 Week 8: Physical Implementation of Qubits: Harmonic Oscillators and Superconductors | ECE 802-730

# Week 8: Physical Implementation of Qubits: Harmonic Oscillators and Superconductors | ECE 802-730

#### 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, 