Approximation Methods | Week 3
Course Name: Approximation Methods
Course Link: Approximation Methods
These are answers of Approximation Methods Coursera Week 3 Quiz
Question 1
Variational solution of 1D harmonic oscillator
Use the following information for Questions 1-2:
Pretend that we don’t know the solution of 1D harmonic oscillator. From the profile of the potential, we expect the ground state wavefunction is likely to have a maximum at x=0 and approaches zero at x=±∞. Based on this observation, we choose (quite luckily!) a gaussian function as the trial wavefunction,
ϕ(x)=Aexp(−x2/w2)
where w is a real, positive number that specifies the width of our gaussian function and A is a complex normalization constant.
Normalize the trial wavefunction and find the constant A.
Enter your answer in terms of mass m, oscillator resonance frequency omega, width parameters w and constant pi and reduced Planck’s constant habr.
Answer: (1/w)^(1/2)*(2/pi)^(1/4)
Question 2
Variational solution of 1D harmonic oscillator
Use the following information for Questions 1-2:
Using the normalized trial wavefunction obtained in Question 1, calculate the energy expectation value,
⟨ϕ∣H∣ϕ⟩ and find the expression for w that minimizes the energy expectation value. Here, H is the Hamiltonian of the 1D harmonic oscillator.
Enter your answer in terms of mass m, oscillator resonance frequency omega, constant pi and reduced Planck’s constant habr.
Answer: sqrt(2)*sqrt((hbar)/(m*omega))
These are answers of Approximation Methods Coursera Week 3 Quiz
Question 3
Coupled delta-function potential well
Use the following information for Questions 3-x:
Consider an infinitely deep and vanishingly narrow potential well, whose potential profile is described by a delta function. Explicitly, the Hamiltonian can be written as
H^=-ℏ2/2m * d2/dx2 V0δ(x)
where V0 is a parameter that specifies the depth of the potential.
This delta-function potential well supports one bound state. Find the energy of the bound state.
Hint: It is convenient to define a parameter κ=√2m|E|/ℏ2. Also, note that the derivative of the wavefunction dψ/dx is not continuous at x=0 because the potential is infinite at that point. The discontinuity of dψ/dx can be obtained by integrating the Schrödinger equation over an infinitesimal interval across x=0.
Enter your answer in terms of mass m, depth parameter V0, constant pi and reduced Planck’s constant habr.
Answer: -m*V0^2/(2*hbar^2)
Question 4
Coupled delta-function potential well
Find the normalized wavefunction in the region x<0.
Enter your answer in terms of mass m, depth parameter V0, parameter kappa, constant pi and reduced Planck’s constant habr.
Answer: sqrt(V0*m/(hbar^2))*exp(kappa*x)
These are answers of Approximation Methods Coursera Week 3 Quiz
Question 5
Coupled delta-function potential well
Find the normalized wavefunction in the region x>0.
Enter your answer in terms of mass m, depth parameter V0, parameter kappa, constant pi and reduced Planck’s constant habr.
Answer: sqrt(V0*m/(hbar^2))*exp(-kappa*x)
These are answers of Approximation Methods Coursera Week 3 Quiz
Question 6
Coupled delta-function potential well
Let us now consider two delta-function potential wells separated by a distance d. The Hamiltonian is given by
H^=-ℏ2/2m * d2/dx2 V0[δ(x+d/2)+δ(x-d/2)]
Use the tight binding method and obtain the two energy eigenvalues and their corresponding eigenfunctions.
What is the larger of the two energy eigenvalues?
Enter your answer in terms of mass m, depth parameter V0, parameter kappa, separation distance d, constant pi and reduced Planck’s constant habr.
Answer:
These are answers of Approximation Methods Coursera Week 3 Quiz
Question 7
Coupled delta-function potential well
Express the wavefunction ϕl corresponding to the larger of the two energy eigenvalues in terms of the unperturbed wavefucntions, ψL and ψR, which are the energy eigenfunctions of left and right well, respectively, i.e.,
ϕl = aψL + bψR
What is the constant a?
Enter your answer in terms of mass m, depth parameter V0, parameter kappa, separation distance d, constant pi and reduced Planck’s constant habr.
Answer: 1/sqrt(2)
These are answers of Approximation Methods Coursera Week 3 Quiz
Question 8
Coupled delta-function potential well
Continuing Question 7, what is the constant b?
Enter your answer in terms of mass m, depth parameter V0, parameter kappa, separation distance d, constant pi and reduced Planck’s constant habr.
Answer: -1/sqrt(2)
These are answers of Approximation Methods Coursera Week 3 Quiz
Question 9
Coupled delta-function potential well
Continuing from Question 6, what is the smaller of the two energy eigenvalues?
Enter your answer in terms of mass m, depth parameter V0, parameter kappa, separation distance d, constant pi and reduced Planck’s constant habr.
Answer:
These are answers of Approximation Methods Coursera Week 3 Quiz
Question 10
Coupled delta-function potential well
Express the wavefunction ϕs corresponding to the smaller of the two energy eigenvalues in terms of the unperturbed wavefucntions, ψL and ψR , which are the energy eigenfunctions of left and right well, respectively, i.e.,
ϕs = cψL + dψR
What is the constant c?
Enter your answer in terms of mass m, depth parameter V0, parameter kappa, separation distance d, constant pi and reduced Planck’s constant habr.
Answer: 1/sqrt(2)
These are answers of Approximation Methods Coursera Week 3 Quiz
Question 11
Coupled delta-function potential well
Continuing Question 10, what is the constant d?
Enter your answer in terms of mass m, depth parameter V0, parameter kappa, separation distance d, constant pi and reduced Planck’s constant habr.
Answer: 1/sqrt(2)
These are answers of Approximation Methods Coursera Week 3 Quiz
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