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Unformatted text preview: Physics 139A Introduction to Quantum Mechanics Spring 2010 Homework Assignment #1 Due Thursday, April 8 Before writing up your solutions, please see the guidelines on the last page of this assignment. 1. Consider the wave function Ψሺݔ, ݐሻ ൌ ܣ݁ ିఒ|௫| ݁ ିఠ௧ where ܣ , ߣ , and ߱ are real positive constants. a. Find the value of A that normalizes the wave function. b. Calculate the expectation values of x and x 2 . c. Find , the standard deviation of x . d. Sketch the probability density and mark the points ۃݔۄ േ ߪ . e. What is the probability for the particle to be found outside the range ۃݔۄ േ ߪ ? 2. See the calculation of the time derivative of ۃݔۄ on page 16 of Griffiths. Do a similar calculate for the time derivative of ۃۄ (see Eqn. 1.35) and show that ݀ ݀ݐ ۃۄ ൌ ۃെ ߲ܸ ߲ݔ ۄ which looks very much like Newton’s 2 nd law. This is an example of Ehrenfest’s theorem, that expectation values should obey the classical laws of physics. 3. A particle of mass m is in the state Ψሺݔ, ݐሻ ൌ ܣ݁ ିൣ൫௫ మ ⁄ ൯ା௧൧ where ܣ and a are positive real constants. a. Find the normalization constant A . b. What must be the potential energy function ܸሺݔሻ for this wave function to satisfy the Schrödinger equation? c. Calculate the expectation values of ݔ, ݔ ଶ , , and ଶ . d. Find ߪ ௫ and ߪ . Show that their product is consistent with the uncertainty principle. 4. Do the following 3 simple proofs: a. For normalizable solutions to the time independent Schrödinger equation the separation constant E must be real. Write ܧ ൌ ܧ ݅Γ and show that if the normalization condition Eqn. 1.20 is to hold for all t , then Γ must be zero. must be zero....
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