Approximation Methods | Week 2

Course Name: Approximation Methods

Course Link: Approximation Methods

These are answers of Approximation Methods Coursera Week 2 Quiz

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Question 1
1D harmonic oscillator under time-varying force
Use the following information for Questions 1-3:
A 1D hormonic oscillator with mass m and resonance frequency ω is subject to a pulse of force, f₀, which is constant in space and has a Gaussian time dependence. The potential can be written as
V(t)=-f₀xexp(−t²/τ²)
where τ is the time constant that defines the width of the pulse.
The oscillator was in the ground state, ∣n=0⟩ initially (at t=−∞). Use the time-dependent perturbation theory to the lowest non-vanishing order to find the probability of finding the system in the first excited state ∣n=1⟩ sufficiently long time after the pulse (i.e.,t=∞).
Enter your answer using mass m, oscillator frequency omega, force f₀, pulse width tau, constant pi and reduced Planck’s constant hbar. Your answer does not have to include all these parameters.
Hint: You may use ⟨m∣x∣n⟩= √ℏ/2mω(√n+1*δ_m,n+1+√n_m,n−1)

Answer: (pi*f0^2*tau^2)/(2*hbar*m*omega)*(exp(-(tau^2*omega^2)/(2)))

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These are answers of Approximation Methods Coursera Week 2 Quiz

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Question 2
1D harmonic oscillator under time-varying force
Find the probability of finding the system in the second excited state ∣n=2⟩ sufficiently long time after the pulse (i.e., t=∞).
Enter your answer using mass m, oscillator frequency omega, force f₀, pulse width tau, constant pi and reduced Planck’s constant hbar. Your answer does not have to include all these parameters.

Answer: 0

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These are answers of Approximation Methods Coursera Week 2 Quiz

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Question 3
1D harmonic oscillator under time-varying force
Find the expectation value of position x^ sufficiently long time after the pulse (i.e., t=∞).
For simplicity, you can use the limiting values for the perturbation coefficients, (t→∞) so that the time dependence arises only from the phase evolution of the unperturbed eigenstates, ∣n⟩.
Enter your answer using mass m, oscillator frequency omega, force f₀, pulse width tau, constant pi and reduced Planck’s constant hbar. Your answer does not have to include all these parameters.

Answer:

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These are answers of Approximation Methods Coursera Week 2 Quiz

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Question 4
1D infinite potential well under time-varying force
Use the following information for Questions 4-6:
A particle with mass m is in an infinite potential well with width L. The particle is subject to a force, f₀ which is uniform in space and is applied suddenly at t=0 and then exponentially decays in time. The time-dependent potential can be written as
V(t)=f₀xe^−t/τ for t>0
where τ is the time constant that defines the decay of the force.
The particle was in the ground state, ∣ψ₁⟩ initially (at t=0). Use the time-dependent perturbation theory to the lowest non-vanishing order to find the probability of finding the system in the first excited state ∣ψ₂⟩ sufficiently long time after the onset of the force (i.e. in the limit of t→∞).
Enter your answer using mass m, well width L, force f₀, pulse width tau, constant pi and reduced Planck’s constant hbar. You may also use omega21 which is defined as ω₂₁=(E₂−E₁)/ℏ to simplify your expression. Your answer does not have to include all these parameters.

Answer: ((16*L/(9*pi^2))^2/(hbar^2)*f0^2)/(tau^(-2)+omega21^2)

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These are answers of Approximation Methods Coursera Week 2 Quiz

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Question 5
1D infinite potential well under time-varying force
Find the probability of finding the system in the second excited state ∣ψ₃⟩ sufficiently long time after the pulse (i.e., t=∞).
Enter your answer using mass m, well width L, force f₀, pulse width tau, constant pi and reduced Planck’s constant hbar. You may also use omega21 which is defined as ω₂₁=(E₂−E₁)/ℏ to simplify your expression. Your answer does not have to include all these parameters.

Answer: 0

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These are answers of Approximation Methods Coursera Week 2 Quiz

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Question 6
1D infinite potential well under time-varying force
Find the expectation value of position x^ sufficiently long time after the pulse (i.e., t=∞). Keep only the terms that are first order in perturbation.
Enter your answer using mass m, well width L, force f0, pulse width tau, constant pi, reduced Planck’s constant hbar and time t. You may also use omega21 which is defined as ω₂₁=(E₂−E₁)/ℏ to simplify your expression. Your answer does not have to include all these parameters.

Answer:

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These are answers of Approximation Methods Coursera Week 2 Quiz

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