ps07 - Quantum Mechanics Problem Sheet 7 (Solutions are to...

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Unformatted text preview: Quantum Mechanics Problem Sheet 7 (Solutions are to be handed in for marking before the lecture on Monday, 7 Mar 2005) 1. In a harmonic oscillator potential V ( x ) = m 2 x 2 / 2 the wave function of the ground state is ( x ) = m h 1 / 4 exp- mx 2 2 h , and the wave function of the first excited state is 1 ( x ) = m h 1 / 4 r 2 m h x exp- mx 2 2 h . (a) Calculate the mean position h x i , the mean square position h x 2 i , the mean momentum h p i , and the mean square momentum h p 2 i in the ground and first excited states. (b) Use your results to determine the standard deviations of position and momentum, x and p , in these two states and check whether the uncertainty relation is satisfied. One of them is called a minimum uncertainty state. Which one, and why? (c) Determine the expectation values of kinetic and potential energies, h T i and h V i for these two states....
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