Coursework focused on A Quantum Engineer’s Guide to Superconducting Qubits.

#### Exercise 1

The Schrödinger equation is given as:

For the lowest energy level state , write the solution of . Then substitute it back in the Schrödinger equation and show it’s a solution. Use the time independent Schrödinger equation:

Which is equivalent to,

If then we know, ; because is a Hermite Polynomial

Therefore,

Plugged back into the time independent Schrödinger equation, we have:

Where, :

Therefore,

#### Exercise 2

The expectation value of a variable is given by . Show for the lowest energy state of the quantum harmonic oscillator that the expected value of the position is zero.

where

Therefore,

#### Exercise 3

What is the Josephson oscillation frequency for a potential drop across a junction? Is this frequency change detectable by standard electronics? What voltage drop would give a Josephson oscillation frequency of ?

If , and

Then,

This frequency isn’t detectable by standard electronics

If

Therefore,