hws04_r - PHYS 115A Homework 4 Solutions Winter 2008...

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PHYS 115A Homework 4 Solutions Winter 2008 revisions added February 12: see ADDITIONAL NOTE below and Problem 1 The message below was emailed to the class before this homework assignment was due: As noted at the beginning of class, the lecture notes are merely notes and should not be trusted as a reference. You should always refer to the textbook when copying equations or results, especially for homework or when studying for exams. There are some errors in Notes 3 that might affect people that ignore the advice above and instead use the notes as a reference. Please make the corrections in your copy of the notes: Notes 3: Simple Quantum Systems, Section 9: The Harmonic Oscillator Eq. 75: First line. There should be a negative sign in front of the second partial derivative term. Eq. 82: 3rd line. The factor of ~ ω 0 at the beginning of the line should not be there. Eq. 84: The positive sign should be a negative sign, otherwise if you solve for E n you don’t get the right eigenvalues for the SHO. Eq. 85: Both negative signs should be positive signs, otherwise if you solve for E n you don’t get the right eigenvalues for the SHO. Eq. 105: n should be ( n + 1) Eq. 106: ( n + 1) should be n ADDITIONAL NOTE (added February 12): There are many places in Notes 3 of the class reader, starting at equation (104), where Ψ n ( x, t ) should be corrected to ψ n ( x ). This is because when the operators a + and a - act on an energy eigenstate Ψ n ( x, t ) = ψ n ( x ) e - iE n t/ ~ , they only affect the spatial wavefunction ψ n ( x ), and they don’t change the time-dependent factor e - iE n t/ ~ .
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