# Approximation Methods | Week 3

**Course Name: Approximation Methods**

**Course Link: Approximation Methods**

#### These are answers of Approximation Methods Coursera Week 3 Quiz

**Question 1Variational solution of 1D harmonic oscillatorUse 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(−x**

^{2}/w

^{2})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 2Variational solution of 1D harmonic oscillatorUse 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 3Coupled delta-function potential wellUse 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 asH^=-ℏ**

^{2}/2m * d

^{2}/dx

^{2}V

_{0}δ(x)where V

_{0}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|/ℏ

^{2}. 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 V

_{0}, constant pi and reduced Planck’s constant habr.

Answer: -m*V0^2/(2*hbar^2)

**Question 4Coupled delta-function potential wellFind the normalized wavefunction in the region x<0.Enter your answer in terms of mass m, depth parameter V**

_{0}, 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 5Coupled delta-function potential wellFind the normalized wavefunction in the region x>0.Enter your answer in terms of mass m, depth parameter V**

_{0}, 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 6Coupled delta-function potential wellLet us now consider two delta-function potential wells separated by a distance d. The Hamiltonian is given byH^=-ℏ**

^{2}/2m * d

^{2}/dx

^{2}V

_{0}[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 V

_{0}, 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 7Coupled delta-function potential wellExpress the wavefunction ϕ**

_{l} corresponding to the larger of the two energy eigenvalues in terms of the unperturbed wavefucntions, ψ

_{L}and ψ

_{R}, which are the energy eigenfunctions of left and right well, respectively, i.e.,ϕ

_{l}= aψ

_{L}+ bψ

_{R}What is the constant a?Enter your answer in terms of mass m, depth parameter V

_{0}, 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 8Coupled delta-function potential wellContinuing Question 7, what is the constant b?Enter your answer in terms of mass m, depth parameter V**

_{0}, 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 9Coupled delta-function potential wellContinuing from Question 6, what is the smaller of the two energy eigenvalues?Enter your answer in terms of mass m, depth parameter V**

_{0}, 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 10Coupled delta-function potential wellExpress the wavefunction ϕ**

_{s}corresponding to the smaller of the two energy eigenvalues in terms of the unperturbed wavefucntions, ψ

_{L}and ψ

_{R}, which are the energy eigenfunctions of left and right well, respectively, i.e.,ϕ

_{s}= cψ

_{L}+ dψ

_{R}What is the constant c?Enter your answer in terms of mass m, depth parameter V

_{0}, 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 11Coupled delta-function potential wellContinuing Question 10, what is the constant d?Enter your answer in terms of mass m, depth parameter V**

_{0}, 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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