Annihilation and Creation Operators

The annihilation and creation operators (or in collection called the harmonic operators) are given by:

\[\begin{split} \hat a = \sqrt\frac{m \omega}{2 \hbar}\left(\hat x + \frac{i}{m\omega}\hat p\right)\\ \hat a^\dagger = \sqrt\frac{m \omega}{2 \hbar}\left(\hat x - \frac{i}{m\omega}\hat p\right) \end{split}\]

The Hamiltonian in terms of the harmonic operators is given by,

\[ \hat H = \frac{1}{2}h \omega \left(2\hat a^\dagger \hat a + 1\right) \]

Motivation

Recall the harmonic oscillator Hamiltonian,

\[ \hat H = \frac{\hat p^2}{2m} + \frac{1}{2}m\omega^2x^2 \]

You may be tempted to factor the Hamiltonian out to be,

\[ \frac{1}{2} m \omega^2 \left(\hat x - \frac{i}{m\omega}\hat p \right)\left(\hat x + \frac{i}{m\omega}\hat p \right) \]

However, this is not \(\hat H\) because \(\hat x\) and \(\hat p\) does not commute (\([\hat x, \hat p] = i\hbar\)). Instead,

\[ \hat H = \frac{1}{2} m \omega^2 \left(\hat x - \frac{i}{m\omega}\hat p \right)\left(\hat x + \frac{i}{m\omega}\hat p \right) - \frac{i \omega}{2}[\hat x, \hat p] \]

In the first term, the two factors along with some constant motivates the definition of \(\hat a,\hat a^\dagger\) such that,

\[ \hat H = \frac{1}{2}h \omega \left(2a^\dagger a + 1\right) \]

Number Operator and Eigenstuff

\[ \hat H = h \omega \left(\hat N+ \frac{1}{2}\right) \]

• \(\hat N\): Number operator, \(\hat N \equiv \hat a^\dagger \hat a\)

The eigenvector of \(\hat H\) are eigenvectors of then number operator. It is very useful to notate the eigenvector as \(\ket n\) and eigenvalue as \(n\).

\[\begin{split} \hat H \ket n = E_n \ket n\\ \hat N \ket n = n \ket n \end{split}\]

Where \(E_n\) is easily determined by applying \(\ket n\),

\[\begin{split} \hat H \ket n = \frac{1}{2}\hbar \omega (2 \hat N \ket n + \ket n) = \frac{1}{2} \hbar \omega (2 n \ket n + \ket n)\\ \hat H \ket n = \underbrace{\left(n + \frac{1}{2} \right) \hbar \omega}_{E_n} \ket n \end{split}\]

Lowering and Raising Operators

\(\hat a\) and \(\hat a^\dagger\) are also known as the lowering and raising operator respectively. This comes from applying the operator and determining the energy eigenvalue.

First using the raising operator, we find the eigenstate of \(\hat N\) goes up by one,

\[\begin{split} \hat N \hat a^\dagger \ket n = \hat a^\dagger \hat a \hat a^\dagger \ket n\\ \hat N \hat a^\dagger \ket n = (n+1) \hat a^\dagger \ket n \end{split}\]

Normalization

Now we established that the lower and raising operator transform the state from \(\ket{n+1}\) to \(\ket{n}\) or reverse respectively, we need to actually define the operation,

\[\begin{split} \hat a \ket n = \sqrt{n} \ket{n - 1}\\ \hat a^\dagger \ket n = \sqrt{n+1}\ket{n+1} \end{split}\]

Eigenstates

We can now define the eigenstates

\[ \ket n = \frac{(\hat a^\dagger)^n}{\sqrt{n!}} \ket 0 \]

Multiple Harmonic Oscillators

To discrete a general system of harmonic oscillator let’s start out with uncoupled harmonic oscillators. The Hamiltonian for each oscillator is,

\[ \hat H_k = \frac{\hat p_k ^2}{2m_k} + \frac{1}{2}m_k \omega_k^2 x_k^2 \]

The harmonic operators \(\hat a_k\) and \(\hat a^\dagger_k\) only applies to the \(k\)th oscillator and does nothing to the rest.

The only commutation rule that is nonzero is,

\[ [\hat a_i, \hat a^\dagger_j] = \delta_{i,j} \]

Equivalently, we can write the Hamiltonian as,

\[ \hat H = \sum_{k} \hbar w_k \left(\hat a_k^\dagger \hat a_k + \frac{1}{2} \right) \]

The general eigenstate \(\ket{n_1, n_2, \ldots}\) called the occupation number representation is then,

\[ \ket {n_1, n_2, \ldots} = \prod_{k} \frac{1}{\sqrt{n_k!}}\hat a_k^\dagger \ket{0,0,\ldots,0} \]