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Fund Quantum Mechanics Lect & HW Solutions 108

Fund Quantum Mechanics Lect & HW Solutions 108 - 1...

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90 CHAPTER 7. TIME EVOLUTION Classical physics understands the wave nature of light well, and not its particle nature. This is the opposite of the situation for an electron, where classical physics understands the particle nature, and not the wave nature. 7.1.2.2 Solution schrodsol-b Question: For the one-dimensional harmonic oscillator, the energy eigenvalues are E n = 2 n + 1 2 ω Write out the coefficients c n (0) e i E n t/ ¯ h for those energies. Now classically, the harmonic oscillator has a natural frequency ω . That means that whenever ωt is a whole multiple of 2 π , the harmonic oscillator is again in the same state as it started out with. Show that the coefficients of the energy eigenfunctions have a natural frequency of
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Unformatted text preview: 1 2 ω ; 1 2 ωt must be a whole multiple of 2 π for the coe±cients to return to their original values. Answer: The coe±cients are c n (0) e − i (2 n +1) 2 ωt Now if ωt is 2 π , the argument of the exponential equals i times an odd multiple of π . That makes the exponential equal to minus one. It takes until ωt = 4 π until the exponential returns to its original value one. 7.1.2.3 Solution schrodsol-c Question: Write the full wave function for a one-dimensional harmonic oscillator. Formulae are in chapter 3.6.2. Answer: Using the give formulae Ψ( x, t ) = ∞ s n =0 c n (0) e − i (2 n +1) 2 ωt h n ( x ) 7.1.3 Energy conservation...
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