# Foundations of Quantum Mechanics | Week 4

Course Name: Foundations of Quantum Mechanics

Course Link: Foundations of Quantum Mechanics

#### These are answers of Foundations of Quantum Mechanics Coursera Week 4 Quiz

Question 1
Expectation value of energy

Use the following information for Questions 1-2:

Consider a particle with mass, m, in an infinite potential well with a width L.
The particle was initially in the first excited state ψ2​. What is the expectation value of energy, ⟨H^⟩?
Express your answer in terms of mass, m, width, L, reduced Planck’s constant, hbar and a constant pi. Note that your answer does not have to include all of these variables.

Question 2
Expectation value of energy

Now suppose 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 energy, ⟨H^⟩?
Express your answer in terms of mass, m, width, L, reduced Planck’s constant, hbar 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 4 Quiz

Question 3
Expectation value of energy eigenstate

Use the following information for Questions 3-8:

Consider the simple harmonic oscillator problem with mass, m, and resonance frequency, ω. The energy eigenvalues are En​=(n+1/2​)ℏω where n=0,1,⋯ and the eigenfunctions are gaussian functions modulated by Hermite polynomials, as discussed in Module 2.
The oscillator is in the first excited state ∣1⟩. What is the expectation value of energy, ⟨H^⟩?
Express your answer in terms of mass, m, resonance frequency, omega, reduced Planck’s constant, hbar and constant pi. Note that your answer does not have to include all of these variables.

Question 4
Expectation value of energy eigenstate

The oscillator is in the first excited state ∣1⟩∣1⟩. What is the expectation value of position, ⟨x^⟩?
Express your answer in terms of mass, m, resonance frequency, omega, 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 4 Quiz

Question 5
Expectation value of energy eigenstate

The oscillator is in the first excited state ∣1⟩∣1⟩. What is the variance of position, ⟨x^2⟩, i.e. expectation value of x^2?
Express your answer in terms of mass, m, resonance frequency, omega, reduced Planck’s constant, hbar and constant pi. Note that your answer does not have to include all of these variables.

Question 6
Expectation value of superposition state

Now suppose the harmonic oscillator is in a superposition state ϕ=1/√2​(∣0⟩+∣1⟩) where ∣0⟩ and ∣1⟩ are the two lowest energy eigenstates, respectively.
What is the expectation value of energy, ⟨H^⟩?
Express your answer in terms of mass, m, resonance frequency, omega, 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 4 Quiz

Question 7
Expectation value of superposition state

The harmonic oscillator is in a superposition state ϕ=1/√2​(∣0⟩+∣1⟩) where ∣0⟩ and ∣1⟩ are the two lowest energy eigenstates, respectively.
What is the expectation value of position, ⟨x^⟩?
Express your answer in terms of mass, m, resonance frequency, omega, reduced Planck’s constant, hbar and constant pi. Note that your answer does not have to include all of these variables.
In order to avoid confusion with autograder, use sqrt() for square root instead of power of 1/21/2. Also, use a single sqrt() function including all relevant variables rather than multiple sqrt() functions multiplied together.

Question 8
Expectation value of superposition state

The harmonic oscillator is in a superposition state ϕ=2​1​(∣0⟩+∣1⟩) where ∣0⟩ and ∣1⟩ are the two lowest energy eigenstates, respectively.
What is the variance of position, ⟨x^2⟩, i.e. expectation value of x^2?
Express your answer in terms of mass, m, resonance frequency, omega, 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 4 Quiz

Question 9
Change of basis

Use the following information for Questions 9-19:

Consider the vector space composed of all linear functions, f(x)=ax+b, defined within the region, −1<x<1. The constants a and b are complex numbers.
Previously, we used ψ1=1/√2​ and ψ2​=√3x/2 as basis set to construct matrix representations for all linear functions defined above (See Questions 8-16 in Module 3 for reference).
This time, we will use ϕ1​=√3x/2+1/2 and ϕ2​=√3x/2-1/2 as a new basis set.
Calculate the inner product of the two functions, ϕ1​(x) and ϕ2​(x).
Answers within 5% error will be considered correct.

Question 10
Change of basis

Let us now construct the vector (matrix representation) for a function g(x)=−2x+1 using ϕ1​(x) and ϕ2​(x) as the basis set. That is, we want to express the function g(x) as,
∣g⟩=(p′ q′​)
What is the value of ′p′?
Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 4 Quiz

Question 11
Change of basis

In the vector given in Question 10, what is the value of the second vector element ′q′?
Answers within 5% error will be considered correct.

Question 12
Change of basis

Now consider a reflection operator R^ which transforms function f(x) to f(−x), i.e., R^f(x)=f(−x).
Find the matrix representation of R^ in the {ϕ1​,ϕ2​} basis. That is, we want to write
R^=(p′ r′ ​q′ s′​)
What is the value of ′p′?
Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 4 Quiz

Question 13
Change of basis

In the matrix R^ in Question 12, what is the value of ′q′?
Answers within 5% error will be considered correct.

Question 14
Change of basis

In the matrix R^ in Question 12, what is the value of ′r′?
Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 4 Quiz

Question 15
Change of basis

In the matrix R^ in Question 12, what is the value of ′s′?
Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 4 Quiz

Question 16
Change of basis

Now we relate the two matrix representations using two different basis sets {ψ1​,ψ2​} and {ϕ1​,ϕ2​}.
Again the basis functions are defined as
ψ1​=1/√​2 and ψ2=√3​​x/2
ϕ1​=√3x/2+1/2​ and ϕ2​=√3​x/2−1/2
Now we construct a unitary transformation matrix that transforms from the {ϕ1​,ϕ2​} basis to {ψ1​,ψ2​} basis. This is, we want to write
1​ ψ2​​)=(u11 ​u21​ ​u12​ u22​​)(ϕ1​ϕ2​​)
What is the value of u11​?
Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 4 Quiz

Question 17
Change of basis

In the matrix in Question 16, what is the value of u12​​?
Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 4 Quiz

Question 18
Change of basis

In the matrix in Question 16, what is the value of u21​​?
Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 4 Quiz

Question 19
Change of basis

In the matrix in Question 16, what is the value of u22?
Answers within 5% error will be considered correct.