There is a 1-d infinite square well of width a, between x = 0 and a....
There is a 1-d infinite square well of width a, between x = 0 and a. We also know that V(x) = 0 for 0<x<a and +∞ elsewhere.
Then we add a perturbation to it so that V(x) = ?_0 for 0 < x < a/2, and the same as before elsewhere.
That is, a flat bump of width a/2 and height ?_0 in the left half of the well.
Please answer following 3 questions.
1) Sketch the potential and indicate all of its relevant parameters
2) Use properties of the eigenfunctions to write the complete expressions for the first order energy correction, then evaluate the integrals.
3) Please write the complete expressions like wave functions, integrals, variables, limits, etc.for calculating the first order correction to the wave functions and second order energy correction.
If that is possible, could you please write your answer with clear handwriting? Thanks!!!
Homework Answers
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There is a 1-d infinite square well of width a, between
x = 0 and a....
Please answer a,b and c. Now, consider a 1-d infinite square well of width a, between...
Please answer a,b and c.
Now, consider a 1-d infinite square well of width a, between x = 0 and a, such that V(x) = 0 for 0<x<a and too elsewhere. A perturbation is then added to it so that V(x) = V. for 0 <x <a/2, and the same as before elsewhere. In other words, a flat bump of width a/2 and height V. in the left half of the well. (a) (5 pts) Carefully sketch the potential and...
X. The first energy correction E) to the 3rd Perturbation of Infinite Square Well. Consider this perturbation to the 1D...
X. The first energy correction E) to the 3rd Perturbation of Infinite Square Well. Consider this perturbation to the 1D infinite square well of width L H1 = eigenenergy E is V(x) A. EL) = EŞV) = V C. EX") " EL) = 1
An infinite square well and a finite square well in 1D with equal width. The potential...
An infinite square well and a finite square well in 1D with
equal width. The potential energies of these wells are
Infinite square well: V(x)=0, from 0 < x < a, also V(x) =
, elsewhere
Finite square well: V(x)= 0, from 0 < x < a, also V(x) =
,
elsewhere
The ground state of both systems have identical particles.
Without solving the energies of ground states, determine which
particle has the higher energy and explain why?
For these graphs, they are wave functions of identical particles that are within an infinite square well and their width is a. a.)What is the most probable value of the energy for each wave function...
For these graphs, they are wave functions of identical particles
that are within an infinite square well and their width is a.
a.)What is the most probable value of the energy for each wave
function and which state has the largest probable energy?
b.) Which of these states has the largest expectation value of
the energy? (this part can be done without calculating the
expectation value of the energy)
Vi Aax V2 0 elsewhere elsewhere a x
Vi Aax V2...
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny, and n, are integers. The corresponding allowed en...
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny, and n, are integers. The corresponding allowed energies are Now let us introduce the perturbation otherwise a) Find the first-order correction to the ground state energy b) Find the first-order correction to the first excited state
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny,...
Consider an infinite well for which the boltom is not lat, as sketched here. I the slope is small, the potential V = er/a may be considered as a per- turbation on the square-well potential over-a/2 &...
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Find parts a and b. Consider the three-dimensional cubic well V = {(0 if 0<x<a, 0<y<a,...
Find parts a and b.
Consider the three-dimensional cubic well V = {(0 if
0<x<a, 0<y<a, 0<z<a), (infinity otherwise).
The stationary states are psi^(0) (x, Y, z) =
(2/a)^(3/2)sin(npix/a)sin(npiy/a) sin(npiz/a), where nx, ny , and
nz are integers.
The corresponding allowed energies are E^0 =
(((pi^2)(hbar^2))/2m(a^2))(nx^2+ny^2+nz^2).
Now let us introduce perturbation V={(V0 if
0<x<(a/2), 0<y<(a/2)), (0 otherwise)
a) Find the first-order correction to the ground state
energy.
b) Find the first-order correction to the first
excited state.
1. Consider the...
2. A particle of mass m in the infinite square well of width a (located at...
2. A particle of mass m in the infinite square well of width a (located at 0 SSa) has as its initial wave function a mixture of two stationary states: v(x,0)Avi(x) +2s (x). (a) Find the probability density of finding the particle at the center of the well, as a function of time. (b) Find the average momentum of the particle at time t.
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a)...
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a) sin ( x) sin (y,) The corresponding energies are n) , 123 Note that the ground state, ?11 is nondegenerate with the energy E00)-E1)-' r' Now introduce the perturbation, given by the shaded region in the figure ma AH,-{Vo, if 0<x otherwise y<a/2 (a) What is the energy of the 1.st excited state of the unperturbed system? What is its degree of degeneracy,v? (b)...
Quantum Mechanics Problem 1. (25) Consider an infinite potential well with the following shape: 0 a/4...
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
Please answer a,b and c.
Now, consider a 1-d infinite square well of width a, between x = 0 and a, such that V(x) = 0 for 0<x<a and too elsewhere. A perturbation is then added to it so that V(x) = V. for 0 <x <a/2, and the same as before elsewhere. In other words, a flat bump of width a/2 and height V. in the left half of the well. (a) (5 pts) Carefully sketch the potential and...
X. The first energy correction E) to the 3rd Perturbation of Infinite Square Well. Consider this perturbation to the 1D infinite square well of width L H1 = eigenenergy E is V(x) A. EL) = EŞV) = V C. EX") " EL) = 1
An infinite square well and a finite square well in 1D with
equal width. The potential energies of these wells are
Infinite square well: V(x)=0, from 0 < x < a, also V(x) =
, elsewhere
Finite square well: V(x)= 0, from 0 < x < a, also V(x) =
,
elsewhere
The ground state of both systems have identical particles.
Without solving the energies of ground states, determine which
particle has the higher energy and explain why?
For these graphs, they are wave functions of identical particles
that are within an infinite square well and their width is a.
a.)What is the most probable value of the energy for each wave
function and which state has the largest probable energy?
b.) Which of these states has the largest expectation value of
the energy? (this part can be done without calculating the
expectation value of the energy)
Vi Aax V2 0 elsewhere elsewhere a x
Vi Aax V2...
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny, and n, are integers. The corresponding allowed energies are Now let us introduce the perturbation otherwise a) Find the first-order correction to the ground state energy b) Find the first-order correction to the first excited state
4. 3D Infinite Square Well Perturbation (20 pts) Consider the three-dimensional infinite cubic well: otherwise The stationary states are where n, ny,...
Consider an infinite well for which the boltom is not lat, as sketched here. I the slope is small, the potential V = er/a may be considered as a per- turbation on the square-well potential over-a/2 < x < a/2. vox) a/2 -a/2 (a) Calculate the ground-state energy, correct to first order in perturbation theory. (b) Calculate the energy of the first excited state, correct to first order in perturbation theory. (c) Calculate the wave function in the ground state,...
Find parts a and b.
Consider the three-dimensional cubic well V = {(0 if
0<x<a, 0<y<a, 0<z<a), (infinity otherwise).
The stationary states are psi^(0) (x, Y, z) =
(2/a)^(3/2)sin(npix/a)sin(npiy/a) sin(npiz/a), where nx, ny , and
nz are integers.
The corresponding allowed energies are E^0 =
(((pi^2)(hbar^2))/2m(a^2))(nx^2+ny^2+nz^2).
Now let us introduce perturbation V={(V0 if
0<x<(a/2), 0<y<(a/2)), (0 otherwise)
a) Find the first-order correction to the ground state
energy.
b) Find the first-order correction to the first
excited state.
1. Consider the...
2. A particle of mass m in the infinite square well of width a (located at 0 SSa) has as its initial wave function a mixture of two stationary states: v(x,0)Avi(x) +2s (x). (a) Find the probability density of finding the particle at the center of the well, as a function of time. (b) Find the average momentum of the particle at time t.
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a) sin ( x) sin (y,) The corresponding energies are n) , 123 Note that the ground state, ?11 is nondegenerate with the energy E00)-E1)-' r' Now introduce the perturbation, given by the shaded region in the figure ma AH,-{Vo, if 0<x otherwise y<a/2 (a) What is the energy of the 1.st excited state of the unperturbed system? What is its degree of degeneracy,v? (b)...
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
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