# 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 1Expectation 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 ψ _{2}. 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.**

Answer: 2*pi^2*(hbar^2)/(m*L^2)

**Question 2Expectation value of energy**

**Now suppose the particle was initially in a superposition state ϕ=1/√2(ψ _{1}+ψ_{2}) where ψ_{1} and ψ_{2} 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.

Answer: 5*pi^2*(hbar^2)/(4*m*L^2)

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

**Question 3Expectation 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 E _{n}=(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.**

Answer: 3*hbar*omega/2

**Question 4Expectation 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.**

Answer: 0

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

**Question 5Expectation 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.**

Answer: 3*hbar/(2*m*omega)

**Question 6Expectation 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.**

Answer: hbar*omega

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

**Question 7Expectation 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.**

Answer: sqrt(hbar/(2*m*omega))

**Question 8Expectation value of superposition state**

**The harmonic oscillator is in a superposition state ϕ=21(∣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.**

Answer: hbar/(m*omega)

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

**Question 9Change 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 ψ _{1}=1/√2 and ψ_{2}=√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.**

Answer: 0

**Question 10Change 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.**

Answer: -0.155

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

**Question 11Change 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.**

Answer: -2.155

**Question 12Change 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 writeR^=(p′ r′ q′ s′)What is the value of ′p′?Answers within 5% error will be considered correct.**

Answer: 0

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

**Question 13Change of basis**

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

Answer: -1

**Question 14Change of basis**

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

Answer: -1

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

**Question 15Change of basis**

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

Answer: 0

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

**Question 16Change of basis**

**Now we relate the two matrix representations using two different basis sets {ψ _{1},ψ_{2}} and {ϕ1,ϕ2}.**

Again the basis functions are defined as

ψ_{1}=1/√2 and ψ_{2}=√3x/2

ϕ1=√3x/2+1/2 and ϕ2=√3x/2−1/2

Now we construct a unitary transformation matrix that transforms from the {ϕ1,ϕ2} basis to {ψ_{1},ψ_{2}} basis. This is, we want to write

(ψ_{1} ψ_{2})=(u_{11} u_{21} u_{12} u_{22})(ϕ1ϕ2)

What is the value of u_{11}?

Answers within 5% error will be considered correct.

Answer: 0.707106781

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

**Question 17Change of basis**

**In the matrix in Question 16, what is the value of u _{12}?Answers within 5% error will be considered correct.**

Answer: -0.707106781

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

**Question 18Change of basis**

**In the matrix in Question 16, what is the value of u _{21}?Answers within 5% error will be considered correct.**

Answer: 0.5

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

**Question 19Change of basis**

**In the matrix in Question 16, what is the value of u _{22}?Answers within 5% error will be considered correct.**

Answer: 0.5

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

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