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Consider a particle in a one-dimensional box, where the potential the potential...
Particle with a speed bump Consider our old friend the 1D particle in the box, except...
Particle with a speed bump Consider our old friend the 1D particle in the box, except now with a speed bump in the box so the potential now is given by L and L < x < L 0, Vo 0 x V (x) otherwise (a) Calculate the first order correction to the ground state (n = 1) and first excited state (n = 2) energies (b) Calculate the first order correction to the ground state wave function in terms...
4. (20 points). δ. unction perturbation. Consider a particle in a one-dimensional infinite square well with...
4. (20 points). δ. unction perturbation. Consider a particle in a one-dimensional infinite square well with boundaries at -a and-a. We introduce the following 6-function perturbation at x=0: a. Compute the first-order corrections to the energies of the particle induced by the ν' perturbation b. Recall that you solved this problem exactly in problem set 4 (Griffiths 2.43). Compare your perturbation theory result to the exact solution.
Consider a particle confined in two-dimensional box with infinite walls at x 0, L;y 0, L. the dou...
quantum mechanics
Consider a particle confined in two-dimensional box with infinite walls at x 0, L;y 0, L. the doubly degenerate eigenstates are: Ιψη, p (x,y))-2sinnLx sinpry for 0 < x, y < L elsewhere and their eigenenergies are: n + p, n, p where n, p-1,2, 3,.... Calculate the energy of the first excited state up to the first order in perturbation theory due to the addition of: 2 2
Consider a particle confined in two-dimensional box with infinite...
4. (20 points). 5-function perturbation. Consider a particle in a one-dimensional infinite square well with boundaries...
4. (20 points). 5-function perturbation. Consider a particle in a one-dimensional infinite square well with boundaries at x--a and x-a. We introduce the following δ-function perturbation at V'(x) 00(z). a. Compute the first-order corrections to the energies of the particle induced by the perturbation b. Recall that you solved this problem exactly in problem set 4 (Griffiths 2.43). Compare your perturbation theory result to the exact solution
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...
Instead of assuming that a one-dimensional particle has no energy (v(x)=0), consider the case of a...
Instead of assuming that a one-dimensional
particle has no energy (v(x)=0), consider the case of a
one-dimensional particle which has finite, but constant, energy
V(x)= V sub zero.. Show that the ID particle in a box wave
functions. n(x)= A sin ((pi n x)/a). Also solve the Schrödinger
equation for this potential, and determine the energies En
Problem 2: Particle in a Box with Non-Zero Energy (2 points) Instead of assuming that a one-dimensional particle has no energy (V(x) =...
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,...
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)...
1. Imagine a version of the particle in a box where the potential is given by:...
1. Imagine a version of the particle in a box where the potential is given by: b-1 b-1 oootherwise where b is any real number greater than or equal to 2 a) Assuming that > Vo for all n, use the WKB approximation to find the energies. Give your final answer in terms of b, b, and E b) What happens in either extreme, as b approaches 2 or o°? Does the WKB approximation give the exact answers in these...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
Particle with a speed bump Consider our old friend the 1D particle in the box, except now with a speed bump in the box so the potential now is given by L and L < x < L 0, Vo 0 x V (x) otherwise (a) Calculate the first order correction to the ground state (n = 1) and first excited state (n = 2) energies (b) Calculate the first order correction to the ground state wave function in terms...
4. (20 points). δ. unction perturbation. Consider a particle in a one-dimensional infinite square well with boundaries at -a and-a. We introduce the following 6-function perturbation at x=0: a. Compute the first-order corrections to the energies of the particle induced by the ν' perturbation b. Recall that you solved this problem exactly in problem set 4 (Griffiths 2.43). Compare your perturbation theory result to the exact solution.
quantum mechanics
Consider a particle confined in two-dimensional box with infinite walls at x 0, L;y 0, L. the doubly degenerate eigenstates are: Ιψη, p (x,y))-2sinnLx sinpry for 0 < x, y < L elsewhere and their eigenenergies are: n + p, n, p where n, p-1,2, 3,.... Calculate the energy of the first excited state up to the first order in perturbation theory due to the addition of: 2 2
Consider a particle confined in two-dimensional box with infinite...
4. (20 points). 5-function perturbation. Consider a particle in a one-dimensional infinite square well with boundaries at x--a and x-a. We introduce the following δ-function perturbation at V'(x) 00(z). a. Compute the first-order corrections to the energies of the particle induced by the perturbation b. Recall that you solved this problem exactly in problem set 4 (Griffiths 2.43). Compare your perturbation theory result to the exact solution
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...
Instead of assuming that a one-dimensional
particle has no energy (v(x)=0), consider the case of a
one-dimensional particle which has finite, but constant, energy
V(x)= V sub zero.. Show that the ID particle in a box wave
functions. n(x)= A sin ((pi n x)/a). Also solve the Schrödinger
equation for this potential, and determine the energies En
Problem 2: Particle in a Box with Non-Zero Energy (2 points) Instead of assuming that a one-dimensional particle has no energy (V(x) =...
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,...
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)...
1. Imagine a version of the particle in a box where the potential is given by: b-1 b-1 oootherwise where b is any real number greater than or equal to 2 a) Assuming that > Vo for all n, use the WKB approximation to find the energies. Give your final answer in terms of b, b, and E b) What happens in either extreme, as b approaches 2 or o°? Does the WKB approximation give the exact answers in these...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
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