PHYSICS 137A Midterm: physics137A-sp2013-mt2-Debuhr-soln

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turquoisegnu128

8 Jan 2019

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Physics 137a quantum mechanics spring 2013. Consider the quantum simple harmonic oscillator (sho) with frequency . The normalized eigen- vectors of the sho will be designated |ni or as n(x). Explain brie y. (b) if the current state of the system is | (t)i = c1 |1i + c3 |3i + c4 |4i, where ci are complex. |3i + 3 (c) if the current state of the system is | (t)i = i (d) compute the commutator of the number operator n = a+ a. 4 |4i, what is the probability with the raising operator: | (t)i? of measuring the energy to be /2. Solutions (a) the raising operator is not hermitian, so it is not an observable. (b) we just need to use the rule that a. = c1 1|0i + c3 3|2i + c4 4|3i . (c) the energy eigenstate associated with the energy /2 is the ground state, |0i.

Related Questions

Consider an harmonic oscillator with spring constant k and massm.
- Find the classical frequency of this oscillator as a function ofthe spring constant and m.

In quantum mechanics, the spectrum of possible energies for thisoscillator is given by: En = hf (n + 1/2) where n is a naturalnumber: n = 0, 1, 2, 3, ..., h is Planckâs constant and f is thefrequency.
- Find the ground state energy as a function of the spring constantand the mass.
- What is the difference of energy between the second excited stateand the ground state?
- Assume now that the oscillator carries a unit of charge, what isthe wavelength of the photon emitted as a function of k and m whenthe oscillator goes from the first excited state down to the groundstate?
- What is the wavelength of the emitted photon of the previousquestion if k = 10eV/ Â°A2, where Â°A denotes Angstrom and eV:electron volt. Is the emitted photon in the visible spectrum of ahuman eye?

thank you so much for any help!

Consider an harmonic oscillator with spring constant k and massm.
- Find the classical frequency of this oscillator as a function ofthe spring constant and
m.
In quantum mechanics, the spectrum of possible energies for thisoscillator is given by:
En = hf (n + 1/2) where n is a natural number: n = 0, 1, 2, 3, ...,h is Planckâs constant
and f is the frequency.
- Find the ground state energy as a function of the spring constantand the mass.
- What is the di?erence of energy between the second excited stateand the ground
state?
- Assume now that the oscillator carries a unit of charge, what isthe wavelength of
the photon emitted as a function of k and m when the oscillatorgoes from the ?rst
excited state down to the ground state?
- What is the wavelength of the emitted photon of the previousquestion if k = 10eV /Â°A
2
,
where Â°A denotes Angstrom and eV: electron volt. Is the emittedphoton in the visible
spectrum of a human eye?

A Particle in a box
In order to answer the following questions you will need toconsideran infinite potential
well of length L, in which a particle has been confined.

1. For principal quantum number n=7, write code to calculatethenormalization constant.
Then, plot the probability density function for this state (usingL= 10).
2. Using the properly normalized wave equation from Question1,write code to determine
the expectation values <x> and <x2> . Is <x>oneof the most probable positions? Explain
why.
3. Using the properly normalized wave equation from Question1,write code to determine
the expectation values <p> and <p2> . Show the valueyouobtain for <p2> is equal to
2mE7 (where E7 is the energy of the n = 7 level). Remember that pisa differential
operator so you will have to use the diff command.
4. The uncertainty of a quantum measurement is defined on page222,Equation 6.41 (3rd
edition SMM). Using this equation calculate the uncertainties oftheposition and
momentum of a particle in the n = 5 state. Compute the productofthese uncertainties and
compare this with the prediction of the Heisenberguncertaintyprinciple.
Quantum oscillator
5. Repeat calculations in the previous question for the groundstatewave function of the
quantum oscillator. Comment on the obtained result.

PHYSICS 137A Midterm: physics137A-sp2013-mt2-Debuhr-soln

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