An electron bounces elastically in 1D between two infinite potential walls separated by a distance L. The electronis in its lowest possible energy state.(a) What is the energy of this state?(b) The separation between the walls is slowly (a.k.a. 'adiabatically') increased to 2L. That the process occursslowly means that the electron slowly adapts to continue to occupy the ground state of this new well of width 2L.What is the change in energy that the electron experiences?(c) With the walls again at a distance of L, imagine now that the separation is abruptly increased from L to 2L.This means that, at the moment when the change is made, the wavefunction is unchanged for a < L and zero (yes!)for x > L. Write a (normalized) expression for 1(x) at this very moment, and draw it for the intervalx = [0, 2L]. What is the expectation value of the energy for this ₁(a)? I'm calling it 1(a) not (x) tomake it clear that it represents the ground state.(d) Show that the above 1(r) is no longer a solution for the Schroedinger equation for a box ofseparation 2L.(e) ₁(x) can, nevertheless, be expressed as a linear combination (or a 'weighed sum', or a 'superposition') ofwavefunctions that satisfy the Schroedinger equation for a box of separation 2L:₁(x) = a₁01(x) + a202(x) + a303(x) + ...(9)Here, 1(2) is the ground state wavefunction solution for the 2L separation. Write a (normalized) expressionfor 1(x), and draw it for the interval x = [0, 2L].

Answered: in its lowest possible energy state. )…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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An electron bounces elastically in 1D between two infinite potential walls separated by a distance L. The electron
is in its lowest possible energy state.
(a) What is the energy of this state?
(b) The separation between the walls is slowly (a.k.a. 'adiabatically') increased to 2L. That the process occurs
slowly means that the electron slowly adapts to continue to occupy the ground state of this new well of width 2L.
What is the change in energy that the electron experiences?
(c) With the walls again at a distance of L, imagine now that the separation is abruptly increased from L to 2L.
This means that, at the moment when the change is made, the wavefunction is unchanged for a < L and zero (yes!)
for x > L. Write a (normalized) expression for 1(x) at this very moment, and draw it for the interval
x = [0, 2L]. What is the expectation value of the energy for this ₁(a)? I'm calling it 1(a) not (x) to
make it clear that it represents the ground state.
(d) Show that the above 1(r) is no longer a solution for the Schroedinger equation for a box of
separation 2L.
(e) ₁(x) can, nevertheless, be expressed as a linear combination (or a 'weighed sum', or a 'superposition') of
wavefunctions that satisfy the Schroedinger equation for a box of separation 2L:
₁(x) = a₁01(x) + a202(x) + a303(x) + ...
(9)
Here, 1(2) is the ground state wavefunction solution for the 2L separation. Write a (normalized) expression
for 1(x), and draw it for the interval x = [0, 2L].

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The particle in a box model (also known as the infinite potential well or the infinite square well) in quantum mechanics depicts a particle free to travel in a small space surrounded by impenetrable barriers.

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