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Ladder operator

Raising/lowering vs. creation/annihilation
There is some confusion regarding the relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory. The creation operator ai† increments the number of particles in state i, while the corresponding annihilation operator ai decrements the number of particles in state i. This clearly satisfies the requirements of the above definition of a ladder operator: the incrementing or decrementing of the eigenvalue of another operator (in this case the particle number operator).

Confusion arises because the term ladder operator is typically used to describe an operator that acts to increment or decrement a quantum number describing the state of a system. To perform the same operation with creation/annihilation operators would require the use of an annihilation operator to remove a particle from the initial state and a creation operator to add a particle to the final state.

Quantum harmonic oscillator
The most well known application of the ladder operator approach was developed by Paul Dirac to obtain the eigenvalues of the quantum harmonic oscillator. We define the Hermitian adjoint operators a and a†,



\begin{align} a &= \sqrt{\dfrac{m\omega}{2\hbar}}\left(x + \dfrac{i}{m\omega}p\right)\\ a^\dagger &= \sqrt{\dfrac{m\omega}{2\hbar}}\left(x - \dfrac{i}{m\omega}p\right) \end{align} $$

The operator a is not Hermitian since it is not equal to its adjoint, a†.

From the definition of the harmonic oscillator Hamiltonian,


 * $$\hat{H} = \dfrac{\hat{p}^2}{2m} + \dfrac{1}{2}m\omega^2x^2$$,

and the canonical commutation relation,


 * $$\left[\hat{x},\hat{p}\right] = i\hbar$$,

we can easily derive the relations



\begin{align} \left[H,x\right] &= -\dfrac{i\hbar p}{m},\\ \left[H,p\right] &= i\hbar m\omega^2x. \end{align} $$

From these we can obtain the results



\begin{align} &\left[H,a\right] = -\hbar\omega a,\\ &\left[H,a^\dagger\right] = +\hbar\omega a. \end{align} $$