Question

3. Consider the (normalized) eigenvalue problem for the Schroedinger equa-\ntion ?" - [Vo(x) + ?V?(x)]? = ?E?, x ? ? where ?(??) = ?(?) = 0. The potentials are continuous functions. This exercise examines so-called logarithmic perturbation expansions to find the corrections to the energy (Imbo and Sukhatme, 1984). To do this, Preprint submitted to Blackboard (TBS) February 16, 2021 2 it's assumed that the unperturbed state (? = 0) is nonzero, specifically a nondegenerate ground state. (a) Assuming ? ~ ??(x) + ???(x) + ?²??(x) and E ~ E? + ?E? + ?²E?, find what problem the first term in each of these expansions satisfies. In this problem, assume ???? ??²dx = 1, and ???? |V?(x)|dx finite. (b) Letting ? = e^(??(x)), find the problem ?(x) satisfies. (c) Expand ?(x) for small ? and from this find E? and E? in terms of ?? and the perturbing potential. (d) For a harmonic oscillator (V? = ?²x² with ? > 0) with perturbing potential V? = ?x exp(??x²) (where ? and ? are positive), show that $\frac{??}{?} \sim -\frac{1}{4} (\frac{??}{?} + (\frac{?}{2?})^{2})^{1/2}$

          3. Consider the (normalized) eigenvalue problem for the Schroedinger equa-\ntion
?" - [Vo(x) + ?V?(x)]? = ?E?, x ? ?
where ?(??) = ?(?) = 0. The potentials are continuous functions.
This exercise examines so-called logarithmic perturbation expansions to
find the corrections to the energy (Imbo and Sukhatme, 1984). To do this,
Preprint submitted to Blackboard (TBS)
February 16, 2021
2
it's assumed that the unperturbed state (? = 0) is nonzero, specifically a
nondegenerate ground state.
(a) Assuming ? ~ ??(x) + ???(x) + ?²??(x) and E ~ E? + ?E? + ?²E?, 
find what problem the first term in each of these expansions satisfies.
In this problem, assume
???? ??²dx = 1, and ???? |V?(x)|dx finite.
(b) Letting ? = e^(??(x)), find the problem ?(x) satisfies.
(c) Expand ?(x) for small ? and from this find E? and E? in terms of
?? and the perturbing potential.
(d) For a harmonic oscillator (V? = ?²x² with ? > 0) with perturbing
potential V? = ?x exp(??x²) (where ? and ? are positive), show that
$\frac{??}{?} \sim -\frac{1}{4} (\frac{??}{?} + (\frac{?}{2?})^{2})^{1/2}$

3. Consider the (normalized) eigenvalue problem for the Schroedinger equa-?" - [Vo(x) + ?V?(x)]? = ?E?, x ? ?
where ?(??) = ?(?) = 0. The potentials are continuous functions.
This exercise examines so-called logarithmic perturbation expansions to
find the corrections to the energy (Imbo and Sukhatme, 1984). To do this,
Preprint submitted to Blackboard (TBS)
February 16, 2021
2
it's assumed that the unperturbed state (? = 0) is nonzero, specifically a
nondegenerate ground state.
(a) Assuming ? ??(x) + ???(x) + ?²??(x) and E E? + ?E? + ?²E?,
find what problem the first term in each of these expansions satisfies.
In this problem, assume
???? ??²dx = 1, and ???? |V?(x)|dx finite.
(b) Letting ? = e^(??(x)), find the problem ?(x) satisfies.
(c) Expand ?(x) for small ? and from this find E? and E? in terms of
?? and the perturbing potential.
(d) For a harmonic oscillator (V? = ?²x² with ? > 0) with perturbing
potential V? = ?x exp(??x²) (where ? and ? are positive), show that
(??)/(?)∼ -(1)/(4) ((??)/(?) + ((?)/(2?))^2)^1/2

Added by James N.

Question

Please give Ace some feedback