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finished prob 11 chap 9

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@ -39,19 +39,17 @@ The expecation values ❬x̂❭ and ❬p̂❭ are of interest.
⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠ ⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠
❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞ x̂ ⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞. ❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞ x̂ ⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞.
⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠ ⎝ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠ ⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠ ⎝ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠
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. 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.
x̂ ≐ √⎛_͟ħ͟ ⎞⎛ 0 √1 ⎞ p̂ ≐ √⎛͟ħ͟m͟ω͟⎞⎛ 0 -ι√1 ⎞
⎝2mω⎠⎝ √1 0 ⎠, ⎝ 2 ⎠⎝ ι√1 0 ⎠.
x̂ ≐ p̂ ≐ 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.
⎛ 0 √1 ⎞ ⎛ 0 -ι√1 ⎞
⎝ √1 0 ⎠, ⎝ ι√1 0 ⎠.
❬0❙x̂❙0❭ = x₀₀ = ❬1❙x̂❙1❭ = x₁₁ = 0.
❬0❙x̂❙1❭ = x₀₁ = ❬1❙x̂❙0❭ = x₁₀ = √1 = 1.
❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙x̂❙0❭ + ⎞ ❬x̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙x̂❙0❭ + ⎞
⎜ ⎝ ħ ⎠ ⎝ ħ ⎠ ⎟ ⎜ ⎝ ħ ⎠ ⎝ ħ ⎠ ⎟
@ -64,20 +62,19 @@ The matrix representations of the position operator x̂ and momentum operator p
⎜ ⎟ ⎜ ⎟
⎜ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙x̂❙1❭⎞⎟ ⎜ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙x̂❙1❭⎞⎟
⎝ ⎝ ⎝ ħ 2⎠⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠⎠. ⎝ ⎝ ⎝ ħ 2⎠⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠⎠.
❬0❙x̂❙0❭ = x₀₀ = ❬1❙x̂❙1❭ = x₁₁ = 0.
❬0❙x̂❙1❭ = x₀₁ = ❬1❙x̂❙0❭ = x₁₀ = √⎛_͟ħ͟ ⎞.
⎝2mω⎠
Substituting the matrix elements: Substituting the matrix elements:
❬x̂❭ = ⅖⎛exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + ⎞ = ⅖⎛exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + ⎞ ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ ⎞
⎜ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎟ ⎜ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎟ ⎝mω⎠⎝ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎝ ħ ⎝ ħ 2⎠⎠ ⎠;
⎜ ⎟ ⎜ ⎟
⎜ exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞⎟ ⎜ exp⎛ι͟⎛⎛E͟₁͟−͟E͟₀͟⎞ - _͟π͟⎞⎞⎟
⎝ ⎝ ħ ⎝ ħ 2⎠⎠⎠ ⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠⎠;
❬x̂❭ = ⅖⎛ exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞ ⎞
⎝ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎝ ħ ⎝ ħ 2⎠⎠ ⎠;
❬x̂❭ = ⅖⎛ exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛E͟₁͟−͟E͟₀͟⎞t - _͟π͟⎞⎞ ⎞ ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛E͟₁͟−͟E͟₀͟⎞t - _͟π͟⎞⎞ ⎞
⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎠. ⎝mω⎠⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎠.
For the Harmonic Oscillator, Eₙ = ħω(n + ½). For the Harmonic Oscillator, Eₙ = ħω(n + ½).
@ -85,15 +82,75 @@ For the Harmonic Oscillator, Eₙ = ħω(n + ½).
E₀−E₁ = ħω(0 - 1) = -ħω; E₀−E₁ = ħω(0 - 1) = -ħω;
E₁−E₀ = ħω. E₁−E₀ = ħω.
❬x̂❭ = ⅖⎛ exp⎛ι⎛⎛−͟ħ͟ω͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞ ⎞ ❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞⎛ exp⎛ι⎛⎛−͟ħ͟ω͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞ ⎞
⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎠. ⎝mω⎠⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎠.
❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟)⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - π⎞⎞ ⎞
⎝mω⎠ 2 ⎝ ⎝ ⎝ ħ ⎠ ⎠ ⎝ ⎝⎝ ħ⎠ ⎠⎠ ⎠.
❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟)⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ - exp⎛ι⎛ħ͟ω͟⎞t⎞ ⎞
⎝mω⎠ 2 ⎝ ⎝ ⎝ ħ ⎠ ⎠ ⎝ ⎝ ħ⎠ ⎠ ⎠.
This is a sine function.
❬x̂❭ = ⅕ √⎛͟2͟͟ħ͟⎞ exp(ι͟π͟) 2ι sin⎛−͟ħ͟ω͟t͟⎞ = ⅕ √⎛͟8͟ħ͟⎞ ι² -sin(ωt).
⎝mω⎠ 2 ⎝ ħ ⎠ ⎝mω⎠
(𝐜,x̂) (𝐜,x̂)
❬x̂❭ = ⅖⎛ ι(-ωt + _͟π͟)⎞ + exp⎛ι⎛ωt - _͟π͟⎞⎞ ⎞ ❬x̂❭ = ⅕ √⎛͟8͟ħ͟⎞ sin(ωt).
⎝ e 2 ⎠ ⎝ ⎝ 2⎠⎠ ⎠. ⎝mω⎠
The expectation value progresses with the time parameter t. ω is a characteristic parameter of the system. It is related to the steepness of the parabolic potential curve. The expectation value progresses periodically with the time parameter t.
A very similar argument can be made for the momentum operator.
❬p̂❭ = ⅕⎛exp⎛ι͟E͟₀͟t⎞ exp⎛-ι͟E͟₀͟t⎞❬0❙p̂❙0❭ + ⎞
⎜ ⎝ ħ ⎠ ⎝ ħ ⎠ ⎟
⎜ ⎟
⎜ exp⎛ι͟E͟₀͟t⎞ 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞ ❬0❙p̂❙1❭ + ⎟
⎜ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎟
⎜ ⎟
⎜ exp⎛-ι͟E͟₀͟t⎞ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞ ❬1❙p̂❙0❭ + ⎟
⎜ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎟
⎜ ⎟
⎜ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙p̂❙1❭⎞⎟
⎝ ⎝ ⎝ ħ 2⎠⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠⎠.
❬0❙p̂❙0❭ = p₀₀ = ❬1❙p̂❙1❭ = p₁₁ = 0.
❬0❙p̂❙1❭ = p₀₁ = -ι√⎛͟ħ͟m͟ω͟⎞.
⎝ 2 ⎠
❬1❙p̂❙0❭ = p₁₀ = ι√⎛͟ħ͟m͟ω͟⎞.
⎝ 2 ⎠
❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞⎛-exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + exp⎛-ι͟E͟₀͟t + ι⎛E͟₁͟t - _͟π͟⎞⎞
⎝ 2 ⎠⎜ ⎝ ħ ⎝ 2 ħ ⎠⎠ ⎝ ħ ⎝ ħ 2⎠⎠.
❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞⎛exp⎛ι⎛⎛−͟ħ͟ω͟⎞t - _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞⎞
⎝ 2 ⎠⎝ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠⎠.
❬p̂❭ = ⅖ ι √⎛͟ħ͟m͟ω͟⎞ exp⎛−͟ι͟π͟⎞ ⎛ exp⎛ι⎛−͟ħ͟ω͟⎞t⎞ + exp⎛ι⎛ħ͟ω͟⎞t⎞ ⎞
⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ⎝ ⎝ ħ⎠ ⎠ ⎝ ⎝ ħ⎠ ⎠ ⎠.
This is a cosine.
(𝐜,p̂)
❬p̂❭ = ⅘ ι √⎛͟ħ͟m͟ω͟⎞ exp⎛−͟ι͟π͟⎞ cos(ωt) = ⅕ √(8ħmω) cos(ωt).
⎝ 2 ⎠ ⎝ 2 ⎠
Ehrenfest's theorem states
❬p̂❭ = m d͟❬͟x̂͟❭͟.
dt
m d͟❬͟x̂͟❭͟ = ⅕ mω √⎛͟8͟ħ͟⎞ cos(ωt) = ⅕ √(8mωħ) cos(ωt) = ❬p̂❭.
dt ⎝mω⎠
So, the theorem holds for this case.

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