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

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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 ) = 2 x 2 / 2 the wave function of the ground state is φ 0 ( x ) = π ¯ h 1 / 4 exp - mωx 2 h , and the wave function of the first excited state is φ 1 ( x ) = π ¯ h 1 / 4 r 2 ¯ h x exp - mωx 2 h . (a) Calculate the mean position h ˆ x i , the mean square position h ˆ x 2 i , the mean momentum h ˆ p i , and the mean square momentum h ˆ p 2 i in the ground and first excited states. (b) Use your results to determine the standard deviations of position and momentum, Δˆ x and Δˆ 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 ˆ T i and h ˆ V i for these two states. For each of them compare h ˆ T i and h ˆ V i .
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