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### Week 01 Quiz Answers

Q1. **Double-slit experiment**

Use this information to answer Question 1-2:

*Following the standard analysis of double slit experiment (see, for example, *https://en.wikipedia.org/wiki/Double-slit_experiment*), the interference pattern on the screen is described by*

*|\psi|^2 \propto \cos^2\big(\frac{\pi dz}{D\lambda}\big) = \frac{1}{2}\Big[ 1+\cos\big(\frac{2\pi dz}{D\lambda}\big)\Big] ∣ψ∣2∝cos2(Dλπdz)=21[1+cos(Dλ2πdz)]*

*where d = slit separation, D = distance between slit and screen, z = position on the screen and \lambdaλ = wavelength of electron.*

We conduct the double slit experiment using an electron beam. Find the fringe spacing (distance between two adjacent bright spots) for the case where d=0.5 \: nm*d*=0.5*nm*, D=1\:m*D*=1*m* and the incident electron energy of 1 MeV.

Give the answer in unit of m*m*. Answers within 5% error will be considered correct.

Q2. What if we use a neutron beam with the same energy, 1 MeV, and same values for all other parameters? Find the fringe spacing in that case.

Give the answer in unit of m*m*. Answers within 5% error will be considered correct.

Q3. **Semiconductor quantum well**

A semiconductor quantum dot (a nanoparticle of semiconductor) exhibits a larger bandgap than the bulk semiconductor due to the quantum confinement effect. This bandgap shift can be estimated by the ground state energy of infinite potential well problem. What is the dependence of bandgap energy on the size, L*L*, of the quantum dot ?

- \propto L^{-2}∝
*L*−2 - \propto L^{-6}∝
*L*−6 - \propto L^2∝
*L*2 - \propto \exp(-L)∝exp(−
*L*)

Q4. A color center is a commonly observed defect in ionic crystals and is composed of an electron trapped in a vacancy. It can be modeled as an electron in a three-dimensional infinite potential well with a side, d *d*,

As we survey similar crystals (e.g. alkali metal halides) which have the same crystal structure but with different lattice constants, predict the dependence of absorption peak wavelength, \lambda_{abs} *λabs* on the lattice constant, d *d*.

- \propto d^{-2}∝
*d*−2 - \propto d^{2}∝
*d*2 - \propto d^{6}∝
*d*6 - \propto \exp({-d})∝exp(−
*d*)

Q5. **1D infinite potential well**

Use this information to answer Question 5-6:

*Consider an infinite potential well defined as*

*V(x)=0 ,\:\:\: -5\:nm < x < +5\:nm V(x)=0,−5nm<x<+5nm*

*V(x)=\infty ,\:\:\: elsewhere V(x)=∞,elsewhere*

Suppose an electron is in the n = 3*n*=3 state in this infinite potential well. What is the probability of finding the electron within 1 nm region at the center of the potential well (i.e. -0.5\:nm\leq x \leq +0.5\:nm−0.5*nm*≤*x*≤+0.5*nm*) ?

Answers within 5% error will be considered correct.

Q6. **1D infinite potential well**

Suppose now an electron is in the n=4*n*=4 state . What is the probability of finding the electron within 1 nm region at the center of the potential well (i.e. -0.5\:nm\leq x \leq +0.5\:nm−0.5*nm*≤*x*≤+0.5*nm*) ?

Answers within 5% error will be considered correct.

Q7. Use this information to answer Question 7-8:

*Suppose an electron in an infinite potential well with width, LL, has a wavefunction,*

*\phi(z)=Az(z-L) ϕ(z)=Az(z−L) for 0<z<L 0<z<L*

Normalize this wavefunction and derive an expression for the constant A*A* in terms of L*L*.

In order to avoid confusion with autograder, use a single sqrt*sqrt* function containing all necessary variables.

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Q8. The eigenfunctions, \psi_n*ψn*, of the infinite potential well form a complete, orthonormal basis set and we can express the wavefunction \phi*ϕ* as a linear combination of \psi_n*ψn*’s. Find the coefficient a_1*a*1 of eigenfunction \psi_1*ψ*1 in the expansion of \phi*ϕ*.

Answers within 5% error will be considered correct.

### Week 02 Quiz Answers

Q1. **Finite potential well**

Use this information to answer Question 1-2:

*Consider an electron in a finite potential with a depth of V_0 = 0.3 \:eVV0=0.3eV and a width of 10 \:nm10nm.*

Find the lowest energy level.

Give your answer in unit of eV*eV*. Answers within 5% error will be considered correct.

Note that in the lecture titled “Finite Potential Well”, the potential well is defined from -L−*L* to L*L*, which makes the well width 2L2*L*.

Q2. **Finite potential well**

Find the second lowest energy level.

Give your answer in unit of eV*eV*. Answers within 5% error will be considered correct.

Q3. **Potential step**

Consider a potential step with a height of V_0 = 0.2\: eV*V*0=0.2*eV*. An electron is incident from the lower potential side (where the potential energy is set to equal to zero) with an energy E = 0.3\: eV*E*=0.3*eV*.

Calculate the reflection probability.

Q4. **Potential step **

Consider a potential step with a height of V_0 = 0.5\: eV*V*0=0.5*eV*. An electron is incident from the lower potential side (where the potential energy is set to equal to zero) with an energy E = 0.3\: eV*E*=0.3*eV*.

Find the penetration depth which is defined as the distance into the barrier region at which the probability decreases to 1/e1/*e* of its initial value at the potential discontinuity, z = 0*z*=0.

Give your answer in unit of nm*nm*. Answers within 5% error will be considered correct.

Q5. **Potential step**

A non-classical behavior found in the potential step problem is that there is a finite reflection even when the potential step is negative. That is, an electron may reflect at the boundary even when it is incident in the region of higher potential, as shown below.

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/hnLzZzrKQ_my82c6ytP5IQ_30aec9897bbe47d88d887622898627e6_negative_potential_step.png?expiry=1657929600000&hmac=mb8MqqQRNjTd5l_9T2cuo9MvR6_qBviriqOCPxT37AU>

Calculate the reflection probability when E = 0.1\: eV*E*=0.1*eV* and V_0 = 0.5\: eV*V*0=0.5*eV*.

Answers within 5% error will be considered correct.

Q6. **Potential barrier**

Consider a potential barrier with height V_0 = 1\: eV*V*0=1*eV* and width L = 1\: nm*L*=1*nm*. Find the energy of incident electron at which the reflection probability is 95%.

Give your answer in unit of eV*eV*. Answers within 5% error will be considered correct.

Q7. **Negative potential barrier**

Consider a negative potential barrier with V_0 = -1\: eV*V*0=−1*eV* and L = 1\: nm*L*=1*nm*. Find the lowest energy of incident electron at which the transmission probability becomes unity.

Give your answer in unit of eV*eV*. Answers within 5% error will be considered correct.

Note this is a finite potential well problem with electron energy greater than the well depth.

### Week 03 Quiz Answers

Q1. **Measurement**

Suppose an electron is in a spin state that can be described by;

|\phi\rangle = \frac{\sqrt{3}}{2} |+\rangle + \frac{1}{2}|-\rangle∣*ϕ*⟩=23∣+⟩+21∣−⟩

where ++ and –– are eigenstates of S_z*Sz* with eigenvalue +\frac{\hbar}{2} +2ℏ and -\frac{\hbar}{2}−2ℏ.

If we measure z-component of spin of this electron, what is the probability of measuring spin up, +\frac{\hbar}{2} +2ℏ?

Answers within 5% error will be considered correct.

Q2. **Measurement**

Use the following information for Questions 2-3:

*Suppose there are two quantum mechanical observables cc and dd represented by operators \hat{C}C^ and \hat{D}D^, respectively. Both operators have two eigenstates, \phi_1ϕ1 and \phi_2ϕ2 for \hat{C}C^ and \psi_1ψ1 and \psi_2ψ2 for \hat{D}D^. Furthermore, the two sets of eigenstates are related to each other as below.*

*\phi_1 = \frac{1}{13}(5\psi_1 + 12\psi_2) ϕ1=131(5ψ1+12ψ2)*

*\phi_2 = \frac{1}{13}(12\psi_1 – 5\psi_2) ϕ2=131(12ψ1−5ψ2)*

*The system was found to be in state \phi_1ϕ1 initially.*

If we measure \hat{D}*D*^, what is the probability of finding the system in \psi_2*ψ*2?

Answers within 5% error will be considered correct.

Q3. **Measurement**

Once again, the system is in state \phi_1*ϕ*1 initially*. *This time, we perform two successive measurements in which we first measure \hat{D}*D*^ and then \hat{C}*C*^ again. What is the probability of finding the system is still in \phi_1*ϕ*1 ?

Answers within 5% error will be considered correct.

Q4. **Expectation value and measurement**

Use the following information for Questions 4-7:

*Consider an infinite potential well with a width L=10\:nmL=10nm is located in the region* 0<z<L0<*z*<*L*. *Suppose an electron in that infinite potential well is described by wavefunction*

*\Phi(z) = Az^2(L-z) Φ(z)=Az2(L−z) for 0 < z < L0<z<L*

Normalize the wavefunction and determine the constant A*A*. Give your answer in the standard SI unit. Answers within 5% error will be considered correct.

Q5. **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.

Q6. **Expectation value and measurement**

If we measure the energy of this electron, what is the probability of measuring the ground state energy, E_1=\frac{\pi^2\hbar^2}{2mL^2}*E*1=2*mL*2*π*2ℏ2? Answers within 5% error will be considered correct.

Q7. **Expectation value and measurement**

If we measure the energy of this electron, what is the probability of measuring the first excited state energy, E_2=\frac{4\pi^2\hbar^2}{2mL^2}*E*2=2*mL*24*π*2ℏ2? Answers within 5% error will be considered correct.

Q8. **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 f(x)=ax+b, defined within the region, -1 < x < 1−1<x<1. The constants aa and bb are complex numbers.*

Consider a function \psi_1(x) = c*ψ*1(*x*)=*c* where c*c* is a complex number. Normalize this function and determine c*c*

Answers within 5% error will be considered correct.

Q9. **Vector space and matrix representation**

Consider another function \psi_2(x) = dx*ψ*2(*x*)=*dx* where d*d* is a complex number. Normalize this function and determine the constant d*d*.

Answers within 5% error will be considered correct.

Q10. **Vector space and matrix representation**

Calculate the inner product of the two functions, \psi_1(x)*ψ*1(*x*) and \psi_2(x)*ψ*2(*x*).

Answers within 5% error will be considered correct.

Q11. **Vector space and matrix representation**

Let us now construct the vector (matrix representation) for a function g(x) = -2x + 1 *g*(*x*)=−2*x*+1 using \psi_1(x)*ψ*1(*x*) and \psi_2(x)*ψ*2(*x*) as the basis set. That is, we want to express the function g(x)*g*(*x*) as,

|g\rangle=

(*pq*)

∣*g*⟩=(*pq*)

What is the value of p*p*?

Answers within 5% error will be considered correct.;

Q12. **Vector space and matrix representation**

In the vector given in Question 11, what is the value of the second vector element q*q*?

Answers within 5% error will be considered correct.;

Q13. **Vector space and matrix representation**

Now consider a reflection operator \hat{R}*R*^ which transforms function f(x)*f*(*x*) to f(-x)*f*(−*x*), i.e., \hat{R}f(x) = f(-x)*R*^*f*(*x*)=*f*(−*x*).

Find the matrix representation of \hat{R}*R*^ in the \{\psi_1,\psi_2\}{*ψ*1,*ψ*2} basis. That is, we want to write

\hat{R}=

(*prqs*)

*R*^=(*pr**qs*)

What is the value of p*p*?

Answers within 5% error will be considered correct.

**Vector space and matrix representation**

In the matrix \hat{R}*R*^ in Question 13, what is the value of q*q*?;

Answers within 5% error will be considered correct.

15. **Vector space and matrix representation**

In the matrix \hat{R}*R*^ in Question 13, what is the value of r*r*?

Answers within 5% error will be considered correct.

16. **Vector space and matrix representation**

In the matrix \hat{R}*R*^ in Question 13, what is the value of s*s*?

Answers within 5% error will be considered correct.

### Week 04 Quiz Answers

Q1. **Expectation value of energy**

Use the following information for Questions 1-2:

*Consider a particle with mass, mm, in an infinite potential well with a width LL.*

The particle was initially in the first excited state \psi_2*ψ*2. What is the expectation value of energy, \langle\hat{H}\rangle⟨*H*^⟩?

Express your answer in terms of mass, m*m*, width, L*L*, reduced Planck’s constant, hbar*hbar* and a constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q2. **Expectation value of energy**

Now suppose the particle was initially in a superposition state \phi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)*ϕ*=21(*ψ*1+*ψ*2) where \psi_1*ψ*1 and \psi_2*ψ*2 are the two lowest energy eigenstates, respectively.

What is the expectation value of energy, \langle\hat{H}\rangle⟨*H*^⟩?

Express your answer in terms of mass, m*m*, width, L*L*, reduced Planck’s constant, hbar*hbar* and a constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q3. **Expectation value of energy eigenstate**

Use the following information for Questions 3-8:

*Consider the simple harmonic oscillator problem with mass, mm, and resonance frequency, \omegaω. The energy eigenvalues are E_n = (n+\frac{1}{2})\hbar\omegaEn=(n+21)ℏω where n=0,\:1,\: \cdotsn=0,1,⋯ and the eigenfunctions are gaussian functions modulated by Hermite polynomials, as discussed in Module 2.*

The oscillator is in the first excited state |1\rangle∣1⟩. What is the expectation value of energy, \langle\hat{H}\rangle⟨*H*^⟩?

Express your answer in terms of mass, m*m*, resonance frequency, omega*omega*, reduced Planck’s constant, hbar*hbar* and constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q4. **Expectation value of energy eigenstate**

The oscillator is in the first excited state |1\rangle∣1⟩. What is the expectation value of position, \langle\hat{x}\rangle⟨*x*^⟩?

Express your answer in terms of mass, m*m*, resonance frequency, omega*omega*, reduced Planck’s constant, hbar*hbar* and constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q5. **Expectation value of energy eigenstate**

The oscillator is in the first excited state |1\rangle∣1⟩. What is the variance of position, \langle\hat{x}^2\rangle⟨*x*^2⟩, i.e. expectation value of \hat{x}^2*x*^2?

Express your answer in terms of mass, m*m*, resonance frequency, omega*omega*, reduced Planck’s constant, hbar*hbar* and constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q6. **Expectation value of superposition state**

Now suppose the harmonic oscillator is in a superposition state \phi = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)*ϕ*=21(∣0⟩+∣1⟩) where |0\rangle∣0⟩ and |1\rangle∣1⟩ are the two lowest energy eigenstates, respectively.

What is the expectation value of energy, \langle\hat{H}\rangle⟨*H*^⟩?

*m*, resonance frequency, omega*omega*, reduced Planck’s constant, hbar*hbar* and constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q7. **Expectation value of superposition state**

The harmonic oscillator is in a superposition state \phi = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)*ϕ*=21(∣0⟩+∣1⟩) where |0\rangle∣0⟩ and |1\rangle∣1⟩ are the two lowest energy eigenstates, respectively.

What is the expectation value of position, \langle\hat{x}\rangle⟨*x*^⟩?

*m*, resonance frequency, omega*omega*, reduced Planck’s constant, hbar*hbar* and constant pi*pi*. Note that your answer does not have to include all of these variables.

In order to avoid confusion with autograder, use sqrt()*sqrt*() for square root instead of power of 1/21/2. Also, use a single sqrt()*sqrt*() function including all relevant variables rather than multiple sqrt()*sqrt*() functions multiplied together.

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Q8. **Expectation value of superposition state**

The harmonic oscillator is in a superposition state \phi = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)*ϕ*=21(∣0⟩+∣1⟩) where |0\rangle∣0⟩ and |1\rangle∣1⟩ are the two lowest energy eigenstates, respectively.

What is the variance of position, \langle\hat{x}^2\rangle⟨*x*^2⟩, i.e. expectation value of \hat{x}^2*x*^2?

*m*, resonance frequency, omega*omega*, reduced Planck’s constant, hbar*hbar* and constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q9. **Change of basis**

Use the following information for Questions 9-19:

*Consider the vector space composed of all linear functions, f(x) = ax + b f(x)=ax+b, defined within the region, -1 < x < 1−1<x<1. The constants aa and bb are complex numbers.*

Previously, we used \psi_1=\frac{1}{\sqrt{2}}*ψ*1=21 and \psi_2=\frac{\sqrt{3}}{2}x*ψ*2=23*x* as basis set to construct matrix representations for all linear functions defined above (See Questions 8-16 in Module 3 for reference).

This time, we will use \phi_1 = \frac{\sqrt{3}}{2}x+\frac{1}{2}*ϕ*1=23*x*+21 and \phi_2 = \frac{\sqrt{3}}{2}x-\frac{1}{2}*ϕ*2=23*x*−21 as a new basis set.

Calculate the inner product of the two functions, \phi_1(x)*ϕ*1(*x*) and \phi_2(x)*ϕ*2(*x*).

Answers within 5% error will be considered correct.;

Q10. **Change of basis**

Let us now construct the vector (matrix representation) for a function g(x) = -2x + 1 *g*(*x*)=−2*x*+1 using \phi_1(x)*ϕ*1(*x*) and \phi_2(x)*ϕ*2(*x*) as the basis set. That is, we want to express the function g(x)*g*(*x*) as,

|g\rangle=

(*p*′*q*′)

∣*g*⟩=(*p*′*q*′)

What is the value of p’*p*′?

Answers within 5% error will be considered correct.

Q11. **Change of basis**

In the vector given in Question 10, what is the value of the second vector element q’*q*′?

Answers within 5% error will be considered correct.

Q12. **Change of basis**

Now consider a reflection operator \hat{R}*R*^ which transforms function f(x)*f*(*x*) to f(-x)*f*(−*x*), i.e., \hat{R}f(x) = f(-x)*R*^*f*(*x*)=*f*(−*x*).

Find the matrix representation of \hat{R}*R*^ in the \{\phi_1,\phi_2\}{*ϕ*1,*ϕ*2} basis. That is, we want to write

\hat{R}=

(*p*′*r*′*q*′*s*′)

*R*^=(*p*′*r*′*q*′*s*′)

What is the value of p’*p*′?

Answers within 5% error will be considered correct.

Q13. **Change of basis**

In the matrix \hat{R}*R*^ in Question 12, what is the value of q’*q*′?

Answers within 5% error will be considered correct.

Q14. **Change of basis**

In the matrix \hat{R}*R*^ in Question 12, what is the value of r’*r*′?

Answers within 5% error will be considered correct.;

Q15. **Change of basis**

In the matrix \hat{R}*R*^ in Question 12, what is the value of s’*s*′?

Answers within 5% error will be considered correct.

Q16. **Change of basis**

Now we relate the two matrix representations using two different basis sets \{\psi_1,\psi_2\}{*ψ*1,*ψ*2} and \{\phi_1,\phi_2\}{*ϕ*1,*ϕ*2}.

Again the basis functions are defined as

\psi_1=\frac{1}{\sqrt{2}}*ψ*1=21 and \psi_2=\frac{\sqrt{3}}{2}x*ψ*2=23*x*

\phi_1 = \frac{\sqrt{3}}{2}x+\frac{1}{2}*ϕ*1=23*x*+21 and \phi_2 = \frac{\sqrt{3}}{2}x-\frac{1}{2}*ϕ*2=23*x*−21

Now we construct a unitary transformation matrix that transforms from the \{\phi_1,\phi_2\}{*ϕ*1,*ϕ*2} basis to \{\psi_1,\psi_2\}{*ψ*1,*ψ*2} basis. This is, we want to write

(*ψ*1*ψ*2)

=

(*u*11*u*21*u*12*u*22)

(*ϕ*1*ϕ*2)

(*ψ*1*ψ*2)=(*u*11*u*21*u*12*u*22)(*ϕ*1*ϕ*2)

What is the value of u_{11}*u*11?

Answers within 5% error will be considered correct.

Q17. **Change of basis**

In the matrix in Question 16, what is the value of u_{12}*u*12?

Answers within 5% error will be considered correct.

18. **Change of basis**

In the matrix in Question 16, what is the value of u_{21}*u*21?

Answers within 5% error will be considered correct.

19. **Change of basis**

In the matrix in Question 16, what is the value of u_{22}*u*22?

Answers within 5% error will be considered correct.

### Week 05 Quiz Answers

Q1. **Time evolution of expectation value**

Use the following information for Questions 1-3:

*Consider a particle with mass, mm, in an infinite potential well with a width LL. Here we choose the coordinates such that the center of the well is x=0x=0 and the walls are located at x=\pm\frac{L}{2}x=±2L.*

*V(x)=0 ,\:\:\: -\frac{L}{2} < x < +\frac{L}{2} V(x)=0,−2L<x<+2L*

*V(x)=\infty ,\:\:\: elsewhere V(x)=∞,elsewhere*

Now suppose the particle was initially in a superposition state \phi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)*ϕ*=21(*ψ*1+*ψ*2) where \psi_1*ψ*1 and \psi_2*ψ*2 are the two lowest energy eigenstates, respectively.

What is the expectation value of energy, \langle\hat{H}\rangle⟨*H*^⟩ as a function of time?

Express your answer in terms of mass, m*m*, width, L*L*, reduced Planck’s constant, hbar*hbar*, time, t*t* and a constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q2. **Time evolution of expectation value**

Again the particle was initially in a superposition state \phi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)*ϕ*=21(*ψ*1+*ψ*2) where \psi_1*ψ*1 and \psi_2*ψ*2 are the two lowest energy eigenstates, respectively.

What is the expectation value of position, \langle\hat{x}\rangle⟨*x*^⟩ initally, at t = 0*t*=0?

Express your answer in terms of mass, m*m*, width, L*L*, reduced Planck’s constant, hbar*hbar* and constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q3. **Time evolution of expectation value**

Continuing from Question 2, what is the expectation value of position, \langle\hat{x}\rangle⟨*x*^⟩ as a function of time, t*t*?

Express your answer in terms of the initial position, x*x*_0, (i.e. the position expectation value at t=0*t*=0), the ground and first excited state energies, E*E*_1 and E*E*_2, time t*t* and the reduced Planck’s constant, hbar*hbar*. Note that your answer does not have to include all of these variables.

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Q4. **Time evolution of expectation value**

Use the following information for Questions 4-7:

*Consider the 1D simple harmonic oscillator problem.*

Suppose the system was initially in a superposition state \phi = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)*ϕ*=21(∣0⟩+∣1⟩) where |n\rangle∣*n*⟩ is the energy eigenstate of 1D harmonic oscillator with energy E_n=(n+\frac{1}{2})\hbar\omega*En*=(*n*+21)ℏ*ω*.

What is the expectation value of position, \langle\hat{x}\rangle⟨*x*^⟩ as a function of time?

Express your answer in terms of mass, m*m*, oscillator frequency, omega*omega*, reduced Planck’s constant, hbar*hbar*, time, t*t* and a constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q5. **Time evolution of expectation value**

Again, assuming that the system was initially in a superposition state \phi = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)*ϕ*=21(∣0⟩+∣1⟩), derive an expression for \langle\hat{x}^2\rangle⟨*x*^2⟩ .

Express your answer in terms of mass, m*m*, oscillator frequency, omega*omega*, reduced Planck’s constant, hbar*hbar*, time, t*t* and a constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q6. **Time evolution of expectation value**

This time, suppose the harmonic oscillator was initially in a superposition state \phi = \frac{1}{\sqrt{2}}(|0\rangle + |2\rangle)*ϕ*=21(∣0⟩+∣2⟩) where |n\rangle∣*n*⟩ is the energy eigenstate of 1D harmonic oscillator with energy E_n=(n+\frac{1}{2})\hbar\omega*En*=(*n*+21)ℏ*ω*.

What is the expectation value of position, \langle\hat{x}\rangle⟨*x*^⟩ as a function of time?

Express your answer in terms of mass, m*m*, oscillator frequency, omega*omega*, reduced Planck’s constant, hbar*hbar*, time, t*t* and a constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q7. **Time evolution of expectation value**

Again, assuming that the harmonic oscillator was initially in a superposition state \phi = \frac{1}{\sqrt{2}}(|0\rangle + |2\rangle)*ϕ*=21(∣0⟩+∣2⟩), derive an expression for the variace of position, \langle\hat{x}^2\rangle⟨*x*^2⟩, as a function of time.

*m*, oscillator frequency, omega*omega*, reduced Planck’s constant, hbar*hbar*, time, t*t* and a constant pi*pi*. Note that your answer does not have to include all of these variables.

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Q8. **Time evolution of quantum state**

Use the following information for Questions 8-15:

\hat{A}*A*^ is a Hermitian operator with two eigenvectors, |a\rangle∣*a*⟩ and |b\rangle∣*b*⟩, with eigenvalues a*a* and b*b*, respectively (a\neq b*a*=*b*).

The Hamiltonian of a quantum system can be expressed as

\hat{H}=\sigma(|a\rangle\langle b|+|b\rangle\langle a|) *H*^=*σ*(∣*a*⟩⟨*b*∣+∣*b*⟩⟨*a*∣)

where \sigma*σ* is a real positive number.

What is the bigger of the two eigenvalues of the Hamiltonian operator?

Express your answer in terms of a*a*, b*b* and sigma*sigma*. Note that your answer does not have to include all of these variables.

Hint: Express the Hamiltonian operator in a matrix form (see Module 3 Video 4 Matrix Representation) and then diagonalize it to obtain eigenvectors.

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9. What is the smaller of the two eigenvalues of the Hamiltonian operator?

Express your answer in terms of a*a*, b*b* and sigma*sigma*. Note that your answer does not have to include all of these variables.

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10. We can express the eigenvectors of the Hamiltonian using |a\rangle∣*a*⟩ and |b\rangle∣*b*⟩ as the basis vectors. That is,

|p\rangle=c|a\rangle+d|b\rangle∣*p*⟩=*c*∣*a*⟩+*d*∣*b*⟩ where c*c* and d*d* are complex numbers.

What is the value of coefficient c*c* for the eigenvector corresponding to the larger of the two eigenvalues?

Express your answer in terms of a*a*, b*b*, sigma*sigma* and imaginary unit i*i*. Note that your answer does not have to include all of these variables.

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Q11. Continuing the problem of Question 10, what is the value of coefficient d*d* for the eigenvector corresponding to the larger of the two eigenvalues?

Express your answer in terms of a*a*, b*b*, sigma*sigma* and imaginary unit i*i*. Note that your answer does not have to include all of these variables.

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Q12. . Continuing the problem of Question 10, this time we want to express the eigenvector of Hamiltonian corresponding to the smaller of the two eigenvalues in the basis of \{|a\rangle, |b\rangle\}{∣*a*⟩,∣*b*⟩}. That is,

|q\rangle=f|a\rangle+g|b\rangle∣*q*⟩=*f*∣*a*⟩+*g*∣*b*⟩ where f*f* and g*g* are complex numbers.

What is the value of coefficient f*f* for the eigenvector corresponding to the smaller of the two eigenvalues?

Express your answer in terms of a*a*, b*b*, sigma*sigma* and imaginary unit i*i*. Note that your answer does not have to include all of these variables.

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Q13. Continuing Question 12, what is the value of coefficient g*g* for the eigenvector corresponding to the smaller of the two eigenvalues?

*a*, b*b*, sigma*sigma* and imaginary unit i*i*. Note that your answer does not have to include all of these variables.

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Q14,. Suppose the system is initially in state |a\rangle∣*a*⟩ at t=0*t*=0. Now we want to express the quantum state at a time t>0*t*>0 in the basis of \{|a\rangle,|b\rangle\}{∣*a*⟩,∣*b*⟩} . That is,

|r(t)\rangle=k|a\rangle+l|b\rangle∣*r*(*t*)⟩=*k*∣*a*⟩+*l*∣*b*⟩ where k*k* and l*l* are complex numbers.

And at t=0*t*=0, k=1*k*=1 and l=0*l*=0.

Write down the expression for k*k* in terms of a*a*, b*b*, sigma*sigma*, imaginary unit i*i*, reduced Planck’s constant, hbar*hbar*, and time t*t*. Note that your answer does not have to include all of these variables

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Q15. Continuing Question 14, write down the expression for l*l* in terms of a*a*, b*b*, sigma*sigma*, imaginary unit i*i*, reduced Planck’s constant, hbar*hbar*, and time t*t*. Note that your answer does not have to include all of these variables.

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Q16. **Particle current**

Recall the 1D potential barrier problem whose solution is given as

\psi(x)=Te^{ik_0x} *ψ*(*x*)=*Teik*0*x* in region III (the region past the barrier)

where T =e^{-ik_0L}\frac{2i\mu}{(1+\mu^2)\sin{kL}+2i\mu\cos{kL}}*T*=*e*−*ik*0*L*(1+*μ*2)sin*kL*+2*iμ*cos*kL*2*iμ*, k_0 = \sqrt{\frac{2mE}{\hbar}}*k*0=ℏ2*mE*, k = \sqrt{\frac{2m(E-V_0)}{\hbar}}*k*=ℏ2*m*(*E*−*V*0), and \mu=\frac{k}{k_0}*μ*=*k*0*k*. V_0*V*0 and L*L* are the height and width of the potential barrier and E*E* is the energy of the incident particle.

Write down the expression for the particle current in region III.

Express your answer in terms of m*m*, k*k*, k0*k*0, mu*mu*, T*T*, L*L*, imaginary unit i*i* and reduced Planck’s constant hbar*hbar*. Note that your answer does not have to include all of these variables.

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### Week 06 Quiz Answers

Q1. **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+\rangle∣sx+⟩ and |sy+\rangle∣sy+⟩. That is, the spin state can be expressed as*

*|\chi\rangle = a(|sx+\rangle+|sy+\rangle)∣χ⟩=a(∣sx+⟩+∣sy+⟩)*

*where |sx+\rangle∣sx+⟩ and |sy+\rangle∣sy+⟩ are the eigenstates of \hat{S_x}Sx^ and \hat{S_y}Sy^ operators with eigenvalues +\frac{\hbar}{2}+2ℏ, respectively.*

First, normalize the vector and determine the constant a*a*.

Express your answer in terms of imaginary unit i*i*, constant pi*pi* and reduced Planck’s constant, hbar*hbar*. Note that your answer does not have to include all of these variables.

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Q2. **Pure and mixed spin states**

For the state defined in Question 1, find the expectation value for the x-component of spin, \langle\hat{S_x}\rangle⟨*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*i*, constant pi*pi* and reduced Planck’s constant, hbar*hbar*. Note that your answer does not have to include all of these variables.

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Q3. **Pure and mixed spin states**

For the state defined in Question 1, find the expectation value for the y-component of spin, \langle\hat{S_y}\rangle⟨*Sy*^⟩.

Express your answer in terms of imaginary unit i*i*, constant pi*pi* and reduced Planck’s constant, hbar*hbar*. Note that your answer does not have to include all of these variables.

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Q4. **Pure and mixed spin states**

For the state defined in Question 1, find the expectation value for the z-component of spin, \langle\hat{S_z}\rangle⟨*Sz*^⟩.

*i*, constant pi*pi* and reduced Planck’s constant, hbar*hbar*. Note that your answer does not have to include all of these variables.

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Q5. **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+\rangle∣*sx*+⟩ and |sy+\rangle∣*sy*+⟩.

Find the ensemble average [\hat{S_x}][*Sx*^] for the x-component of spin.

This represents the average value you will obtain when measuring x-component of spin many times.

*i*, constant pi*pi* and reduced Planck’s constant, hbar*hbar*. Note that your answer does not have to include all of these variables.

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Q6. **Pure and mixed spin states**

For the mixed state defined in Question 5, find the ensemble average [\hat{S_y}][*Sy*^] for the y-component of spin.

*i*, constant pi*pi* and reduced Planck’s constant, hbar*hbar*. Note that your answer does not have to include all of these variables.

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Q7. **Pure and mixed spin states**

For the mixed state defined in Question 5, find the ensemble average [\hat{S_z}][*Sz*^] for the z-component of spin.

*i*, constant pi*pi* and reduced Planck’s constant, hbar*hbar*. Note that your answer does not have to include all of these variables.

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Q8. **Two-electron system** Consider two electrons in the same spin state, interacting through a short range potential given by

V(x_1,x_2)=V(|x_1-x_2|)=\{

−*V*0,|*x*1−*x*2|<*a*0,elsewhere

*V*(*x*1,*x*2)=*V*(∣*x*1−*x*2∣)={−*V*0,∣*x*1−*x*2∣<*a*0,elsewhere

where V_0*V*0 is a real, positive number and a*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,

\Big(-\frac{\hbar^2}{2m_1}\frac{\partial^2}{\partial x_1^2}-\frac{\hbar^2}{2m_2}\frac{\partial^2}{\partial x_2^2}+V(x_1,x_2)\Big)\Psi(x_1,x_2)=E\Psi(x_1,x_2)(−2*m*1ℏ2∂*x*12∂2−2*m*2ℏ2∂*x*22∂2+*V*(*x*1,*x*2))Ψ(*x*1,*x*2)=*E*Ψ(*x*1,*x*2)

and separate it into two uncoupled equations – one for the center of mass coordinate, X=\frac{m_1x_1+m_2x_2}{m_1+m_2}*X*=*m*1+*m*2*m*1*x*1+*m*2*x*2 and the other for the relative coordinate, x=x_1-x_2*x*=*x*1−*x*2. 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 V_0 = 10\: eV*V*0=10*eV* and a = 1\:nm*a*=1*nm*.

Give your answer in unit of eV*eV*. Answers within 5% errors will be considered correct.

*Hint: Note that the energy would have a negative value between -10\: eV−10eV and 0\: eV0eV.*

Q9. **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 LL. One particle is in the ground state *(*n=1n=1*)* and the other particle is in the first excited state *(*n=2n=2*)*.*

First, suppose the two particles are distinguishable and calculate \langle(x_1-x_2)^2\rangle⟨(*x*1−*x*2)2⟩, the expectation value of (x_1-x_2)^2(*x*1−*x*2)2, where x_1*x*1 and x_2*x*2 are the positions of particle 1 and 2, respectively.

Give your answer in unit of L^2*L*2. Answers within 5% error will be considered correct.

Q10. Now suppose the two particles are indistinguishable bosons and calculate \langle(x_1-x_2)^2\rangle⟨(*x*1−*x*2)2⟩, the expectation value of (x_1-x_2)^2(*x*1−*x*2)2, where x_1*x*1 and x_2*x*2 are the positions of particle 1 and 2, respectively.

Give your answer in unit of L^2*L*2. Answers within 5% error will be considered correct.

Q11. This time, suppose the two particles are indistinguishable fermions and calculate \langle(x_1-x_2)^2\rangle⟨(*x*1−*x*2)2⟩, the expectation value of (x_1-x_2)^2(*x*1−*x*2)2, where x_1*x*1 and x_2*x*2 are the positions of particle 1 and 2, respectively.

Give your answer in unit of L^2*L*2. Answers within 5% error will be considered correct.

**Review: **

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