Please solve with the
explanations of notations
Homework Answers
please note, first i calculate
energy and then corresdponding wave function.
Add Answer to:
1. The two dimensional Harmonic Oscillator has the Hamiltonian n, n'>denotes the state In> of the...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D:...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
3. Consider a system whose Hamiltonian H, admits two eigenstates y, (with eigenvalues F) and v,...
3. Consider a system whose Hamiltonian H, admits two eigenstates y, (with eigenvalues F) and v, (with eigenvalues E,). Assume E, E, and they are () orthogonal, (ifi) normalized and (ii) non-degenerate. After the perturbation is on the diagonal matrix elements become zero ie, <4, l H'l Ψ)-(4, I H'ly,)-0, while the off diagonal equals to a constant value ie. (v, l H'l%)-(wil H'ly)-c Using the 2nd order perturbation theory evaluate the energy of the perturbed system.
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through...
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through the lens of perturbation theory. The Hamiltonian of the oscillator is given by * 2m + mw?f? + cî, and, as solved for previously, it has eigenenergies of En = hwan + mwra and eigenstates of (0) = N,,,a1 + role of (rc)*/2, where Do = 42 and a=(mw/h) (a) By treating the term cî as a perturbation, show that the first-order correction to...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
5.8 Solve for the eigenfunctions and eigenvalues of the Hamiltonian of the two-dimensional isotropic harmonic oscillator in polar coordinates. Bibliography 5.8 Solve for the eigenfunctions an...
5.8 Solve for the eigenfunctions and eigenvalues of the Hamiltonian of the two-dimensional isotropic harmonic oscillator in polar coordinates. Bibliography
5.8 Solve for the eigenfunctions and eigenvalues of the Hamiltonian of the two-dimensional isotropic harmonic oscillator in polar coordinates. Bibliography
10. A harmonic oscillator with the Hamiltonian H t 2m dr? mooʻr is now subject to...
10. A harmonic oscillator with the Hamiltonian H t 2m dr? mooʻr is now subject to a 2 weak perturbation: H-ix. You are asked to solve the ground state of the new Hamiltonian - À + in two ways. (a) Solve by using the time-independent perturbation theory. Find the lowest non- vanishing order correction to the energy of the ground state. And find the lowest non vanishing order correction to the wavefunction of the ground state. (b) Find the wavefunction...
2. (20 pts) Degenerate Perturbation Theory. A system with Hamiltonian H has two degenerate eigenstates l ψ )and lp : Ea...
2. (20 pts) Degenerate Perturbation Theory. A system with Hamiltonian H has two degenerate eigenstates l ψ )and lp : Ea h petturbationHi-h E, :}lifts the degeneracy. The matrix given is in the basis Ambatas nlist 0 Eindthe "good" states, the two eigenstates Ιψ%)-α ws> +Pr IOS)ofHL and the sorresponding eigenvalues AEF which resolve the degeneracy
2. (20 pts) Degenerate Perturbation Theory. A system with Hamiltonian H has two degenerate eigenstates l ψ )and lp : Ea h petturbationHi-h E,...
QM HAmiltonian, perturbation 2A where the first term is the dot product of two vectors of...
QM HAmiltonian, perturbation
2A where the first term is the dot product of two vectors of operators S1 and 52, W where the first term is the dot product of two vectors of operators Š and S2, which would book. The relative sign in the second term is due to the opposite electric charges of the par a) ) Find the energy eigenvalues and eigenstates of this system. V There are two states with the same energy. Suppose you perturbed...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a pert...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a perturbation Hi(t)--eEXe t2 (e, E and rare constants) be applied between - and too. What is the probability that the oscillator will be in the state n) (of the unperturbed oscillator) as t-> oo?(15%) (b) The bottom of an infinite well is changed to have the shape V(x)-ε sin® for 0Sxa. Calculate the energy shifts for all the excited states...
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and =...
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥg = îl + Ħ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... a)...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
3. Consider a system whose Hamiltonian H, admits two eigenstates y, (with eigenvalues F) and v, (with eigenvalues E,). Assume E, E, and they are () orthogonal, (ifi) normalized and (ii) non-degenerate. After the perturbation is on the diagonal matrix elements become zero ie, <4, l H'l Ψ)-(4, I H'ly,)-0, while the off diagonal equals to a constant value ie. (v, l H'l%)-(wil H'ly)-c Using the 2nd order perturbation theory evaluate the energy of the perturbed system.
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through the lens of perturbation theory. The Hamiltonian of the oscillator is given by * 2m + mw?f? + cî, and, as solved for previously, it has eigenenergies of En = hwan + mwra and eigenstates of (0) = N,,,a1 + role of (rc)*/2, where Do = 42 and a=(mw/h) (a) By treating the term cî as a perturbation, show that the first-order correction to...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
5.8 Solve for the eigenfunctions and eigenvalues of the Hamiltonian of the two-dimensional isotropic harmonic oscillator in polar coordinates. Bibliography
5.8 Solve for the eigenfunctions and eigenvalues of the Hamiltonian of the two-dimensional isotropic harmonic oscillator in polar coordinates. Bibliography
10. A harmonic oscillator with the Hamiltonian H t 2m dr? mooʻr is now subject to a 2 weak perturbation: H-ix. You are asked to solve the ground state of the new Hamiltonian - À + in two ways. (a) Solve by using the time-independent perturbation theory. Find the lowest non- vanishing order correction to the energy of the ground state. And find the lowest non vanishing order correction to the wavefunction of the ground state. (b) Find the wavefunction...
2. (20 pts) Degenerate Perturbation Theory. A system with Hamiltonian H has two degenerate eigenstates l ψ )and lp : Ea h petturbationHi-h E, :}lifts the degeneracy. The matrix given is in the basis Ambatas nlist 0 Eindthe "good" states, the two eigenstates Ιψ%)-α ws> +Pr IOS)ofHL and the sorresponding eigenvalues AEF which resolve the degeneracy
2. (20 pts) Degenerate Perturbation Theory. A system with Hamiltonian H has two degenerate eigenstates l ψ )and lp : Ea h petturbationHi-h E,...
QM HAmiltonian, perturbation
2A where the first term is the dot product of two vectors of operators S1 and 52, W where the first term is the dot product of two vectors of operators Š and S2, which would book. The relative sign in the second term is due to the opposite electric charges of the par a) ) Find the energy eigenvalues and eigenstates of this system. V There are two states with the same energy. Suppose you perturbed...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a perturbation Hi(t)--eEXe t2 (e, E and rare constants) be applied between - and too. What is the probability that the oscillator will be in the state n) (of the unperturbed oscillator) as t-> oo?(15%) (b) The bottom of an infinite well is changed to have the shape V(x)-ε sin® for 0Sxa. Calculate the energy shifts for all the excited states...
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥg = îl + Ħ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... a)...
Most questions answered within 3 hours.
-
Water is poured into a container that has a leak. The mass m of
the water...
asked 1 minute ago -
Shakespeare - whether you like his writing or not - is well
known for his writing...
asked 10 minutes ago -
A certain genetic characteristic of a particular plant can
appear in one of three forms (phenotypes)....
asked 26 minutes ago -
summarize Robert s. McNamara and the evolution of modern
management and the key point.
asked 25 minutes ago -
how many double bond equivalents are contained in a compound
with a molecular formula of C6H2F3I
asked 31 minutes ago -
A sample of pond sediment was diluted by placing a 0.1 g into
9.9 ml of...
asked 32 minutes ago -
A student performed an experiment using the same setup as the
one used in the lab...
asked 46 minutes ago -
predict the major product of nitration of bromobenzene, followed by
SnCl2 reduction
asked 55 minutes ago -
A standing wave with five antinodes is excited in a string of
length 2.5 m where...
asked 49 minutes ago -
An apnea monitor for adults consists of a flexible coil that
wraps around the chest, When...
asked 49 minutes ago -
Suppose the method of tree ring dating gave the following dates
A.D. for an archaeological excavation...
asked 1 hour ago -
Network security
1. explain succinctly how a Denial of Service attack may occur
on an implementation...
asked 1 hour ago