mirror of
https://asciireactor.com/otho/phy-4600.git
synced 2025-01-18 19:25:04 +00:00
20 lines
564 B
Plaintext
20 lines
564 B
Plaintext
|
Translation Operators
|
|||
|
|
|
|||
|
|
Matrix mechanical expression:
|
|||
|
|
|
|||
|
|
T̂(a𝓍 î) ❙x,y,z❭ = ❙x+a𝓍,y,z❭
|
|||
|
|
T̂(a𝓎 î) ❙x,y,z❭ = ❙x,y+a𝓎,z❭
|
|||
|
|
T̂(a𝓏 î) ❙x,y,z❭ = ❙x,y,z+a𝓏❭
|
|||
|
|
|
|||
|
|
Position space form:
|
|||
|
|
|
|||
|
|
T(a𝓍 î) = exp(-ι p𝓍 a𝓍/ħ )
|
|||
|
|
|
|||
|
|
Translation operators commute since momentum operators commute.
|
|||
|
|
|
|||
|
|
But, [x̂ᵢ,p̂ⱼ]=ιħδᵢⱼ
|
|||
|
|
|
|||
|
|
T̂(𝐚) = exp(-ι p𝓍 a𝓍/ħ ) exp(-ι p𝓎 a𝓎/ħ ) exp(-ι p𝓏 a𝓏/ħ )
|
|||
|
|
= exp(-ι 𝐩 𝐚/ħ )
|
|||
|
|
|
|||
|
|
❬r❙p̂❙Ψ❭ = ħ/ι ∇ ❬r❙Ψ❭
|