# 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 1
Measurement

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/2
If 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.

Question 2
Measurement

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, ϕ1​ and ϕ2​ for C^ and ψ1​ and ψ2​ for D^. Furthermore, the two sets of eigenstates are related to each other as below.
ϕ1=1/13​(5ψ1​+ψ2​)
ϕ2=1/13​(12ψ1​−5ψ2​)
The system was found to be in state ϕ1​ initially.
If we measure D^, what is the probability of finding the system in ψ2​?
Answers within 5% error will be considered correct.

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

Question 3
Measurement

Once again, the system is in state ϕ1​ 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 ϕ1?
Answers within 5% error will be considered correct.

Question 4
Expectation 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 wavefunction
Φ(z)=Az^2(L−z) for 0<z<L
Normalize the wavefunction and determine the constant A.

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

Question 5
Expectation 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.

Question 6
Expectation value and measurement

If we measure the energy of this electron, what is the probability of measuring the ground state energy, E1​=π^2ℏ^2/2mL^2​?
Answers within 5% error will be considered correct.

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

Question 7
Expectation value and measurement

If we measure the energy of this electron, what is the probability of measuring the first excited state energy, E2​=4π^2ℏ^2/2mL^2​?
Answers within 5% error will be considered correct.

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 ψ1​(x)=c where c is a complex number. Normalize this function and determine c
Answers within 5% error will be considered correct.

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

Question 9
Vector space and matrix representation

Consider another function ψ2​(x)=dx where d is a complex number. Normalize this function and determine the constant d.
Answers within 5% error will be considered correct.

Question 10
Vector space and matrix representation

Calculate the inner product of the two functions, ψ1​(x) and ψ2​(x).
Answers within 5% error will be considered correct.

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

Question 11
Vector space and matrix representation

Let us now construct the vector (matrix representation) for a function g(x)=−2x+1 using ψ1​(x) and ψ2​(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.

Question 12
Vector 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.

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

Question 13
Vector 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 write
R^=(p r ​q s​)
What is the value of p?
Answers within 5% error will be considered correct.

Question 14
Vector 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.

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

Question 15
Vector 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.

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

Question 16
Vector 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.