation value of L z ? d) Assume that we measure L z and the result of the measurement is-~ . What is the state of the system after the measurement? Determine the product L x L y after the measurement. 3. And the nal problem (Total: 20 pts) a) Work out the following commutators: [ L z , p x ] = i ~ p y , [ L z , p y ] =-i ~ p x , [ L z , p z ] = 0 , b) Use the results obtained in part a) to evaluate the commutator [ L z , p 2 ]. c) Use that [ L x , p 2 ] = [ L y , p 2 ] = [ L z , p 2 ] (you do not have to show these relations) and the result obtained in part b) to show that the Hamiltonian H = ( p 2 / 2 m ) + V commutes with L 2 and L z , provided that V = V ( r ) depends only on r ....
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