Homework7 - Physics 401 - Homework #7 1) Orthogonality of...

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Physics 401 - Homework #7 1) Orthogonality of the stationary states of the harmonic oscillator (three points). Since energy is an observable quantity, it is represented by a Hermitian operator ( H ˆ ), and this guarantees that the energy eigenstates (or stationary states) are orthogonal. We would like to demonstrate that the stationary states are, in fact, orthogonal, for a few specific cases. To do this, show that the ground state, first excited state, and second excited state of the harmonic oscillator have zero overlap, working in the position basis. In other words, prove that 0 2 0 2 1 1 0 using the spatial wavefunctions of these states:  2 / 4 / 1 0 2 ! 2 1 ) ( e H n m x n n n , x m 0 2 4 2 1 2 2 1 0 H H H Hint: By examining the even-odd properties of the integrands, you can easily show that some of these integrals are zero without any real work. 2) (3 points) Suppose we prepare a simple harmonic oscillator with the following wavefunction:  2 / ) ( 9 exp
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Homework7 - Physics 401 - Homework #7 1) Orthogonality of...

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