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3890ps3_09

# 3890ps3_09 - Chem 3890 Physical Chemistry I Fall 2009...

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Unformatted text preview: Chem 3890 Physical Chemistry I Fall 2009 Chemistry 3890 Problem Set 3 Fall 2009 Due: in class, Friday, September 25 Last revised: September 20, 2009 Additions/corrections: ✑ Reminder : To clear global variables, use Remove["Global ` * "] Problem 1 Define a wavefunction ψ ( x ) = A e ikx e- ( x- x ) 2 / 2 (1) where k and x are real, and A is a (real) normalization constant. 1. Calculate the normalization constant A for general ( k,x ). Use the normalized ψ for the rest of the problem. 2. Calculate ( x ) , ( x 2 ) and σ x for general ( k,x ). Do your answers depend on k ? 3. Calculate ( p ) , ( p 2 ) and σ p for general ( k,x ). (Do not use the numerical value for planckover2pi1 ; keep your answer in symbolic form.) 4. Calculate the uncertainty product σ x σ p for general ( k,x ). Does your expression depend on the parameters k or x ? As you can see, the wavefunctions ψ ( x ) are very useful in that the mean (expectation) values of position and momentum can be “dialled in” through the choice of parameters x and k . Consider two normalized wavefunctions ψ a ( x ) and ψ b ( x ) of the form (1), having ( x ) = 2, ( p ) = 0 ( ψ a ( x )) ✑ and ( x ) = 2, ( p ) = 4 planckover2pi1 ( ψ b ( x )), respectively. 5. Make plots of the real part (Re[ ψ ]), the imaginary part (Im[ ψ ]), and the associated probability density ρ ( x ) for each state. Is there any way that the states ψ a and ψ b can be distinguished on the basis of expectation values of functions of the position variable x (for example, evaluation of higher moments ( x n ) )? If yes, explain how; if no, explain why....
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3890ps3_09 - Chem 3890 Physical Chemistry I Fall 2009...

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