Discussion3 - Note 2 1 sin 1 cos 2 2 x x =-Physics 486 Fall...

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Physics 486 Fall 2007 Discussion Problems Week 3 1). A particle of mass m is in an eigenstate of an infinite square well potential that extends from x=0 to x=L. (a). If this is a classical particle that is moving back and forth inside the well, what is the probability of measuring the particle in the leftmost quarter of the well? (b). Now, consider a quantum particle confined to this potential well in a particular eigenstate n. Derive an expression (in terms of the quantum number n) for the probability of finding the particle in the of the well. (c). If the quantum particle is in the ground state (n=1), calculate the probability of finding the particle in the leftmost quarter of the well. How does this result compare with the classical result? (d). How do the classical and quantum probabilities compare for very large n? Is the so-called correspondence principle satisfied? Obtain an expression for the quantum probability and use it to justify your answer.
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Unformatted text preview: Note: ( ) 2 1 sin 1 cos 2 2 x x =-Physics 486 Fall 2007 2). Modeling quantum particles in an infinite square well potential is very useful for making quick estimates of a large number of real systems. Consider the following example: The nuclear potential that binds protons and neutrons in the nucleus is often approximated by a square well potential. Assume that a proton (m p c 2 =938 MeV) is confined to an infinite square well potential of width L = 10-5 nm (a typical nuclear dimension). (a). Calculate the ground state energy of the proton. (b). Calculate the energy of the first excited state of the proton in this potential. (c). Calculate the energy and wavelength associated with a photon that is emitted when the proton makes a transition from the first excited state (n=2) to the ground state (n=1) of this potential. With what part of the electromagnetic spectrum is this wavelength associated?...
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This note was uploaded on 09/20/2010 for the course PHYS 486 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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Discussion3 - Note 2 1 sin 1 cos 2 2 x x =-Physics 486 Fall...

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