adding to rotation lecture notes

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caes 2016-03-16 19:06:42 -04:00
parent a9a0d33b50
commit a41c0748e2

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@ -12,6 +12,7 @@ Rotational Invariance
= [1 - ι/ħ (x p𝓎 - y p𝓍)dϕ] ❙x,y,z❭ = [1 - ι/ħ (x p𝓎 - y p𝓍)dϕ] ❙x,y,z❭
(x p𝓎 - y p𝓍) ≝ L̂𝓏 (angular momentum in z axis) (x p𝓎 - y p𝓍) ≝ L̂𝓏 (angular momentum in z axis)
𝓏 is the z component of L̂ = 𝐫×𝐩̂ 𝓏 is the z component of L̂ = 𝐫×𝐩̂
@ -39,7 +40,9 @@ Rotational Invariance
= [L̂𝓏,p̂𝓍²] + [L̂𝓏,p̂𝓎²] + [L̂𝓏,p̂𝓏²] = [L̂𝓏,p̂𝓍²] + [L̂𝓏,p̂𝓎²] + [L̂𝓏,p̂𝓏²]
= p̂𝓍[L̂𝓏,p̂𝓍] + [L̂𝓏,p̂𝓍]p̂𝓍 + p̂𝓎[L̂𝓏,p̂𝓎] + [L̂𝓏,p̂𝓎]p̂𝓎 + p̂𝓏[L̂𝓏,p̂𝓏] + [L̂𝓏,p̂𝓏]p̂𝓏` = p̂𝓍[L̂𝓏,p̂𝓍] + [L̂𝓏,p̂𝓍]p̂𝓍
+ p̂𝓎[L̂𝓏,p̂𝓎] + [L̂𝓏,p̂𝓎]p̂𝓎
+ p̂𝓏[L̂𝓏,p̂𝓏] + [L̂𝓏,p̂𝓏]p̂𝓏 (f.n. 1)
= ιħp̂𝓍𝓎 + ιħp̂𝓎𝓍 - ιħp̂𝓎𝓍 - ιħp̂𝓍𝓎 = 0 = ιħp̂𝓍𝓎 + ιħp̂𝓎𝓍 - ιħp̂𝓎𝓍 - ιħp̂𝓍𝓎 = 0
@ -56,3 +59,31 @@ Rotational Invariance
[L̂²,Ĥ] = 0, so L̂² is also a constant of motion. [L̂²,Ĥ] = 0, so L̂² is also a constant of motion.
(f.n. 1)
[L̂𝓏,p̂𝓍²] = [L̂𝓏,p̂𝓍𝓍]
= L̂𝓏𝓍𝓍 - p̂𝓍𝓍𝓏
= ?
= p̂𝓍[L̂𝓏,p̂𝓍] + [L̂𝓏,p̂𝓍]p̂𝓍
= (p̂𝓍𝓏𝓍 - p̂𝓍𝓍𝓏) + (L̂𝓏𝓍𝓍 - p̂𝓍𝓏𝓍)
= L̂𝓏𝓍𝓍 - p̂𝓍𝓍𝓏
✓ = [L̂𝓏,p̂𝓍²] = [L̂𝓏,p̂𝓍𝓍]