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first-order energy correction in case of 1-D delta-function

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Recall that a 1-D delta-function potential well of the form V (x) =
−B delta(x) had exactly one bound state, with a double-tailed exponential wave function.
(a) Apply a harmonic oscillator perturbation of the form V ′(x) = (m omega^2 x^2)/2. Calculate the
ground-state energy for this perturbed system to first order.
(b) Imagine instead that the harmonic oscillator is the original unperturbed system, and
the delta-function potential is the perturbation. Calculate the ground state energy for
this system to first order.
(c) Do your results agree with each other? Discuss why or why not.

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Solution Summary

Problem here was to calculate first-order energy correction when we have 1-D delta-function potential well as unperturbed system and some form of harmonic oscillator as perturbation and vice versa, when we have ground state of harmonic oscillator as unperturbed system and delta-function potential as perturbation.

Solution Preview

Recall that a 1-D delta-function potential well of the form had exactly one bound state with a double-tailed exponential wave function.

(a) Apply a harmonic oscillator perturbation of the form Calculate the
ground-state energy for this perturbed system to first order.

(b) Imagine instead that the harmonic oscillator is the original unperturbed system, and
the delta-function potential is the perturbation. Calculate the ground state energy for
this system to first order.

(c) Do your results agree with each other? Discuss why or why not.

Solution.

1-D delta-function potential well is defined as and it is well known (see for example Ref. 1 and 2) that bound state wave function has next form:

,

where and ...

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