Consider an electron trapped in a one-dimensional harmonic potential and it is subjected to an external electric field to the right with a magnitude of Ɛ. The Hamiltonian for this system is given as: ÂĤ = ² + + eɛâ . %3D 2m 2 ) Treating eɛâ as a perturbation, and using non-degenerate perturbation theory, calculate the first order shift in energy for the n-th quantum state, where |Øn) are the exact energy eigenstates of the unperturbed system (i.e. when the electric field is zero). a b) ( ) Repeat part (a) except now calculate the shift in energy accurate to second order.
Consider an electron trapped in a one-dimensional harmonic potential and it is subjected to an external electric field to the right with a magnitude of Ɛ. The Hamiltonian for this system is given as: ÂĤ = ² + + eɛâ . %3D 2m 2 ) Treating eɛâ as a perturbation, and using non-degenerate perturbation theory, calculate the first order shift in energy for the n-th quantum state, where |Øn) are the exact energy eigenstates of the unperturbed system (i.e. when the electric field is zero). a b) ( ) Repeat part (a) except now calculate the shift in energy accurate to second order.
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Transcribed Image Text:Consider an electron trapped in a one-dimensional harmonic potential and it is subjected to an
external electric field to the right with a magnitude of Ɛ. The Hamiltonian for this system is given as:
1
mw²£2
+ eƐ£ .
2m
2
) Treating eɛ£ as a perturbation, and using non-degenerate perturbation theory, calculate the first
order shift in energy for the n-th quantum state, where |Pn) are the exact energy eigenstates of the
unperturbed system (i.e. when the electric field is zero).
b) (
c) (
quadratic form, and you can solve that problem exactly. Solve for the new energy eigenfunctions given
by Pn (x) = (x|Vn). As a function of ɛ, compare the energy spectrum when there is an electric field to
the case when there is no electric field. Also specify how n(x) is related to pn(x) mathematically, and
comment on what this result means physically. Specifically, does this result makes intuitive sense?
) Repeat part (a) except now calculate the shift in energy accurate to second order.
) Notice that by completing the square, the Hamiltonian given above can be transformed to take a
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