*Exploring Quantum Physics – Week 4, Lecture 7*

**Classical Harmonic Oscillator**

- Hook’s force:
- Newton’s 2nd law:
- Rewrite as: , where

- Solving for x:

**Phase portrait of a harmonic oscillator**

- Choose initial condition s.t. ,
- Velocity:
- Energy:
- Expanding
*x*and*v*:

- Expanding

**Going quantum**

- In most situations we can approximate the lowest energy potential as a quadratic, since

- Classical energy:
- Usual transformation:

**Quantum Hamiltonian**

## Creation and annihilation operators

Making the Hamiltonian dimensionless:

Using :

*where *

Let be the **creation operator**

and be the **annihilation operator**.

**Summary:**

## Generating the energy spectrum of the quantum harmonic oscillator

**Commutation relations**

**Generating the spectrum**

, so

latex \hat{a}^\dagger \hat{a} ( \hat{a}^\dagger | n \rangle) = \hat{a}^\dagger (1 + \hat{a}^\dagger \hat{a}) |n\rangle = \hat{a}^\dagger ( n + 1) | n \rangle = (n + 1)(hat{a}^\dagger | n \rangle)$

Annihilation:

Ground state:

## Harmonic oscillator wave-functions

**Deriving ground-state wave-function**

For the ground state,

Expanding operators:

Let

**Result**

Where *C* is the normalizing coefficient:

**Excited states**

## Summary

- Hamiltonian

- Energy spectrum

- Relation between (normalized) eigenfunctions

- Ground-state wave function:

- Excited-state wave functions:

, etc.

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