Several coupled quantum oscillators

Two coupled quantum harmonic oscillators

Source: eigenchris video.
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The states live in a tensor product Hilbert space:

| Ψ (t) ⟩ \in H_{1} \otimes H_{2}

with basis of position given by

| q_{1}, q_{2} ⟩ = | q_{1} ⟩ \otimes | q_{2} ⟩

The operators of the individual oscillators promote to new operators:

{\hat{q}}_{1} ⟹ {\hat{q}}_{1} \otimes 1 {\hat{q}}_{2} ⟹ 1 \otimes {\hat{q}}_{2}

{\hat{p}}_{1} ⟹ {\hat{p}}_{1} \otimes 1 {\hat{p}}_{2} ⟹ 1 \otimes {\hat{p}}_{2}

The Hamiltonian is

\hat{H} = \frac{{\hat{p}}_{1}^{2}}{2 m} + \frac{1}{2} k {\hat{q}}_{1}^{2} + \frac{{\hat{p}}_{2}^{2}}{2 m} + \frac{1}{2} k {\hat{q}}_{2}^{2} + \frac{1}{2} κ ({\hat{q}}_{2} - {\hat{q}}_{1})^{2},

obtained by quantizing the classical coupled oscillators Hamiltonian.
We can express it in terms of the corresponding Ladder Operators with the change

{\hat{q}}_{i} \equiv \sqrt{\frac{ℏ}{2 m ω}} ({\hat{a}}_{i} + {\hat{a}}_{i}^{†}), {\hat{p}}_{i} \equiv \frac{i}{\sqrt{2}} \sqrt{ℏ m ω} ({\hat{a}}_{i} - {\hat{a}}_{i}^{†})

So the Hamiltonian becomes:

\hat{H} = ℏ ω ({\hat{a}}_{1}^{†} {\hat{a}}_{1} + {\hat{a}}_{2}^{†} {\hat{a}}_{2} + 1) - κ \frac{ℏ}{2 m ω} ({\hat{a}}_{1} + {\hat{a}}_{1}^{†}) ({\hat{a}}_{2} + {\hat{a}}_{2}^{†})

which is still very complicated.
If we perform a variable transformations into normal modes:

{\hat{q}}_{1} = \frac{1}{\sqrt{2}} ({\hat{Q}}_{1} + {\hat{Q}}_{2}), {\hat{q}}_{2} = \frac{1}{\sqrt{2}} ({\hat{Q}}_{1} - {\hat{Q}}_{2})

{\hat{p}}_{1} = \frac{1}{\sqrt{2}} ({\hat{P}}_{1} + {\hat{P}}_{2}), {\hat{p}}_{2} = \frac{1}{\sqrt{2}} ({\hat{P}}_{1} - {\hat{P}}_{2})

we obtain the transformed Hamiltonian:

\hat{H} = \frac{1}{2 m} {\hat{P}}_{1}^{2} + \frac{1}{2} k {\hat{Q}}_{1}^{2} + \frac{1}{2 m} {\hat{P}}_{2}^{2} + \frac{1}{2} (k + 2 κ) {\hat{Q}}_{2}^{2}

which corresponds to two decoupled quantum harmonic oscillators.

We can use now the corresponding ladder operators for the normal modes ${\hat{A}}_{k}, {\hat{A}}_{k}^{†}$ for these decoupled oscillators:

{\hat{Q}}_{k} \equiv \sqrt{\frac{ℏ}{2 m Ω_{k}}} ({\hat{A}}_{k} + {\hat{A}}_{k}^{†}), {\hat{P}}_{k} \equiv \frac{i}{\sqrt{2}} \sqrt{ℏ m Ω_{k}} ({\hat{A}}_{k} - {\hat{A}}_{k}^{†})

to obtain the Hamiltonian:

\hat{H} = ℏ Ω_{1} ({\hat{A}}_{1}^{†} {\hat{A}}_{1} + \frac{1}{2}) + ℏ Ω_{2} ({\hat{A}}_{2}^{†} {\hat{A}}_{2} + \frac{1}{2})

Infinite chain of quantum oscillators

Coming from several coupled oscillators#Infinite discrete chain of oscillators.
To transition from the classical description to the quantum version of the infinite discrete chain of oscillators, we promote the normal coordinates $Q_{k} (t)$ and their conjugate momenta to quantum operators. Each normal coordinate $Q_{k} (t)$ behaves as an independent harmonic oscillator. In quantum mechanics, the corresponding classical variables become operators satisfying the commutation relations:

[{\hat{Q}}_{k}, {\hat{P}}_{k^{'}}] = i ℏ δ_{k, k^{'}},

where ${\hat{P}}_{k}$ is the conjugate momentum of ${\hat{Q}}_{k}$ .
To express the quantum theory in terms of creation and annihilation operators, we define:

{\hat{Q}}_{k} = \sqrt{\frac{ℏ}{2 m ω_{k}}} ({\hat{A}}_{k} + {\hat{A}}_{- k}^{†}),

{\hat{P}}_{k} = - i \sqrt{\frac{ℏ m ω_{k}}{2}} ({\hat{A}}_{k} - {\hat{A}}_{- k}^{†}) .

The classical Hamiltonian for the system is:

H = \frac{1}{2} \int_{- π}^{π} (| P_{k} |^{2} + ω_{k}^{2} | Q_{k} |^{2}) d k .

After quantization, the Hamiltonian becomes:

\hat{H} = \frac{1}{2} \int_{- π}^{π} ({\hat{P}}_{k}^{†} {\hat{P}}_{k} + ω_{k}^{2} {\hat{Q}}_{k}^{†} {\hat{Q}}_{k}) d k .

Substituting the expressions for ${\hat{Q}}_{k}$ and ${\hat{P}}_{k}$ in terms of ladder operators, we obtain:

\hat{H} = \int_{- π}^{π} ℏ ω_{k} ({\hat{A}}_{k}^{†} {\hat{A}}_{k} + \frac{1}{2}) d k .

This is the Hamiltonian for a collection of independent quantum harmonic oscillators, each labeled by the wave number $k$ .
It can be shown to exist a so called vacuum state $| 0 ⟩$ , defined as the state annihilated by all annihilation operators:

{\hat{A}}_{k} | 0 ⟩ = 0 \forall k .

The vacuum state represents the ground state of the system, where no quanta (phonons) are present.

Excited states of the system are constructed by applying creation operators ${\hat{a}}_{k}^{†}$ to the vacuum state. For example:

A single-phonon state with wave number $k$ is: $| k ⟩ = {\hat{A}}_{k}^{†} | 0 ⟩ .$
A multi-phonon state is: $| k_{1}, k_{2}, \dots, k_{n} ⟩ = {\hat{A}}_{k_{1}}^{†} {\hat{A}}_{k_{2}}^{†} \dots {\hat{A}}_{k_{n}}^{†} | 0 ⟩ .$

The operators ${\hat{A}}_{k}^{†}$ and ${\hat{A}}_{k}$ create and annihilate phonons with wave number $k$ , a kind of "particle of pure momentum and no position", analogous to photons in quantum electrodynamics. They let us to establish an initial setup whose time evolution is given by Schrödinger equation.

Finally, recall that in the classical setup

q_{j} = \int_{- π}^{π} Q_{k} e^{i j k} d k,

{\hat{q}}_{j} = \int_{- π}^{π} {\hat{Q}}_{k} e^{i j k} d k .

Substituting ${\hat{Q}}_{k}$ we obtain:

{\hat{q}}_{j} = \int_{- π}^{π} \sqrt{\frac{ℏ}{2 m ω_{k}}} ({\hat{A}}_{k} + {\hat{A}}_{- k}^{†}) e^{i j k} d k .

And then

{\hat{q}}_{j} = \int_{- π}^{π} \sqrt{\frac{ℏ}{2 m ω_{k}}} {\hat{A}}_{k} e^{i j k} d k + \int_{- π}^{π} \sqrt{\frac{ℏ}{2 m ω_{k}}} {\hat{A}}_{- k}^{†} e^{i j k} d k,

and changing the variable $k \to - k$ in the second integral, while taking into account that $ω_{- k} = ω_{k}$ because of the dispersion relation being symmetric, we get:

{\hat{q}}_{j} = \int_{- π}^{π} \sqrt{\frac{ℏ}{2 m ω_{k}}} {\hat{A}}_{k} e^{i j k} d k + \int_{- π}^{π} \sqrt{\frac{ℏ}{2 m ω_{k}}} {\hat{A}}_{k}^{†} e^{- i j k} d k,

and finally

{\hat{q}}_{j} = \int_{- π}^{π} \sqrt{\frac{ℏ}{2 m ω_{k}}} ({\hat{A}}_{k} e^{i j k} + {\hat{A}}_{k}^{†} e^{- i j k}) d k .