The way in which a [Harmonic Oscillator State Vector](Harmonic%20Oscillator%20State%20Vector.md) is transformed into a lower or higher energy state is through the application of a _raising operator_ $\hat{a}^\dagger$ or a _lowering_ operator $\hat{a},$ which are referred to as [ladder operators](Ladder%20operators.md).
Given a Harmonic oscillator state $|n\rangle$,
$\hat{a}|n\rangle=\sqrt{n}|n-1\rangle$
and
$\hat{a}^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle$
and from these expressions follow the [number operator](number%20operator%20of%20a%20Harmonic%20Oscillator.md),
# Commutation relation
The [commutation relation](Commutators%20in%20quantum%20mechanics.md) is given as $[\hat{a},\hat{a}^\dagger]=1.$
The commutation relation needs to follow from the [position-momentum representation](Harmonic%20Oscillator%20Ladder%20Operators.md#Proof%20of%20the%20commutation%20relation) below. Its proof is given [here](Harmonic%20Oscillator%20Ladder%20Operators.md#Proof%20of%20the%20commutation%20relation).
# As bosonic creation and annihilation operators
This notion of ladder operators is equivalent to so called [creation and annihilation operations](creation%20and%20annihilation%20operators.md) that give rise to [bosonic](Boson.md) [fock states.](fock%20state.md)
# Correspondence with position and momentum representations
In situations where we want to apply the ladder operators in [position and momentum representations](Harmonic%20Oscillator%20State%20Vector.md#Position%20and%20momentum%20representations) of harmonic oscillator state vectors we have to rewrite the [ladder operators](Harmonic%20Oscillator%20Ladder%20Operators.md).
## position space representation
The [harmonic oscillator ladder operator](Harmonic%20Oscillator%20Ladder%20Operators.md) in terms of the [position](Position%20Operator.md) and [Momentum Operator](Momentum%20Operator.md) For a [[quantum harmonic oscillator Hamiltonian]] are expressed below. The following operators are applied to [position space](Harmonic%20Oscillator%20State%20Vector.md#Position%20and%20momentum%20representations) harmonic oscillator states (or [wave function](Stationary%20quantum%20harmonic%20oscillator%20wave%20function.md)s).
$\hat{a}=\frac{1}{(2m\hbar\omega)^{1/2}}(m\omega \hat{x}+i\hat{p}_x)$ ^13295f
$\hat{a}^{\dagger}=\frac{1}{(2m\hbar\omega)^{1/2}}(m\omega \hat{x}-i\hat{p}_x)$ ^7b40ce
Likewise by manipulating these expressions we can also rewrite the position and momentum operators as given [here](Harmonic%20Oscillator%20Position%20and%20Momentum%20Operators.md).
## momentum space representation
Similar to what we defined [above](Harmonic%20Oscillator%20Ladder%20Operators.md#position%20space%20representation) we may apply arising and lowering operators to the harmonic oscillator in [momentum space](Harmonic%20Oscillator%20State%20Vector.md#Position%20and%20momentum%20representations) with the following operators
$\hat{a}=\sqrt{\frac{m\omega}{2\hbar}}\bigg(i\hbar\frac{\partial}{\partial \hat{p}}+\frac{i}{m\omega}\hat{p}\bigg)$
and
$\hat{a}^{\dagger}=\sqrt{\frac{m\omega}{2\hbar}}\bigg(-i\hbar\frac{\partial}{\partial \hat{p}}-\frac{i}{m\omega}\hat{p}\bigg)$
These may be derived from the [position space representation](Harmonic%20Oscillator%20Ladder%20Operators.md#position%20space%20representation) using the [position-momentum commutation relation](Position-Momentum%20Commutators.md).
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# Proofs and Examples
## Proof of the commutation relation
Since position and momentum are [conjugate observables](conjugate%20observables%20in%20quantum%20mechanics.md) the [commutation relation](Harmonic%20Oscillator%20Ladder%20Operators.md#Commutation%20relation) must follow from the [position-momentum commutation relation](Position-Momentum%20Commutators.md) and we show this below:
$[\hat{a},\hat{a}^\dagger]=\frac{m\omega}{2\hbar}\bigg(\bigg(\hat{x}+\frac{i\hat{p}}{m\omega}\bigg)\bigg(\hat{x}-\frac{i\hat{p}}{m\omega}\bigg)-\bigg(\hat{x}-\frac{i\hat{p}}{m\omega}\bigg)\bigg(\hat{x}+\frac{i\hat{p}}{m\omega}\bigg)\bigg)$$=\frac{m\omega}{2\hbar}\bigg(\hat{x}^2-i\frac{\hat{x}\hat{p}}{m\omega}+i\frac{\hat{p}\hat{x}}{m\omega}+\frac{\hat{p}^2}{m^2\omega^2}-\hat{x}^2-i\frac{\hat{x}\hat{p}}{m\omega}+i\frac{\hat{p}\hat{x}}{m\omega}-\frac{\hat{p}^2}{m^2\omega^2}\bigg)$$=\frac{m\omega}{2\hbar}\bigg(-2i\frac{\hat{x}\hat{p}}{m\omega}+2i\frac{\hat{p}\hat{x}}{m\omega}\bigg)=\frac{i}{\hbar}[\hat{p},\hat{x}]=\frac{i}{\hbar}(-i\hbar)=1$
where in the last step we used the fact that $[\hat{p},\hat{x}]=-[\hat{x},\hat{p}]=-i\hbar,$ which enforces the canonical commutation relation.
#QuantumMechanics/StationaryStateQuantumSystems/QuantumHarmonicOscillators
#QuantumMechanics/QuantumDynamics
#QuantumMechanics/QuantumHarmonicOscillators