Lesson: Harmonic Oscillator

Introduction

A harmonic oscillator is a system that undergoes periodic motion around an equilibrium point. It is a fundamental concept in physics and quantum mechanics, and has applications in various fields, including quantum computing.

Classical Harmonic Oscillator

Consider a mass-spring system, where a mass m is attached to a spring with spring constant k. When the mass is displaced from its equilibrium position, it experiences a restoring force F = -kx (Hooke's law). This force causes the mass to oscillate with a frequency:

f = (1/2π) * √(k/m)

Quantum Harmonic Oscillator

In quantum mechanics, the classical harmonic oscillator is quantized. The system can only exist in certain discrete energy levels, given by:

E = (n + 1/2) * hf

where:

• n is the energy level quantum number (0, 1, 2, ...)
• h is Planck's constant
• f is the classical oscillation frequency

Operators and States

In quantum mechanics, the operators corresponding to position and momentum for a harmonic oscillator are:

• Position operator: x̂ = x̂_0 (â+ + â-)
• Momentum operator: p̂ = -iħ (â+ - â-)

where:

• x̂_0 is the classical equilibrium position
• â+ and â- are the creation and annihilation operators, respectively
• ħ is the reduced Planck's constant

The eigenstates of the harmonic oscillator are known as Fock states. The ground state (n=0) is the state with the lowest energy, and subsequent states have higher energies.

Applications in Quantum Computing

Harmonic oscillators are used in quantum computing platforms, such as superconducting qubits and trapped ions. By controlling the energy levels of the oscillator, quantum gates can be implemented to perform calculations.

Learning Resources

• Griffiths, D. J. (2005). Introduction to quantum mechanics (2nd ed.). Pearson Education.
• Shankar, R. (1994). Principles of quantum mechanics (2nd ed.). Plenum Press.
• Quantum Harmonic Oscillator: https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator