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Electron Trapped in Infinite Potential Well

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Hi, I've attached a homework problem I need help with as a JPEG picture file. I've done a few slightly easier ones before, and I'm about to try doing this one, but I still make stupid mistakes that I don't catch and would appreciate your help with the problem. Thank you very much!

1. An electron is trapped in an infinite potential well with L = 2nm. Suppose the wave function of the electron is given: (see attachment for wave function)

a. What is the value of N so that the wave function is normalized. What is its units?
b. If we make an energy measurement, what are the probabilities we will find the electron in the ground state (n=1) and the first excited state (n=2)
c. (Extra credit) What is the probability that the electron is in the nth state?
d. What is the average energy of the electron.

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The solution is attached below in two files. the files are identical in content, only differ in format. The first ...

Solution Summary

The solution shows in detail how to obtain the normalization factor for a wave function in the form x(x-2), then goes on to show how the function is decomposed into eigenfunctions series, how to obtain the probabilities from the decomposition and finally how to find the average energy of the system.

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Energy of Particles in the Nucleus

See the attached file.

a) Suppose that the potential seen by a neutron in a nucleus can be represented (in one dimension) by a square well of width 10^12 cm with very walls. What is the minimum kinetic energy of the neutron in this potential, in Mev?
b) Can an electron be confined in a nucleus? Answer the question using the following outline or some other method?

i) Using the same assumption as in a) - that the nucleus can be represented as a one-dimensional infinite square well of width 10^12cm - calculate the minimum kinetic energy, in MeV. of an electron bound within the nucleus.

ii) Calculate the approximate coulomb potential energy, in MeV, of an electron at the surface of the nucleus, compared with its potential energy at infinity. Take the nuclear charge to be +50e.

iii) Is the potential energy calculated in ii) sufficient to bind the electron of the kinetic energy calculated in i)?

"An introduction to Quantum Physics" by French and Taylor.

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