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

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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.

Answer: 0.75

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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.

Answer: 0.852

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

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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.

Answer: 0.749

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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.
Give your answer in the standard SI unit. Answers within 5% error will be considered correct.

Answer: 105000000000000000000000000000

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

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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.

Answer: 0.00000000000000000000088

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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.

Answer: 0.8765

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

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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.

Answer: 0.1225

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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.

Answer: 0.707

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

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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.

Answer: 1.224

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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.

Answer: 0

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

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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.

Answer: 1.412

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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.

Answer: -1.633

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

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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.

Answer: 1

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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.

Answer: 0

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

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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.

Answer: 0

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

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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.

Answer: -1

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

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