Phonons and Photons

We call the state $\vert n\rangle$ of the harmonic oscillator with energy $\hbar\omega(n+1/2)$ a state with $n$ quanta $\hbar \omega$ of energy plus the zero point energy $\hbar \omega/2$. These quanta are called phonons for systems where massive particles have oscillatory degrees of freedom, the state $\vert n\rangle$ is a $n$-phonon state.

$$\vert n\rangle \longleftrightarrow \mbox{\rm$n$--phonon state} .$$ (275)

The ladder operator $a^{\dagger }$ operates as

$$\fbox{$ \begin{array}{rcl} \displaystyle a^{\dagger}\vert n\rangle&=&\sqrt{n+1}\vert n+1\rangle \end{array}$\ }$$ (276)

and creates a state with one more phonon which is why it is called a creation operator. In the same way, the operator $a$,

$$\fbox{$ \begin{array}{rcl} \displaystyle a\vert n\rangle&=&\sqrt{n}\vert n-1\rangle \end{array}$\ }$$ (277)

leads to a state with one phonon less (it destroys one phonon) and is called a annihilation operator.

In a similar manner, the oscillatory degrees of freedom of the electromagnetic field (light) lead to a Hamiltonian like the one of the harmonic oscillator. The corresponding states are called $n$-photon states. This is one of the topics of Quantum Mechanics II, the theory of light, and many-body theory. It is there where operators like the $a$ and $a^+$ show their full versatility and power.