ProblemSet3 - Chem 120A Problem Set 3 Due 03/17/06 1. In...

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Chem 120A Problem Set 3 Due 03/17/06 1. In physical space, the mean momentum (in 1-D) may be written as: h ˆ p i = - i ¯ h Z - Ψ * ( x ) d dx Ψ ( x ) dx (1) a) Show that the mean momentum, h ˆ p i , is zero if Ψ ( x ) is an eigenstate of parity (Hint: first show that Ψ * is also an eigenstate of parity witht the same eigenvalue) b) Show that h ˆ p i = 0 as long as Ψ * ( x ) = Ψ ( x ) even if Ψ ( x ) is NOT an eigenstate of parity (assume Ψ ( x ) is a bound state). What physical conclusion can you obtain from your proof? 2. Obtain the mean position, h ˆ x i , for a particle moving in a 1-D harmonic oscillator potential, when the particle is in the state with normalized wavefunction: Ψ ( x ) = ( α 4 π ) 1 / 4 ( 2 α x 2 - 1 ) e - α x 2 / 2 (2) (There is an EASY way to do this problem. ....) 3. In class we introduced the following two ”chiral” states: | L i = 1 2 [ | 0 i + | 1 i ] (3) | R i = 1 2 [ | 0 i-| 1 i ] (4) where | 0 i and | 1 i are the ground and 1st excited eigenstates, respectively, of the 1-D Harmonic Oscil-
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ProblemSet3 - Chem 120A Problem Set 3 Due 03/17/06 1. In...

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