量子力学（2.5） 生肉

Posted 2023-03-05 myjs999

uin tux Schrodinger Citon:
ih dΨ/dt=-h^2/2m d^2/(dx2 ) Ψ+V(x,t)Ψ
inim Ψ tuicon x hor t. sit fe moicon t, uin som’e ψ. Schrodinger Citon moicon t clre cuu:
Eψ=-h^2/2m d^2/(dx2 ) ψ+V(x)ψ
inim E bid de Kitn da de Dawses.
bojs p=-ih d/dx, uin poc dy als e:
Eψ=ψ/2m+V(x)ψ
sit uin ne Som H, hen
H=p^2/2m+V(x)
lme Nada’Banap Fuine, V(x)=1/2 mω^2 x^2, hen
H=1/2m (p^2+m2 ω^2 x^2 )
kam kt x hor p inim imi Boc, dewoa Pade. uin boj Pade topiep.
uin kam a_+≔1/√2ℏmω (-ip+mωx), a_-≔1/√2ℏmω (ip+mωx). kt a_+ hor a_- cin duo li Suoc uc ip.
por uin inboc a_- a_+=1/2ℏmω (p^{2+imωpx-imωxp+m}2 ω^2 x^2 )=H/ℏω+i/2ℏ[p,x].
uin boj [p,x]=-iℏ. hen a_- a_+=H/ℏω+1/2.
inbocs a_+ a_- uin gocha va =H/ℏω-1/2. mielo [a_-,a_+ ]=1.
uin tui kt Hψ=Eψ.
inbocs uin gocha H(a_+ ψ)=(E+ℏω)(a_+ ψ), hor H(a_- ψ)=(E-ℏω)(a_- ψ).
uin di’em Taibon Pade, cuilo Anade hor Mhoade.
ψ_0 gol a_- ψ_0=0.
inbocs’e gochauim ψ_0 (x)=(mω/πℏ)^(1/4) e^(-mω/2ℏ x^2 ).
ψ_n (x)=A_n a_+^n ψ_0.E_n (x)=ℏω(n+1/2)

e Maobom uin boj A_n=1/√n!.

Dirac Delta
uin ondi boj kt δ a mmdd li 0, hor a 0 li timp. dewoa, ∫δ(x)dx=1.
uin ondi boj kt ∫f(x)δ(x-a)dx=∫f(a)δ(x-a)dx=f(a)∫δ(x)dx=f(a).

uin ondi boj kt da Dawses’damo ba Wond(di’e Wonddawses),
Heigoc ondi lini
(da Heigoc’nacit,) Heises tonakog
Heises Wo e Nii d woda(poc ana van Dawses)

uin tux ri Boc. Boc cuu 0, 1, 2, 3, -1, -2, zo Kasiboc(Bociboc), fem d kasi(boci).
Boc cuu 1/2 zo Clreinshboc, fem d clrein.
Boc cuu Pi, E, zo Liniboc, fem d lini(roap).
Boc cuu (1+i) zo Roc(Kuxiboc). fem d kuxi(roa).
mn ma Galciboc, Muukachboc.

Problem 11, P116(chrome)

Avlo, uin boja Roc z ma kt |z|^2≥(Im(z))2=(1/2i (z-z^* ))^2. gro poc mak fe ba Pei dablo duilo.
uin ondi mak kt σ_A^2 σ_B^2=⟨A-⟨A⟩│A-⟨A⟩ ⟩⟨B-⟨B⟩│B-⟨B⟩ ⟩
≥⟨A-⟨A⟩│B-⟨B⟩ ⟩^2
≥(1/2i (⟨A-⟨A⟩│B-⟨B⟩ ⟩-⟨B-⟨B⟩│A-⟨A⟩ ⟩))^2
≥(1/2i (⟨AB⟩-⟨B⟩⟨A|-⟨A⟩|B⟩+⟨A⟩⟨B⟩-⟨BA⟩+⟨A⟩⟨B|+⟨B⟩|A⟩-⟨A⟩⟨B⟩))^2
=(1/2i (⟨AB⟩-⟨BA⟩))^2

=(1/2i ⟨[A,B]⟩)^2
(inim lawar kt ⟨A│B⟩=⟨AB⟩. nour fumes?)
Bingo!