# Foundations of Quantum Mechanics | Week 5

Course Name: Foundations of Quantum Mechanics

Course Link: Foundations of Quantum Mechanics

#### These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 1
Time evolution of expectation value

Use the following information for Questions 1-3:

Consider a particle with mass, m, in an infinite potential well with a width L. Here we choose the coordinates such that the center of the well is x=0 and the walls are located at x=±L/2​.
V(x)=0,−L/2​<x<+L​/2
V(x)=∞, elsewhere
Now suppose the particle was initially in a superposition state ϕ=​1/√​(ψ1​+ψ2​) where ψ1​ and ψ2​ are the two lowest energy eigenstates, respectively.
What is the expectation value of energy, ⟨H^⟩ as a function of time?
Express your answer in terms of mass, m, width, L, reduced Planck’s constant, hbar, time, t and a constant pi. Note that your answer does not have to include all of these variables.

Question 2
Time evolution of expectation value

Again the particle was initially in a superposition state ϕ=1/√2​(ψ1​+ψ2​) where ψ1​ and ψ2​ are the two lowest energy eigenstates, respectively.
What is the expectation value of position, ⟨x^⟩ initally, at t=0?
Express your answer in terms of mass, m, width, L, reduced Planck’s constant, hbar and constant pi. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 3
Time evolution of expectation value

Continuing from Question 2, what is the expectation value of position, ⟨x^⟩ as a function of time, t?
Express your answer in terms of the initial position, x_0, (i.e. the position expectation value at t=0), the ground and first excited state energies, E_1 and E_2, time t and the reduced Planck’s constant, hbar. Note that your answer does not have to include all of these variables.

Question 4
Time evolution of expectation value

Use the following information for Questions 4-7:

Consider the 1D simple harmonic oscillator problem.
Suppose the system was initially in a superposition state ϕ=1/√2​(∣0⟩+∣1⟩) where ∣n⟩ is the energy eigenstate of 1D harmonic oscillator with energy En​=(n+1/2​)ℏω.
What is the expectation value of position, ⟨x^⟩ as a function of time?
Express your answer in terms of mass, m, oscillator frequency, omega, reduced Planck’s constant, hbar, time, t and a constant pi. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 5
Time evolution of expectation value

Again, assuming that the system was initially in a superposition state ϕ=1/√2​(∣0⟩+∣1⟩), derive an expression for ⟨x^2⟩.
Express your answer in terms of mass, m, oscillator frequency, omega, reduced Planck’s constant, hbar, time, t and a constant pi. Note that your answer does not have to include all of these variables.

Question 6
Time evolution of expectation value

This time, suppose the harmonic oscillator was initially in a superposition state ϕ=2​1​(∣0⟩+∣2⟩) where ∣n⟩ is the energy eigenstate of 1D harmonic oscillator with energy En=(n+1/2​)ℏω.
What is the expectation value of position, ⟨x^⟩ as a function of time?
Express your answer in terms of mass, m, oscillator frequency, omega, reduced Planck’s constant, hbar, time, t and a constant pi. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 7
Time evolution of expectation value

Again, assuming that the harmonic oscillator was initially in a superposition state ϕ=2​1​(∣0⟩+∣2⟩), derive an expression for the variace of position, ⟨x^2⟩, as a function of time.
Express your answer in terms of mass, m, oscillator frequency, omega, reduced Planck’s constant, hbat, time, t and a constant pi. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 8
Time evolution of quantum state

Use the following information for Questions 8-15:

A^ is a Hermitian operator with two eigenvectors, ∣a⟩ and ∣b⟩, with eigenvalues a and b, respectively (a≠b).
The Hamiltonian of a quantum system can be expressed as
H^=σ(∣a⟩⟨b∣+∣b⟩⟨a∣)
where σ is a real positive number.
What is the bigger of the two eigenvalues of the Hamiltonian operator?
Express your answer in terms of a, b and sigma. Note that your answer does not have to include all of these variables.
Hint: Express the Hamiltonian operator in a matrix form (see Module 3 Video 4 Matrix Representation) and then diagonalize it to obtain eigenvectors.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 9
What is the smaller of the two eigenvalues of the Hamiltonian operator?
Express your answer in terms of a, b and sigma. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 10
We can express the eigenvectors of the Hamiltonian using ∣a⟩ and ∣b⟩ as the basis vectors. That is,
∣p⟩=c∣a⟩+d∣b⟩ where c and d are complex numbers.
What is the value of coefficient c for the eigenvector corresponding to the larger of the two eigenvalues?
Express your answer in terms of a, b, sigma and imaginary unit i. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 11
Continuing the problem of Question 10, what is the value of coefficient d for the eigenvector corresponding to the larger of the two eigenvalues?
Express your answer in terms of a, b, sigma and imaginary unit i. Note that your answer does not have to include all of these variables.

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Question 12
Continuing the problem of Question 10, this time we want to express the eigenvector of Hamiltonian corresponding to the smaller of the two eigenvalues in the basis of {∣a⟩,∣b⟩}. That is,
∣q⟩=f∣a⟩+g∣b⟩ where f and g are complex numbers.
What is the value of coefficient f for the eigenvector corresponding to the smaller of the two eigenvalues?
Express your answer in terms of a, b, sigma and imaginary unit i. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 13
Continuing Question 12, what is the value of coefficient g for the eigenvector corresponding to the smaller of the two eigenvalues?
Express your answer in terms of a, b, sigma and imaginary unit i. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 14
Suppose the system is initially in state ∣a⟩ at t=0. Now we want to express the quantum state at a time t>0 in the basis of {∣a⟩,∣b⟩} . That is,
∣r(t)⟩=k∣a⟩+l∣b⟩ where k and l are complex numbers.
And at t=0, k=1 and l=0.
Write down the expression for k in terms of a, b, sigma, imaginary unit i, reduced Planck’s constant, hbar, and time t. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 15
Continuing Question 14, write down the expression for l in terms of a, b, sigma, imaginary unit i, reduced Planck’s constant, hbar, and time t. Note that your answer does not have to include all of these variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 5 Quiz

Question 16
Particle current

Recall the 1D potential barrier problem whose solution is given as
ψ(x)=Te^(ik0​x) in region III (the region past the barrier)
where T=e^(−ik0​L)*2iμ/(1+μ2)sinkL+2iμcoskL​, k0​=√2mE​/ℏ​, k=√2m(E−V0​)​​/ℏ, and μ=k/​k0​. V0​ and L are the height and width of the potential barrier and E is the energy of the incident particle.
Write down the expression for the particle current in region III.
Express your answer in terms of m, k, k0, mu, T, L, imaginary unit i and reduced Planck’s constant hbar. Note that your answer does not have to include all of these variables.