# ps06 - Homework(6 for Physics 316 Modern Physics...

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Unformatted text preview: Homework (6) for Physics 316, Modern Physics I (Hoffstaetter/Drasco/Thibault) Due Date: Friday, 03/11/05 - 9:55 in 132 Rockefeller Hall Exercise 1: Compute the position uncertainty Δ x in the ground state Φ ( x ) = 1 p √ πa e- ξ 2 2 , ξ = x a , a = r ~ mω . (1) for the harmonic oscillator potential. Use Δ x = p < ( x- < x > ) 2 > = √ < x 2 >- < x > 2 and the expectation values < x > = Z ∞-∞ x | Φ ( x ) | 2 dx , < x 2 > = Z ∞-∞ x 2 | Φ ( x ) | 2 dx . (2) Exercise 2: What is the expectation value h V i and the uncertainty Δ V = p h V 2 i-h V i 2 of the harmonic oscillator potential V = 1 2 mω 2 x 2 for the ground state wave function? Show that also for any other stationary state Φ n , h V i is half the energy E n = ~ ω ( n + 1 2 ). This corresponds to the fact that the energy of a classical harmonic oscillator is on average split in equal parts between kinetic an potential energy....
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