Harmonic oscillators, creation and annihilation operators

A.4 Harmonic oscillators, creation and annihilation operators

For a particle of mass $m$ in a one-dimensional harmonic oscillator potential $\frac{1}{2} k x^{2} \equiv \frac{1}{2} m ω^{2} x^{2}$ where $ω$ is the classical frequency of oscillation, the Hamiltonian is

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

The energy levels are $E_{n} = (n + \frac{1}{2}) ℏ ω$ , $n \geq 0$ , and, deﬁning the length scale $x_{0} = \sqrt{ℏ ∕ m ω}$ , the wave functions are

ϕ_{0} (x) = {(π x_{0}^{2})}^{- 1 ∕ 4} e^{- x^{2} ∕ 2 x_{0}^{2}} ϕ_{n} (x) = \frac{1}{\sqrt{2^{n} n!}} H_{n} (\frac{x}{x_{0}}) ϕ_{0} (x)

where the Hermite polynomials are $H_{0} (z) = 1; H_{1} (z) = 2 z; H_{2} (z) = 4 z^{2} - 2; H_{3} (z) = 8 z^{3} - 12 z; H_{4} (z) = 16 z^{4} - 48 x^{2} + 12$ . The Mathematica functions for obtaining them are HermiteH[n, z].

In bra-ket notation, we will represent the state with quantum umber $n$ as $| n ⟩$ , with $ϕ_{n} (x) = ⟨ x | n ⟩$ .

If we deﬁne the annihilation and creation operators

â = \frac{1}{\sqrt{2}} (\frac{\hat{x}}{x_{0}} + i \frac{x_{0}}{ℏ} \hat{p}) and â^{†} = \frac{1}{\sqrt{2}} (\frac{\hat{x}}{x_{0}} - i \frac{x_{0}}{ℏ} \hat{p})

which satisfy $[â, â^{†}] = 1$ , we have

\hat{H} = ℏ ω (â^{†} â + \frac{1}{2}) [â, H] = ℏ ω â and [â^{†}, H] = - ℏ ω â^{†}

from which it follows that

â | n ⟩ = \sqrt{n} | n - 1 ⟩ and â^{†} | n ⟩ = \sqrt{n + 1} | n + 1 ⟩ .

This suggests an interpretation of the states of the system in which the quanta of energy are primary, with $â^{†}$ and $â$ respectively creating and annihilating a quantum of energy. Further notes on creation and annihilation operators can be found here.

For a particle in a two-dimensional potential $\frac{1}{2} m ω_{x}^{2} x^{2} + \frac{1}{2} m ω_{y}^{2} y^{2}$ , we deﬁne $x_{0} = \sqrt{ℏ ∕ m ω_{x}}$ and $y_{0} = \sqrt{ℏ ∕ m ω_{y}}$ , and the wavefunction of the particle will be determined by two quantum numbers $n_{x}$ and $n_{y}$

\begin{array}{rcl} ϕ_{0, 0} (x, y) & = & {(π x_{0} y_{0})}^{- 1 ∕ 2} e^{- x^{2} ∕ 2 x_{0}^{2}} e^{- y^{2} ∕ 2 y_{0}^{2}} \\ ϕ_{n_{x}, n_{y}} (x, y) & = & \frac{1}{\sqrt{2^{n_{x}} n_{x}!}} H_{n_{x}} (\frac{x}{x_{0}}) \frac{1}{\sqrt{2^{n_{y}} n_{y}!}} H_{n_{y}} (\frac{y}{y_{0}}) ϕ_{0, 0} (x, y) \end{array}

In bra-ket notation, we will represent the state with quantum numbers $n_{x}$ and $n_{y}$ as $| n_{x}, n_{y} ⟩$

Creation operators $â_{x}$ and $â_{x}^{†}$ can be constructed from $\hat{x}$ and $\hat{p_{x}}$ as above, and we can construct a second set of operators $â_{y}$ and $â_{y}^{†}$ from $ŷ$ and $\hat{p_{y}}$ (using $y_{0}$ as the scale factor) in the same way. Then $â_{x}^{†}$ and $â_{y}^{†}$ act on $| n_{x}, n_{y} ⟩$ to increase $n_{x}$ and $n_{y}$ respectively, and $â_{x}$ and $â_{y}$ to decrease them, and both of the latter annihilate the ground state. So for instance

â_{x} | n_{x}, n_{y} ⟩ = \sqrt{n_{x}} | n_{x} - 1, n y ⟩ and â_{y}^{†} | n_{x}, n_{y} ⟩ = \sqrt{n_{y} + 1} | n_{x}, n_{y} + 1 ⟩ .

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