Foundations of Quantum Mechanics | Week 6

Course Name: Foundations of Quantum Mechanics

Course Link: Foundations of Quantum Mechanics

These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 1
Pure and mixed spin states
Use the following information for Questions 1-7:

Consider an electron in a pure spin state composed of an equal superposition of ∣sx+⟩ and ∣sy+⟩. That is, the spin state can be expressed as
∣χ⟩=a(∣sx+⟩+∣sy+⟩)
where ∣sx+⟩ and ∣sy+⟩ are the eigenstates of Sx​^​ and Sy​^​ operators with eigenvalues +ℏ/2, respectively.
First, normalize the vector and determine the constant a.
Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant, hbar. Note that your answer does not have to include all of these variables.

Answer: 1/sqrt(3)

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Question 2
Pure and mixed spin states

For the state defined in Question 1, find the expectation value for the x-component of spin, ⟨Sx​^​⟩.
This represents the average value you will obtain when measuring x-component of spin many times.
Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant, hbar. Note that your answer does not have to include all of these variables.

Answer: hbar/3

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 3
Pure and mixed spin states

For the state defined in Question 1, find the expectation value for the y-component of spin, ⟨Sy​^​⟩.
Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant, hbar. Note that your answer does not have to include all of these variables.

Answer: hbar/3

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 4
Pure and mixed spin states

For the state defined in Question 1, find the expectation value for the z-component of spin, ⟨Sz​^​⟩.
Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant, hbar. Note that your answer does not have to include all of these variables.

Answer: hbar/6

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 5
Pure and mixed spin states

This time, let us consider a maxed spin state in which the electrons have equal probabilities of being in the pure states ∣sx+⟩ and ∣sy+⟩.
Find the ensemble average [Sx​^​] for the x-component of spin.
This represents the average value you will obtain when measuring x-component of spin many times.
Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant, hbar. Note that your answer does not have to include all of these variables.

Answer: hbar/4

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 6
Pure and mixed spin states

For the mixed state defined in Question 5, find the ensemble average [Sy​^​] for the y-component of spin.
Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant, hbar. Note that your answer does not have to include all of these variables.

Answer: hbar/4

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 7
Pure and mixed spin states

For the mixed state defined in Question 5, find the ensemble average [Sz​^​] for the z-component of spin.
Express your answer in terms of imaginary unit i, constant pi and reduced Planck’s constant, hbar. Note that your answer does not have to include all of these variables.

Answer: 0

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 8
Two-electron system

Consider two electrons in the same spin state, interacting through a short range potential given by
V(x1​,x2​)=V(∣x1​−x2​∣)={−V0​,∣x1​−x2​∣<a 0,elsewhere​
where V0​ is a real, positive number and a is also a real, positive number that specifies a range within which the two electrons interact with each other.
You can solve this problem by writing down the full Schrödinger equation,
(− ℏ²/2m₁ * ​​∂²/∂x²₁​​ − ℏ²/2m₂ * ​​∂²/∂x²₂+V(x1​,x2​))Ψ(x1​,x2​)=EΨ(x1​,x2​)
and separate it into two uncoupled equations – one for the center of mass coordinate, X=m1x1+m2x2/m1+m2​​ and the other for the relative coordinate, x=x1​−x2​. Let us assume the center of mass momentum is zero and ignore the center of mass equation.

Calculate the lowest possible energy of this two-electron system for the case where V₀​=10eV and a=1nm.
Give your answer in unit of eV. Answers within 5% errors will be considered correct.
Hint: Note that the energy would have a negative value between −10eV and 0eV.

Answer: -9.0042

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 9
System of two particles
Use the following information for Questions 9-11:

Consider a system of two non-interacting particles in an infinite potential well with a width of L. One particle is in the ground state (n=1) and the other particle is in the first excited state (n=2).
First, suppose the two particles are distinguishable and calculate ⟨(x1​−x2​)²⟩, the expectation value of (x1​−x2​)², where x₁ and x₂​ are the positions of particle 1 and 2, respectively.
Give your answer in unit of L2. Answers within 5% error will be considered correct.

Answer: 0.106

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 10
Now suppose the two particles are indistinguishable bosons and calculate ⟨(x1​−x2​)²⟩, the expectation value of (x1​−x2​)², where x₁ and x₂​ are the positions of particle 1 and 2, respectively.
Give your answer in unit of L2. Answers within 5% error will be considered correct.

Answer: 0.04

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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Question 11
This time, suppose the two particles are indistinguishable fermions and calculate ⟨(x1​−x2​)²⟩, the expectation value of (x1​−x2​)², where x₁ and x₂​ are the positions of particle 1 and 2, respectively.
Give your answer in unit of L2. Answers within 5% error will be considered correct.

Answer: 0.17

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These are answers of Foundations of Quantum Mechanics Coursera Week 6 Quiz

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