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Problem 1
A particle of mass m is in the ground state (n=1) of the infinite square well:
Suddenly the well expands to twice its original size - the right wall moving from a to 2a leaving the wave function (momentarily) undisturbed. The energy of particle in now measured. What is the probability of getting the result
(same as the initial energy)?
Solution:
After the wall expands, the new states and energies are:
The result
otherwise
Solutions:
corresponds to
Therefore, the probability of getting this result is given by
Lecture 12. Problem solving
Lecture 12 Page 1
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Problem 2
Calculate <x>, <x2>, <p>, and <p2> for the nth stationary state of the infinite square well.
Solution
otherwise
Solutions:
Lecture 12 Page 2
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Lecture 12 Page 3
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Problem 3
A particle of mass m in the harmonic oscillator potential starts out in the state
for some constant A.
What is the expectation value of the energy?
Energies:
Harmonic oscillator
First three states are
Solution
This function can be expressed as a linear combination of the first three states ofharmonic oscillator.
Lecture 12 Page 4
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Now, we need to find coefficients c by equating same powers of
Normalization gives:
Now it is really easy to find the expectation value of energy:
Lecture 12 Page 5
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Lecture 12 Page 6
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Mathematical formulas
Lecture 12 Page 7