mirror of
https://asciireactor.com/otho/phy-4600.git
synced 2025-01-19 02:15:05 +00:00
100 lines
5.1 KiB
Plaintext
100 lines
5.1 KiB
Plaintext
The harmonic oscillator potential is V(x) = ½kx² = ½mω²x². The hamiltonian is time independent.
|
||
|
||
The initial state vector is
|
||
|
||
Ψ(t=0) = A⎛❙0❭ + 2exp(ι͟π͟)❙1❭⎞
|
||
⎝ 2 ⎠
|
||
|
||
Finding the normalization constant A is simple enough, since
|
||
|
||
1 = A² + (2Aexp(ι͟π͟))² = A²(1 + (2exp(ι͟π͟))²).
|
||
2 2
|
||
|
||
A = √⎛ ______͟1͟______ ⎞ = √⎛ ________͟1͟_______ ⎞
|
||
⎜ 1 + (2exp(ι͟π͟))² ⎟ ⎜ 1 + 4exp(ι͟π͟ - ι͟π͟ ) ⎟,
|
||
⎝ 2 ⎠ ⎝ 2 2 ⎠
|
||
|
||
(𝐚)
|
||
A = √⎛ ____͟1͟____ ⎞ = √⅕.
|
||
⎝ 1 + 4exp(0) ⎠
|
||
|
||
|
||
The time evolution prescription for a time-independent hamiltonian is
|
||
|
||
Ψ(t) = exp(-ι͟E͟ₙ͟t) Ψ(t=0);
|
||
ħ
|
||
|
||
Ψ(t) = exp(-ι͟E͟ₙ͟t) √⅕⎛❙0❭ + 2exp(ι͟π͟)❙1❭⎞.
|
||
ħ ⎝ 2 ⎠
|
||
|
||
(𝐛)
|
||
Ψ(t) = √⅕⎛exp⎛-ι͟E͟₀͟t⎞❙0❭ + 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❙1❭⎞.
|
||
⎝ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠
|
||
|
||
The expecation values ❬x̂❭ and ❬p̂❭ are of interest.
|
||
|
||
❬x̂❭ = ❬Ψ⃰(t)❙x̂❙Ψ(t)❭.
|
||
|
||
Ψ⃰(t) = √⅕⎛exp⎛ι͟E͟₀͟t⎞❬0❙ + 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞❬1❙⎞.
|
||
⎝ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎠
|
||
|
||
❬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 ⎠⎠ ⎠
|
||
|
||
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 ≐ p̂ ≐
|
||
⎛ 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❭ + ⎞
|
||
⎜ ⎝ ħ ⎠ ⎝ ħ ⎠ ⎟
|
||
⎜ ⎟
|
||
⎜ exp⎛ι͟E͟₀͟t⎞ 2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞ ❬0❙x̂❙1❭ + ⎟
|
||
⎜ ⎝ ħ ⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎟
|
||
⎜ ⎟
|
||
⎜ exp⎛-ι͟E͟₀͟t⎞ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞ ❬1❙x̂❙0❭ + ⎟
|
||
⎜ ⎝ ħ ⎠ ⎝ ⎝ ħ 2⎠⎠ ⎟
|
||
⎜ ⎟
|
||
⎜ 2 exp⎛ι⎛E͟₁͟t - _͟π͟⎞⎞2 exp⎛ι⎛_͟π͟ - E͟₁͟t⎞⎞❬1❙x̂❙1❭⎞⎟
|
||
⎝ ⎝ ⎝ ħ 2⎠⎠ ⎝ ⎝ 2 ħ ⎠⎠ ⎠⎠.
|
||
|
||
Substituting the matrix elements:
|
||
|
||
❬x̂❭ = ⅖⎛exp⎛ι͟E͟₀͟t + ι⎛_͟π͟ - E͟₁͟t⎞⎞ + ⎞ = ⅖⎛exp⎛ι⎛⎛E͟₀͟−͟E͟₁͟⎞t + _͟π͟⎞⎞ + ⎞
|
||
⎜ ⎝ ħ ⎝ 2 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⎞⎞ ⎞
|
||
⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎠.
|
||
|
||
For the Harmonic Oscillator, Eₙ = ħω(n + ½).
|
||
|
||
Eₙ−Eₙ′ = ħω(n - n′):
|
||
E₀−E₁ = ħω(0 - 1) = -ħω;
|
||
E₁−E₀ = ħω.
|
||
|
||
❬x̂❭ = ⅖⎛ exp⎛ι⎛⎛−͟ħ͟ω͟⎞t + _͟π͟⎞⎞ + exp⎛ι⎛⎛ħ͟ω͟⎞t - _͟π͟⎞⎞ ⎞
|
||
⎝ ⎝ ⎝⎝ ħ ⎠ 2⎠⎠ ⎝ ⎝⎝ ħ⎠ 2⎠⎠ ⎠.
|
||
|
||
(𝐜,x̂)
|
||
❬x̂❭ = ⅖⎛ ι(-ωt + _͟π͟)⎞ + exp⎛ι⎛ωt - _͟π͟⎞⎞ ⎞
|
||
⎝ e 2 ⎠ ⎝ ⎝ 2⎠⎠ ⎠.
|
||
|
||
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.
|
||
|
||
|
||
|
||
|