HW6_solutions_fa11

HW6_solutions_fa11 - Physics 3220 Homework 6 Solutions Due...

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Physics 3220 Homework 6 Solutions Due Oct. 7, 2011 Physics 3220 HW6.1_solutions Fall 2011 1. Figure out the p-space representation of x_operator. (5 points) In the position-space representation of quantum mechanics, we use a spatial and time dependent wave function and a group of operators for position and momentum, to determine the expectation values of various ensemble measurements. In class, we have been discussing the momentum space representation, where the wave functions are functions of MOMENTUM and time, and there are new representations of the position and momentum operator. In class, we demonstrated that p_op = p in this representation. We asserted in class that x_op = ih d/dp in the momentum representation. Prove it. Show in this representation what the value is for [x, p]. You can go at this by assuming that the operator representation is correct, plug it into the momentum version of the expectation value, and work backwards. Alternatively, you can start with the position expectation value in the position representation, and then work forwards. Here’s a version like we did in class: First, let’s see how the position expectation value is in real space:     * x dx x x x    Next, substitute in the Fourier representation of the position-dependent wave functions:     * 1 2 i px i p x x dx dp p e x dp p e          OK, so how do you get a factor of the position to pop out of the second integral?? Well, you could try by taking the derivative of the plane wave factor. Just to be sure that you know what we’re doing, let’s put all the integrations together on the left, and put the position inside with the plane wave, say like this:     * 1 2 i px i p x x dx dp dp p e p xe       Now you can notice that the factor of position can be reproduced by just taking a derivative of the plane wave portion:     * 1 2 i px i p x d x dx dp dp p e p e i dp       Next, integrate the p_prime integral by parts to shift the d/dp_prime:
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Physics 3220 Homework 6 Solutions Due Oct. 7, 2011 Physics 3220 HW6.2_solutions Fall 2011       i p x i p x i p x dp d dp p e p e dp e i dp i dp           The first term on the right is zero because we assume a normalized momentum wave function that will go to zero at large momentum. The second integral can be reverted to a positive by pulling the complex factor of 1/i up to the top, so we finally have:     * 1 2 i px i p x x dx dp dp p e e i dp        The plane waves pull together with the integral over position to yield the delta function as we saw twice in class:       * x dp dp p i p p dp    
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This note was uploaded on 03/03/2012 for the course PHYS 3220 at Colorado.

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HW6_solutions_fa11 - Physics 3220 Homework 6 Solutions Due...

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