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 Incoming links: Harmonic-chain normal-mode reality condition

A periodic harmonic chain with

N

m_{0}

, spacing

a

and nearest-neighbour spring constant

K

has

H = \sum_{j = 0}^{N - 1} [\frac{p _{j}^{2}}{2 m _{0}} + \frac{K}{2} (u_{j + 1} - u_{j})^{2}], u_{N} = u_{0} .

m_{0} \overset{u}{¨}_{j} = K (u_{j + 1} + u_{j - 1} - 2 u_{j})

u_{j} = N^{- 1/2} \sum_{k} Q_{k} e^{ikja}

p_{j} = N^{- 1/2} \sum_{k} P_{k} e^{ikja}

, with

k = 2 π l / (N a)

ω_{k} = 2 K / m_{0} ∣ sin (ka /2) ∣, H = k \sum [\frac{P _{k} P _{- k}}{2 m _{0}} + \frac{m _{0} ω _{k}^{2}}{2} Q_{k} Q_{- k}] .

Q_{k}^{†} = Q_{- k}

, not that

Q_{k}

and

Q_{- k}

are independent real coordinates. For canonical quantization, impose

[u_{j}, p_{l}] = i ℏ δ_{j l}

, equivalently

[Q_{k}, P_{k^{'}}] = i ℏ δ_{k, - k^{'}}

. For

k \neq = 0

Q_{k} = \frac{ℏ}{2 m _{0} ω _{k}} (a_{k} + a_{- k}^{†}), P_{k} = - i \frac{m _{0} ℏ ω _{k}}{2} (a_{k} - a_{- k}^{†}), [a_{k}, a_{k^{'}}^{†}] = δ_{k k^{'}} .

H = \frac{P _{tot}^{2}}{2 N m _{0}} + k \neq = 0 \sum ℏ ω_{k} (a_{k}^{†} a_{k} + 1/2) .

X = N^{- 1} \sum_{j} u_{j}

, with

[X, P_{tot}] = i ℏ

; it cannot be normalized by dividing by

ω_{0}

. Retaining it gives

u_{j} = X + \frac{1}{N} \sum_{k \neq = 0} \frac{ℏ}{2 m _{0} ω _{k}} (a_{k} e^{ikja} + a_{k}^{†} e^{- ikja}),

with the corresponding momentum expansion and uniform term

P_{tot} / N

. The nonzero modes alone give the site commutator

i ℏ (δ_{j l} - 1/ N)

; the uniform term supplies the missing

1/ N

. Each oscillator occupation is a nonnegative integer, and adding one phonon costs

ℏ ω_{k}

ℏ k

ρ = m_{0} / a

Y = K a

and

L = \frac{1}{2} \int d x [ρ \overset{u}{˙}^{2} - Y (\partial_{x} u)^{2}], Π = ρ \overset{u}{˙}, [u (x), Π (x^{'})] = i ℏ δ (x - x^{'}), c = Y / ρ = a K / m_{0} .

The wave equation and linear low-

k

Z_{osc} = \prod_{k \neq = 0} \frac{1}{2 s i n h ( β ℏ ω _{k} /2 )}, ⟨ a_{k}^{†} a_{k} ⟩ = \frac{1}{e ^{β ℏ ω_{k}} - 1} .

For a long chain, above its finite-size level spacing and below the zone-edge scale, integrating the two acoustic propagation directions gives

C / (N a) = π k_{B}^{2} T / (3ℏ c)

. At high temperature each retained oscillator instead contributes

k_{B}