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.
Answer: 0.75
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
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.
Answer: 0.749
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
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.
Answer: 0.00000000000000000000088
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
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.
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 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
These are answers of Foundations of Quantum Mechanics Coursera Week 3 Quiz
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
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
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.
Answer: 1.412
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
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.
Answer: 1
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.
