The

annihilation (

**a**) and

creation (

**a**^{†}) (aka lowering and raising) operators enable you to go back and forth between the adjacent

eigenfunctions of a given

particle's

wave function.

However, the real fun of these

operators comes into play when they are used together. The two operators are derived from the

factorization of the

Hamiltonian (

**H**) for the

quantum harmonic oscillator problem where

**H = ** kinetic energy

**+** potential energy

**= p**^{2}/2m + √(w/m)x^{2}/2
**= w (√**(

**mw/2**)

**x - ip/√**(

**2mw**)

**)((√**(

**mw/2**)

**x + ip/√**(

**2mw**)

**= hw/2π( ½ + a**^{†}a)
*(w represents the angular frequency of the particle, m is the particle's mass, and x and p denote the position and momentum, respectively)*
One can then use the annihilation and creation operators to represent both the

position and

momentum functions for the particle:

**x = **(

**1/2**)

**√**(

**h/πmw**)

**(a + a**^{†})

p = (

**1/2**)

**√**(

**mwh/π**)

**(a - a**^{†})