# Foundations of Quantum Mechanics | Week 1

Course Name: Foundations of Quantum Mechanics

Course Link: Foundations of Quantum Mechanics

#### These are answers of Foundations of Quantum Mechanics Coursera Week 1 Quiz

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
|ψ|2∝cos2(πdz/Dλ)=1/2[1+cos(2πdz/Dλ)]
where d = slit separation, D = distance between slit and screen, z = position on the screen and λ = wavelength of electron.

Question 1
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=0.5nm, D=1m and the incident electron energy of 1 MeV.
Give the answer in unit of m. Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 1 Quiz

Question 2
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. Answers within 5% error will be considered correct.

Question 3
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, of the quantum dot?

∝L-2
∝L-6
∝L2
∝ exp(-L)

These are answers of Foundations of Quantum Mechanics Coursera Week 1 Quiz

Question 4
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,
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, λabs on the lattice constant, d.

∝d-2
∝d2
∝d6
∝ exp(-d)

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) = ∞, elsewhere

Question 5
Suppose an electron is in the 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 ≤ x ≤ +0.5 nm)?
Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 1 Quiz

Question 6
Suppose now an electron is in the 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 ≤ x ≤ +0.5 nm)?
Answers within 5% error will be considered correct.

These are answers of Foundations of Quantum Mechanics Coursera Week 1 Quiz

Use this information to answer Question 7-8:
Suppose an electron in an infinite potential well with width, L, has a wavefunction,
ϕ(z) = Az(z – L) for 0 < z <L

Question 7
Normalize this wavefunction and derive an expression for the constant A in terms of L.
In order to avoid confusion with autograder, use a single sqrt function containing all necessary variables.

These are answers of Foundations of Quantum Mechanics Coursera Week 1 Quiz

Question 8
The eigenfunctions, ψn, of the infinite potential well form a complete, orthonormal basis set and we can express the wavefunction ϕ as a linear combination of ψn‘s. Find the coefficient a₁ of eigenfunction ψ1 in the expansion of ϕ.
Answers within 5% error will be considered correct.