Approximation Methods | Week 1
Course Name: Approximation Methods
Course Link: Approximation Methods
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 1
Harmonic oscillator under electric field
Use this information for Questions 1-2:
Consider a 1D simple harmonic oscillator under a constant electric field E. The Hamiltonian is given as,
H^=p^2/2m+1/2mω2x2+qEx
The applied electric field is small enough to justify the use of perturbation theory. That is, treat the Hamiltonian without electric field as the unperturbed Hamiltonian and the electric field term as the perturbation.
Obtain the first order correction to the ground state energy.
Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant hbar, constant pi, charge q and applied electric field E. Your answer does not have to include all these parameters.
Answer: 0
Question 2
Harmonic oscillator under electric field
What is the second order correction to the ground state energy?
To simplify the calculation, use only the first non-zero term and ignore all higher order terms in the infinite sum required to evaluate the second order correction.
Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant hbar, constant pi, charge q and applied electric field E. Your answer does not have to include all these parameters.
Answer: -(q^2*E^2)/(2*m*omega^2)
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 3
2D harmonic oscillator
Use the following information for Questions 3-5:
Consider a 2D harmonic oscillator whose Hamiltonian is given as
H^0=p^2x/2m+p^2y/2m+1/2mω2(x2+y2)
What is the energy of the lowest-lying degenerate states?
Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant ℎbar, and constant pi. Your answer does not have to include all these parameters.
Answer: 2*hbar*omega
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 4
2D harmonic oscillator
We now add a perturbation V=amω2xy to the unperturbed Hamiltonian H^0. Here, a is a dimensionless real number, which is small.
Find the first-order correction to the degenerate energy level found in Question 3.
Since the unperturbed energy level is doubly degenerate, it should split into two, if this perturbation lifts the degeneracy. Enter the larger of the two first-order corrections here. If the degeneracy is not lifted, enter 0.
Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant hbar, constant pi and the dimensionless parameter a. Your answer does not have to include all these parameters.
Answer: (a*hbar*omega)/2
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 5
2D harmonic oscillator
Enter the smaller of the two first-order corrections here. If the degeneracy is not lifted, enter 0.
Express your answers in terms of mass m, oscillator resonance frequency omega, reduced Planck’s constant hbar, constant pi and the dimensionless parameter a. Your answer does not have to include all these parameters.
Answer: -(a*hbar*omega)/2
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 6
Two-state system
Use the following information for Questions 6-15:
The Hamiltonian matrix for a two-state system is given by,
H^=(E1(0) Δ Δ E2(0))
where E1(0) and E2(0) are the energies of the unperturbed (Δ=0) states and E1(0)<E2(0).
This problem can be solved exactly by diagonalizing the Hamiltonian matrix to obtain the eigenvalues. What is the smaller of the two energy eigenvalues?
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: (E1+E2)/2-sqrt((E2-E1)^2+4*Delta^2)/2
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 7
Two-state system
What is the larger of the two energy eigenvalues?
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: (E1+E2)/2+sqrt((E2-E1)^2+4*Delta^2)/2
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 8
Two-state system
If Δ is small compared to E2(0)−E1(0), we can use the Taylor expansion of the exact solutions obtained in Questions 6-7 and retain only the lowest order term in Δ. Enter the expression for the smaller of the two energies in this approximation.
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: E1-(Delta^2/(E2-E1))
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 9
Two-state system
Enter the expression for the larger of the two energies in the same approximation used in Question 8.
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: E2+(Delta^2/(E2-E1))
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 10
Two-state system
This time, we will solve the same problem by using the non-degenerate perturbation theory. For this, we express the Hamiltonian as below.
H^=H^0+V
H^0=(E1(0) Δ Δ E2(0)) and V=(0 Δ Δ 0)
Using the non-degenerate perturbation theory, calculate the first order correction for the ground state energy E1(0).
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: 0
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 11
Two-state system
Calculate the first order correction for the excited state energy E2(0).
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: 0
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 12
Two-state system
Calculate the second order correction for the ground state energy E1(0).
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: -(Delta^2)/(E2-E1)
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 13
Two-state system
Calculate the second order correction for the excited state energy E2(0).
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: (Delta^2)/(E2-E1)
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 14
Two-state system
Now let us assume the system is degenerate, E1(0)=E2(0)=E(0). The Hamiltonian is then written as
H^=H^0+V
where H^0=(E1(0) Δ Δ E2(0)) and V=(0 Δ Δ 0)
Using the degenerate perturbation theory, calculate the first order correction for the unperturbed energy E(0).
Since the system is doubly degenerate, it should split into two, if the perturbation lifts the degeneracy. Enter the larger of the two first-order corrections here. If the degeneracy is not lifted, enter 0.
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: Delta
These are answers of Approximation Methods Coursera Week 1 Quiz
Question 15
Two-state system
Enter the smaller of the two first-order corrections here. If the degeneracy is not lifted, enter 0.
Express your answers in terms of the unperturbed energies E1 and E2 (for notational simplicity, you can drop the superscripts for these parameters) and the perturbation Delta, Your answer does not have to include all these parameters.
Answer: -Delta
These are answers of Approximation Methods Coursera Week 1 Quiz
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