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157 lines
7.9 KiB
Plaintext
157 lines
7.9 KiB
Plaintext
The harmonic oscillator potential is V(x) = ½kx² = ½mω²x². The hamiltonian is time independent.
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The initial state vector is
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Ψ(t=0) = A⎛❙0❭ + 2exp(ι͟π͟)❙1❭⎞
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⎝ 2 ⎠
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Finding the normalization constant A is simple enough, since
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1 = A² + (2Aexp(ι͟π͟))² = A²(1 + (2exp(ι͟π͟))²).
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2 2
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A = √⎛ ______͟1͟______ ⎞ = √⎛ ________͟1͟_______ ⎞
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⎜ 1 + (2exp(ι͟π͟))² ⎟ ⎜ 1 + 4exp(ι͟π͟ - ι͟π͟ ) ⎟,
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⎝ 2 ⎠ ⎝ 2 2 ⎠
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(𝐚)
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A = √⎛ ____͟1͟____ ⎞ = √⅕.
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⎝ 1 + 4exp(0) ⎠
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The time evolution prescription for a time-independent hamiltonian is
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Ψ(t) = exp(-ι͟E͟ₙ͟t) Ψ(t=0);
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ħ
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Ψ(t) = exp(-ι͟E͟ₙ͟t) √⅕⎛❙0❭ + 2exp(ι͟π͟)❙1❭⎞.
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ħ ⎝ 2 ⎠
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(𝐛)
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Ψ(t) = √⅕⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞.
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⎝ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠
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The expecation values ❬x̂❭ and ❬p̂❭ are of interest.
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❬x̂❭ = ❬Ψ⃰(t)❙x̂❙Ψ(t)❭.
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Ψ⃰(t) = √⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞.
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⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠
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❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞ x̂ ⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞.
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⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠ ⎝ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠
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The matrix representations of the position operator x̂ and momentum operator p̂ have been developed from the definition of the increment/decrement operators. The matrix elements may be ascertained by inspection.
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x̂ ≐ √⎛_͟ħ͟ ⎞⎛ 0 √1 ⎞ p̂ ≐ √⎛͟ħ͟m͟ω͟⎞⎛ 0 -ι√1 ⎞
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⎝2mω⎠⎝ √1 0 ⎠, ⎝ 2 ⎠⎝ ι√1 0 ⎠.
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ω is a characteristic parameter of the system. It is related to the steepness of the parabolic potential curve. m is the particle's mass.
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❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙x̂❙0❭ + ⎞
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⎜ ⎝ ħ ⎠ ⎝ ħ ⎠ ⎟
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⎜ ⎟
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⎜ exp⎛ι͟E͟₀͟t⎞ 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞ ❬0❙x̂❙1❭ + ⎟
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⎜ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎟
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⎜ ⎟
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⎜ exp⎛-ι͟E͟₀͟t⎞ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞ ❬1❙x̂❙0❭ + ⎟
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⎜ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎟
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⎜ ⎟
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⎜ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙x̂❙1❭⎞⎟
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⎝ ⎝ ⎝ ħ 2⎠⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠⎠.
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❬0❙x̂❙0❭ = x₀₀ = ❬1❙x̂❙1❭ = x₁₁ = 0.
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❬0❙x̂❙1❭ = x₀₁ = ❬1❙x̂❙0❭ = x₁₀ = √⎛_͟ħ͟ ⎞.
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⎝2mω⎠
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Substituting the matrix elements:
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❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ ⎞
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⎝mω⎠⎝ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎝ ħ ⎝ ħ 2⎠⎠ ⎠;
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❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛E͟₁͟−͟E͟₀͟⎞t - _͟π͟⎞⎞ ⎞
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⎝mω⎠⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎠.
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For the Harmonic Oscillator, Eₙ = ħω(n + ½).
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Eₙ−Eₙ′ = ħω(n - n′):
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E₀−E₁ = ħω(0 - 1) = -ħω;
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E₁−E₀ = ħω.
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❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι⎛⎛−͟ħ͟ω͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞ ⎞
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⎝mω⎠⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎠.
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❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟)⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - π⎞⎞ ⎞
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⎝mω⎠ 2 ⎝ ⎝ ⎝ ħ ⎠ ⎠ ⎝ ⎝⎝ ħ⎠ ⎠⎠ ⎠.
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❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟)⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ - exp⎛ι⎛ħ͟ω͟⎞t⎞ ⎞
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⎝mω⎠ 2 ⎝ ⎝ ⎝ ħ ⎠ ⎠ ⎝ ⎝ ħ⎠ ⎠ ⎠.
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This is a sine function.
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❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟) 2ι sin⎛−͟ħ͟ω͟t͟⎞ = ⅕ √⎛͟8͟ħ͟⎞ ι² -sin(ωt).
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⎝mω⎠ 2 ⎝ ħ ⎠ ⎝mω⎠
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(𝐜,x̂)
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❬x̂❭ = ⅕ √⎛͟8͟ħ͟⎞ sin(ωt).
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⎝mω⎠
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The expectation value progresses periodically with the time parameter t.
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A very similar argument can be made for the momentum operator.
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❬p̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙p̂❙0❭ + ⎞
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⎜ ⎝ ħ ⎠ ⎝ ħ ⎠ ⎟
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⎜ ⎟
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⎜ exp⎛ι͟E͟₀͟t⎞ 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞ ❬0❙p̂❙1❭ + ⎟
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⎜ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎟
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⎜ ⎟
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⎜ exp⎛-ι͟E͟₀͟t⎞ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞ ❬1❙p̂❙0❭ + ⎟
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⎜ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎟
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⎜ ⎟
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⎜ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙p̂❙1❭⎞⎟
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⎝ ⎝ ⎝ ħ 2⎠⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠⎠.
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❬0❙p̂❙0❭ = p₀₀ = ❬1❙p̂❙1❭ = p₁₁ = 0.
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❬0❙p̂❙1❭ = p₀₁ = -ι√⎛͟ħ͟m͟ω͟⎞.
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⎝ 2 ⎠
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❬1❙p̂❙0❭ = p₁₀ = ι√⎛͟ħ͟m͟ω͟⎞.
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⎝ 2 ⎠
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❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞⎛-exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞
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⎝ 2 ⎠⎜ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎝ ħ ⎝ ħ 2⎠⎠.
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❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞⎛exp⎛ι⎛⎛−͟ħ͟ω͟⎞t - _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞⎞
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⎝ 2 ⎠⎝ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠⎠.
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❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞ exp⎛−͟ι͟π͟⎞ ⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ + exp⎛ι⎛ħ͟ω͟⎞t⎞ ⎞
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⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ⎝ ⎝ ħ⎠ ⎠ ⎝ ⎝ ħ⎠ ⎠ ⎠.
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This is a cosine.
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(𝐜,p̂)
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❬p̂❭ = ⅘ ι √⎛͟ħ͟m͟ω͟⎞ exp⎛−͟ι͟π͟⎞ cos(ωt) = ⅕ √(8ħmω) cos(ωt).
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⎝ 2 ⎠ ⎝ 2 ⎠
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Ehrenfest's theorem states
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❬p̂❭ = m d͟❬͟x̂͟❭͟.
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dt
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m d͟❬͟x̂͟❭͟ = ⅕ mω √⎛͟8͟ħ͟⎞ cos(ωt) = ⅕ √(8mωħ) cos(ωt) = ❬p̂❭.
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dt ⎝mω⎠
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So, the theorem holds for this case.
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