PS6 - Physics 3316 Spring 2010 Basics of Quantum Mechanics...

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Unformatted text preview: Physics 3316, Spring 2010 Basics of Quantum Mechanics Problem Set 6 (Due in lecture, March 10, 2010) Required Readings : • French and Taylor Ch. 4, Griffiths Ch. 2 Key concepts: Solutions to the time independent Schr¨odinger equation, Manipulation of operators 1 Particle Trapped in a Cavity Revisited Consider a particle, as discussed in lecture, which is trapped between two plates and described by a standing wave pattern of wavelength λ = L/n for a relatively large value of n , say n = 100. Note that “trapped” means that the probability of finding the particle outside of the region 0 < x < L is zero. 1.1 Wave function Write down a continuous normalized wave function ψ ( x ) which corresponds to this state. Hint: Keep in mind that ψ ( x ) = 0 for x ≤ 0 and x ≥ L . 1.2 Momentum Distribution Determine and sketch the probability distribution ˜ P ( k ) for the outcomes of measurement of the momenta of the particle in this state. Explain , physically (using the de Broglie Hypothesis) the approximate value of the momentum associated with the main peak(s) in the distribution. 1.3 Heisenberg Uncertainty Principle in Action Explain why, in this case, the peak(s) of the distribution are not infinitely narrow, and show qualitatively that their width is in accord with the Heisenberg Uncertainty Principle. 2 Coherent States of a Simple Harmonic Oscillator Background: Without the general tools for solving the Time Dependent Schr¨odinger Equation (TDSE) which we will learn in the coming lectures, direct solutions to the TDSE are difficult but not impossible to find. Inwe will learn in the coming lectures, direct solutions to the TDSE are difficult but not impossible to find....
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This note was uploaded on 03/29/2010 for the course PHYS 3318 taught by Professor Flanagan during the Spring '08 term at Cornell.

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PS6 - Physics 3316 Spring 2010 Basics of Quantum Mechanics...

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