Quantum harmonic oscillator

A **quantum harmonic oscillator** is a quantum mechanical system comprising a single, nonrelativistic particle moving in a potential analogous to the classical harmonic oscillator. The Hamiltonian operator for a quantum harmonic oscillator is

H
^

=

p
^

2

2
m

1
2

m

ω
2

x
^

2

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

, where m is the mass of the particle and ω is the angular frequency at which it oscillates.

## Energy eigenstates

The energy eigenstates of the quantum harmonic oscillator can be found by solving the time-independent Schrödinger equation,

−

ℏ
2

2
m

∂
2

ψ

∂

x
2

=

(

E
−

1
2

m

ω
2

x
2

)

ψ

{\displaystyle -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} = \left( E - \frac{1}{2} m \omega^2 x^2 \right) \psi}

, and applying the boundary condition that the probability amplitude must be zero at plus and minus infinity and that the integral of the probability over all space is one.

The solutions to this are

ψ
n

(
x
# )

1

2
n

n
!

(

m
ω

π
ℏ

)

1

/

4

e

−

m
ω

x
2

2
ℏ

H
n

(

m
ω

ℏ

x

)

{\displaystyle \psi\_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} e^{
- \frac{m\omega x^2}{2 \hbar}} H\_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right)}

, where

H
n

{\displaystyle H\_n}

 are the Hermite polynomials. The energies corresponding with these eigenfunctions are

E
n

=
ℏ
ω

(

1
2

n

)

{\displaystyle E\_n = \hbar \omega \left( \frac{1}{2} + n\right)}

.

## Ladder operators

Solving the Schrödinger equation directly for the quantum harmonic oscillator is difficult, and can be done more easily with the harmonic oscillator raising operator and harmonic oscillator lowering operator. These are written as:

* Raising operator

a
^

=

m
ω

2
ℏ

(

x
^

i

m
ω

p
^

)

{\displaystyle \hat a = \sqrt{ \frac{m \omega}{2 \hbar} } \left( \hat x + \frac{i}{m \omega} \hat p \right)}
* Lowering operator

a
^

†

=

m
ω

2
ℏ

(

x
^

−

i

m
ω

p
^

)

{\displaystyle \hat{a}^\dagger = \sqrt{ \frac{m \omega}{2 \hbar} } \left( \hat x - \frac{i}{m \omega} \hat p \right)}

These have the properties that

H
^

a
^

∣
n
# ⟩

(

n
1

)

a
^

∣
n
⟩

{\displaystyle \hat H \hat a \mid n \rangle = \left( n+1 \right) \hat a \mid n \rangle}

 and

H
^

a
^

†

∣
n
# ⟩

(

n
−
1

)

a
^

†

∣
n
⟩

{\displaystyle \hat H \hat{a}^\dagger \mid n \rangle = \left( n-1 \right) \hat{a}^\dagger \mid n \rangle}

; in other words, applying the raising or lowering operator on an energy eigenstate of the harmonic oscillator gives another energy eigenstate with the corresponding eigenvalue being ħω higher or lower, potentially multiplied by some constant.

These constants mean

a
^

∣
n
# ⟩

n

∣
n
1
⟩

{\displaystyle \hat a \mid n \rangle = \sqrt{n} \mid n + 1 \rangle}

, and

a
^

†

∣
n
# ⟩

n
−
1

∣
n
−
1
⟩

{\displaystyle \hat{a}^\dagger \mid n \rangle = \sqrt{n-1} \mid n - 1 \rangle}

. Two properties that come from this are:

* a
^

a
^

†

∣
n
# ⟩

N
^

∣
n
# ⟩
n
∣
n
⟩

{\displaystyle \hat a \hat{a}^\dagger \mid n \rangle = \hat N \mid n \rangle = n \mid n \rangle}

, the harmonic oscillator number operator

* a
^

†

∣
0
# ⟩
0

{\displaystyle \hat{a}^\dagger \mid 0 \rangle = 0}

; the lowering operator applied to the ground state gives zero

This second property lets us find the ground state and all higher states; these are determined by repeatedly applying the raising operator to 0.