# Foundations of Quantum Mechanics | Week 3

**Course Name: Foundations of Quantum Mechanics**

**Course Link: Foundations of Quantum Mechanics**

#### These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz

**Question 1Measurement**

**Suppose an electron is in a spin state that can be described by∣ ϕ⟩=√3/2∣+⟩+1/2∣−⟩where + and – are eigenstates of Sz with eigenvalue +ℏ/2 and -ℏ/2If we measure z-component of spin of this electron, what is the probability of measuring spin up, +ℏ/2?Answers within 5% error will be considered correct.**

Answer: 0.75

**Question 2Measurement**

Use the following information for Questions 2-3:

**Suppose there are two quantum mechanical observables c and d represented by operators C^ and D^, respectively. Both operators have two eigenstates, ϕ _{1} and ϕ_{2} for C^ and ψ_{1} and ψ_{2} for D^. Furthermore, the two sets of eigenstates are related to each other as below.**

ϕ_{1}=1/13(5ψ_{1}+ψ_{2})

ϕ_{2}=1/13(12ψ_{1}−5ψ_{2})

The system was found to be in state ϕ_{1} initially.

If we measure D^, what is the probability of finding the system in ψ_{2}?

Answers within 5% error will be considered correct.

Answer: 0.852

**These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz**

**Question 3Measurement**

**Once again, the system is in state ϕ _{1} initially. This time, we perform two successive measurements in which we first measure D^ and then C^ again. What is the probability of finding the system is still in ϕ_{1}?**

Answers within 5% error will be considered correct.

Answer: 0.749

**Question 4Expectation value and measurement**

Use the following information for Questions 4-7:

**Consider an infinite potential well with a width L=10nm is located in the region 0<z<L. Suppose an electron in that infinite potential well is described by wavefunctionΦ(z)=Az^2(L−z) for 0<z<LNormalize the wavefunction and determine the constant A.Give your answer in the standard SI unit. Answers within 5% error will be considered correct.**

Answer: 105000000000000000000000000000

**These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz**

**Question 5Expectation value and measurement**

**Calculate the expectation value of energy. Give your answer in the standard SI unit. Answers within 5% error will be considered correct.**

Answer: 0.00000000000000000000088

**Question 6Expectation value and measurement**

**If we measure the energy of this electron, what is the probability of measuring the ground state energy, E _{1}=π^2ℏ^2/2mL^2?Answers within 5% error will be considered correct.**

Answer: 0.8765

**These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz**

**Question 7Expectation value and measurement**

**If we measure the energy of this electron, what is the probability of measuring the first excited state energy, E _{2}=4π^2ℏ^2/2mL^2?Answers within 5% error will be considered correct.**

Answer: 0.1225

Question 8**Vector space and matrix representation**

Use the following information for Questions 8-16:

**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.Consider a function ψ _{1}(x)=c where c is a complex number. Normalize this function and determine cAnswers within 5% error will be considered correct.**

Answer: 0.707

**These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz**

**Question 9Vector space and matrix representation**

**Consider another function ψ _{2}(x)=dx where d is a complex number. Normalize this function and determine the constant d.**

Answers within 5% error will be considered correct.

Answer: 1.224

**Question 10Vector space and matrix representation**

**Calculate the inner product of the two functions, ψ _{1}(x) and ψ_{2}(x).**

Answers within 5% error will be considered correct.

Answer: 0

**These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz**

**Question 11Vector space and matrix representation**

**Let us now construct the vector (matrix representation) for a function g(x)=−2x+1 using ψ _{1}(x) and ψ_{2}(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: 1.412

**Question 12Vector space and matrix representation**

**In the vector given in Question 11, what is the value of the second vector element q?Answers within 5% error will be considered correct.**

Answer: -1.633

**These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz**

**Question 13Vector space and matrix representation**

**Now consider a reflection operator R^ which transforms function f(x) to f(−x), i.e., R^f(x) to f(−x).Find the matrix representation of R^ in the {ψ1,ψ2} 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: 1

**Question 14Vector space and matrix representation**

**In the matrix R^ in Question 13, what is the value of q?Answers within 5% error will be considered correct.**

Answer: 0

**These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz**

**Question 15Vector space and matrix representation**

**In the matrix R^ in Question 13, what is the value of r?Answers within 5% error will be considered correct.**

Answer: 0

**These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz**

**Question 16Vector space and matrix representation**

**In the matrix R^ in Question 13, what is the value of s?Answers within 5% error will be considered correct.**

Answer: -1

**These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz**

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