follow ”Modern Quantum Mechanics“（Sakurai）

第一章算符

Hermitian Operator A=A†A = A^\dagA=A†, A=∑a∣a⟩a⟨a∣A = \sum_a |a\rangle a\langle a|A=∑a∣a⟩a⟨a∣
2*2 hermitian matrix, X=a0I+σ⃗⋅a⃗X=a_0I+\vec{\sigma}\cdot\vec{a}X=a0I+σ⋅a
expectancy value: ⟨A⟩=⟨ψ∣A^∣ψ⟩\langle A\rangle = \langle \psi|\hat{A}|\psi\rangle⟨A⟩=⟨ψ∣A^∣ψ⟩
Vector operator, ex. S⃗=Sxx^+Syy^+Szz^\vec{S}=S_x\hat{x}+S_y\hat{y}+S_z\hat{z}S=Sxx^+Syy^+Szz^; Scalar operator, ex. S2=Sx2+Sy2+Sz2S^2 = S_x^2+S_y^2+S_z^2S2=Sx2+Sy2+Sz2
if [A,B]=0[A,B]=0[A,B]=0, ⟨a′′∣B∣a′⟩=0\langle a''|B|a'\rangle=0⟨a′′∣B∣a′⟩=0
Schwarz inequality: ⟨α∣α⟩⟨β∣β⟩≥∣⟨α∣β⟩∣2\langle \alpha|\alpha\rangle\langle \beta|\beta\rangle\ge|\langle \alpha|\beta\rangle|^2⟨α∣α⟩⟨β∣β⟩≥∣⟨α∣β⟩∣2
the Uncertainty Relationship: ⟨(ΔA)2⟩+⟨(ΔB)2⟩≥14∣⟨[A,B]⟩∣2\langle (\Delta A)^2\rangle+\langle (\Delta B)^2\rangle\ge \frac14 |\langle [A,B]\rangle|^2⟨(ΔA)2⟩+⟨(ΔB)2⟩≥41∣⟨[A,B]⟩∣2
when ΔA∣⟩=λΔB∣⟩\Delta A|\rangle = \lambda\Delta B|\rangleΔA∣⟩=λΔB∣⟩, λ\lambdaλ is pure imaginary number, uncertainty ralationship equates.
Unitary operator: U†U=UU†=1U^\dag U = U U^\dag = 1U†U=UU†=1
Uintary operator and Hermitian relationship: U(ϵ)=1+iHϵU(\epsilon)=1+iH\epsilonU(ϵ)=1+iHϵ wher ϵ\epsilonϵ is infinitesmal translation
important
exp⁡(iGλ)Aexp⁡(−iGλ)=A+iλ[G,A]+(iλ)22![G,[G,A]]+…\exp(iG\lambda)A\exp(-iG\lambda)=A + i\lambda [G,A] + \frac{(i\lambda)^2}{2!}[G,[G,A]]+\dots exp(iGλ)Aexp(−iGλ)=A+iλ[G,A]+2!(iλ)2[G,[G,A]]+…
Translation operator and momentum operator: D(x′)=exp⁡(−ip^x′/ℏ)D(x')=\exp(-i\hat{p}x'/\hbar)D(x′)=exp(−ip^x′/ℏ), D(x′)∣x⟩=∣x+x′⟩D(x')|x\rangle = |x+x'\rangleD(x′)∣x⟩=∣x+x′⟩, ⟨x∣D(x′)=⟨x−x′∣\langle x|D(x')=\langle x-x'|⟨x∣D(x′)=⟨x−x′∣, p^=−iℏ∂x\hat{p} = -i\hbar\partial_xp^=−iℏ∂x
relationship between x and p: ⟨x′∣p∣x′′⟩=−iℏ∂x′δ(x′−x′′)\langle x'|p|x''\rangle=-i\hbar \partial_{x'}\delta(x'-x'')⟨x′∣p∣x′′⟩=−iℏ∂x′δ(x′−x′′); ⟨x′∣p′⟩=12πℏexp⁡(ip′x′/ℏ)\langle x'|p' \rangle=\frac{1}{\sqrt{2\pi\hbar}}\exp(ip'x'/\hbar)⟨x′∣p′⟩=2πℏ1exp(ip′x′/ℏ). 3-dimension, ⟨x⃗′∣p⃗′⟩=1(2πℏ)3/2exp⁡(ip⃗′⋅x⃗′/ℏ)\langle \vec{x}'|\vec{p}'\rangle= \frac{1}{(2\pi\hbar)^{3/2}}\exp(i \vec{p}'\cdot \vec{x}'/\hbar)⟨x′∣p′⟩=(2πℏ)3/21exp(ip′⋅x′/ℏ)
Ehrenfest’s Theorem [xi,G(p⃗)]=iℏ∂piG[x_i,G(\vec{p})]=i\hbar \partial_{p_i}G[xi,G(p)]=iℏ∂piG; [pi,F(x⃗)]=−iℏ∂xiF[p_i,F(\vec{x})]=-i\hbar\partial_{x_i}F[pi,F(x)]=−iℏ∂xiF
Gaussian Wave Packet: ⟨x∣ψ⟩=1π1/4dexp⁡[ikx′]exp⁡[−x′2/2d2]\langle x|\psi\rangle=\frac{1}{\pi^{1/4}\sqrt{d}}\exp[ikx']\exp[-x'^2/2d^2]⟨x∣ψ⟩=π1/4d1exp[ikx′]exp[−x′2/2d2], ⟨p∣ψ⟩=dℏπ1/4exp⁡[−(p′−ℏk)2d2/2ℏ2]\langle p|\psi\rangle = \frac{\sqrt{d}}{\sqrt{\hbar}\pi^{1/4}}\exp[-(p'-\hbar k)^2d^2/2\hbar^2]⟨p∣ψ⟩=ℏπ1/4dexp[−(p′−ℏk)2d2/2ℏ2],⟨x⟩=0\langle x\rangle=0⟨x⟩=0, ⟨x2⟩=d2/2\langle x^2\rangle=d^2/2⟨x2⟩=d2/2, ⟨p⟩=ℏk\langle p\rangle=\hbar k⟨p⟩=ℏk, ⟨p2⟩=ℏ2k2+ℏ2/2d2\langle p^2\rangle = \hbar^2k^2+\hbar^2/2d^2⟨p2⟩=ℏ2k2+ℏ2/2d2
coherent state: H=ℏω(a†a+1/2)H=\hbar\omega(a^\dag a+1/2)H=ℏω(a†a+1/2), D(α)=exp⁡(αa†−α∗a)=exp⁡(αa†)exp⁡(−αa)exp⁡(−∣α∣2/2)D(\alpha)=\exp(\alpha a^\dag-\alpha^* a)=\exp(\alpha a^\dag)\exp(-\alpha a)\exp(-|\alpha|^2/2)D(α)=exp(αa†−α∗a)=exp(αa†)exp(−αa)exp(−∣α∣2/2)
a∣n⟩=n∣n⟩a†∣n⟩=n+1∣n⟩a∣α⟩=α∣α⟩,∣α⟩=D(α)∣0⟩=e−∣α∣2/2∑nαnn!∣n⟩⟨x∣0⟩=1π1/4x0exp⁡[−12(x′x0)2],x0=ℏ/mω\begin{aligned} a|n\rangle &= \sqrt{n}|n\rangle \\ a^\dag |n\rangle &= \sqrt{n+1}|n\rangle \\ a |\alpha\rangle &= \alpha |\alpha\rangle,\quad |\alpha\rangle = D(\alpha)|0\rangle = e^{-|\alpha|^2/2}\sum_n\frac{\alpha^n}{\sqrt{n!}}|n\rangle \\ \langle x|0\rangle &= \frac{1}{\pi^{1/4}\sqrt{x_0}}\exp [-\frac{1}{2}(\frac{x'}{x^0})^2],\quad x_0=\sqrt{\hbar/m\omega} \\ \end{aligned} a∣n⟩a†∣n⟩a∣α⟩⟨x∣0⟩=n∣n⟩=n+1∣n⟩=α∣α⟩,∣α⟩=D(α)∣0⟩=e−∣α∣2/2n∑n!αn∣n⟩=π1/4x01exp[−21(x0x′)2],x0=ℏ/mω

第二章动力学

U(t,t0)=exp⁡(−iH(t−t0)/ℏ)U(t,t_0)=\exp(-iH(t-t_0)/\hbar)U(t,t0)=exp(−iH(t−t0)/ℏ)
In the interation picture, i∂tU(t,t0)Ψ(t0)=H(t)U(t,t0)Ψ(t0)i\partial_t U(t,t_0)\Psi(t_0) = H(t) U(t,t_0)\Psi(t_0)i∂tU(t,t0)Ψ(t0)=H(t)U(t,t0)Ψ(t0)
,
U(t,t0)=1−i∫t0tdt1H(t1)U(t1,t0)U(t,t_0)=1 - i \int_{t_0}^t{dt_1\ H(t_1)U(t_1,t_0)} U(t,t0)=1−i∫t0tdt1 H(t1)U(t1,t0)

Iteration get Dyson Series:
U(t,t0)=1+∑n=1∞(−iℏ)n∫t0tdt1∫t0t1dt2⁣⋯∫t0tn−1dtnH(t1)H(t2)…H(tn)U(t,t_0) = 1+\sum_{n=1}^\infty(\frac{-i}{\hbar})^n\int_{t_0}^t dt_1\int_{t_0}^{t_1}dt2\dots \int_{t_0}^{t_{n-1}}dt_n H(t_1)H(t_2)\dots H(t_n) U(t,t0)=1+n=1∑∞(ℏ−i)n∫t0tdt1∫t0t1dt2⋯∫t0tn−1dtnH(t1)H(t2)…H(tn)
p(t)=∣C(t)∣2p(t)= |C(t)|^2p(t)=∣C(t)∣2, C(t) is complex amplitude. C(t)=∑a∣ca∣2exp⁡(−iEat/ℏ)C(t)=\sum_a |c_a|^2\exp(-iE_at/\hbar)C(t)=∑a∣ca∣2exp(−iEat/ℏ).in a quasi-continum spectrum ∑a→∫dEρ(E)\sum_a \rightarrow \int dE\rho(E)∑a→∫dEρ(E), ca→g(E)c_a\rightarrow g(E)ca→g(E)
C(t)=∫dE∣g(E)∣2ρ(E)exp⁡(−iEt/ℏ),∫dE∣g(E)∣2ρ(E)=1C(t) = \int dE |g(E)|^2 \rho(E) \exp(-iEt/\hbar), \quad \int dE |g(E)|^2\rho(E)=1 C(t)=∫dE∣g(E)∣2ρ(E)exp(−iEt/ℏ),∫dE∣g(E)∣2ρ(E)=1
Heisenberg equation of motion:
A(H)dt=1iℏ[A(H),H]\frac{A^{(H)}}{dt} = \frac{1}{i\hbar}[A^{(H)},H] dtA(H)=iℏ1[A(H),H]
Ehrenfest Theorem: md2⟨x⟩/dt2=d⟨p⟩/dt=−⟨∇V(x)⟩m d^2 \langle x\rangle/dt^2= d\langle p\rangle/dt=-\langle \nabla V(x)\ranglemd2⟨x⟩/dt2=d⟨p⟩/dt=−⟨∇V(x)⟩

free space: H=p2/2mH=p^2/2mH=p2/2m, x(t)=x0+pt/mx(t)=x_0+pt/mx(t)=x0+pt/m, p(t)=pp(t)=pp(t)=p

harmonic osscilator:
H=p2/2m+12mω2t2=ℏω(n+12)x(t)=x(0)cos⁡ωt+p(0)mωsin⁡ωtp(t)=−mωx(0)sin⁡ωt+p(0)cos⁡ωta(t)=a(0)exp⁡(−iωt)a†(t)=a†(0)exp⁡(iωt)\begin{aligned} H &= p^2/2m + \frac{1}{2}m\omega^2t^2 = \hbar\omega(n+\frac{1}{2})\\ x(t) &= x(0)\cos\omega t + \frac{p(0)}{m\omega}\sin\omega t\\ p(t) &= -m\omega x(0)\sin\omega t + p(0)\cos\omega t \\ a(t) &= a(0)\exp(-i\omega t)\\ a^\dag(t) &= a^\dag(0)\exp(i\omega t) \end{aligned} Hx(t)p(t)a(t)a†(t)=p2/2m+21mω2t2=ℏω(n+21)=x(0)cosωt+mωp(0)sinωt=−mωx(0)sinωt+p(0)cosωt=a(0)exp(−iωt)=a†(0)exp(iωt)
possibility flux:
ρ(x,t)=∣ψ(x,t)∣2j⃗(x,t)=(ℏ/m)Im(ψ∗∇ψ)∂ρ∂t+∇⋅j⃗=0∫d3(x)j⃗(x,t)=⟨p⟩t/m\begin{aligned} \rho(x,t) &= |\psi(x,t)|^2\\ \vec{j}(x,t) &= (\hbar/m) Im(\psi^*\nabla\psi) \\ \frac{\partial \rho}{\partial t} &+\nabla\cdot\vec{j} = 0\\ \int d^3(x) &\vec{j}(x,t) = \langle p\rangle_t/m \end{aligned} ρ(x,t)j(x,t)∂t∂ρ∫d3(x)=∣ψ(x,t)∣2=(ℏ/m)Im(ψ∗∇ψ)+∇⋅j=0j(x,t)=⟨p⟩t/m
propogator: ψ(x′′,t)=∫d3x′K(x′′,t;x′,t0)ψ(x′,t0)\psi(x'',t)=\int d^3x' K(x'',t;x',t_0)\psi(x',t_0)ψ(x′′,t)=∫d3x′K(x′′,t;x′,t0)ψ(x′,t0)
lim⁡t→t0K(x′′,t;x0,t0)=δ3(x′′−x′)K(x′′,t;x′,t0)=0,for t<0K(x′′,t;x,t0)=⟨x′′,t∣x′,t⟩=⟨x′′∣exp⁡(−iH(t−t0)/ℏ)∣x′⟩K(x′′,t;x′,t0)=m2πiℏ(t−t0)exp⁡[im(x′′−x′)22ℏ(t−t0)]\begin{aligned} \lim_{t\rightarrow t_0} K(x'',t;x_0,t_0) &= \delta^3(x''-x') \\ K(x'',t;x',t_0) &=0 , \text{for } t\lt 0\\ K(x'',t;x,t_0) &= \langle x'',t|x',t\rangle \\ &= \langle x''|\exp(-iH(t-t_0)/\hbar)|x'\rangle \\ K(x'',t;x',t_0) &= \sqrt{\frac{m}{2\pi i\hbar(t-t_0)}}\exp[\frac{im(x''-x')^2}{2\hbar(t-t_0)}] \\ \end{aligned} t→t0limK(x′′,t;x0,t0)K(x′′,t;x′,t0)K(x′′,t;x,t0)K(x′′,t;x′,t0)=δ3(x′′−x′)=0,for t<0=⟨x′′,t∣x′,t⟩=⟨x′′∣exp(−iH(t−t0)/ℏ)∣x′⟩=2πiℏ(t−t0)mexp[2ℏ(t−t0)im(x′′−x′)2]
Feynman Path Integrals:
Scl=∫t1t2dtLcl(x,x˙)S_{cl} = \int_{t_1}^{t_2} dt L_{cl}(x,\dot{x}) Scl=∫t1t2dtLcl(x,x˙)
⟨xNtN∣x1t1⟩=∫x1xND[x(t)]exp⁡(iScl/ℏ)\langle x_N t_N|x_1t_1\rangle = \int_{x_1}^{x_{N}} D[x(t)]\exp(iS_{cl}/\hbar) ⟨xNtN∣x1t1⟩=∫x1xND[x(t)]exp(iScl/ℏ)
electro-magnatic field: B⃗=∇×A⃗\vec{B}=\nabla\times\vec{A}B=∇×A, E⃗=−∇ϕ\vec{E}=-\nabla \phiE=−∇ϕ
H=12m(p−ecA)2+eϕ=Π22m+eϕH = \frac{1}{2m} (p-\frac{e}{c}A)^2+e\phi = \frac{\Pi^2}{2m}+e\phi H=2m1(p−ceA)2+eϕ=2mΠ2+eϕ
Π=mdxdt=p−ecA,[Πi,Πj]=(iℏec)ϵijkBk\Pi = m \frac{dx}{dt} = p-\frac{e}{c}A,\quad [\Pi_i, \Pi_j] = (\frac{i\hbar e}{c})\epsilon_{ijk}B_k Π=mdtdx=p−ceA,[Πi,Πj]=(ciℏe)ϵijkBk
j=ℏmIm(ψ∗∇ψ)−emcA∣ψ∣2,∫d3xj⃗=⟨Π⟩/mj = \frac{\hbar}{m} Im(\psi^*\nabla\psi)-\frac{e}{mc}A|\psi|^2, \int d^3x \vec{j} = \langle \Pi\rangle/m j=mℏIm(ψ∗∇ψ)−mceA∣ψ∣2,∫d3xj=⟨Π⟩/m
gauge tranformation: ϕ→ϕ−1c∂Λ∂t\phi\rightarrow\phi-\frac{1}{c}\frac{\partial \Lambda}{\partial t}ϕ→ϕ−c1∂t∂Λ; A→A+∇ΛA\rightarrow A+\nabla \LambdaA→A+∇Λ; ∣ψ⟩→exp⁡(ieΛ/ℏc)∣ψ⟩|\psi\rangle\rightarrow \exp(ie\Lambda/\hbar c)|\psi\rangle∣ψ⟩→exp(ieΛ/ℏc)∣ψ⟩
spin procession. H=−μ⋅B=ΩSzH = -\mu\cdot B = \Omega S_zH=−μ⋅B=ΩSz, where Ω=−γB\Omega = -\gamma BΩ=−γB, γ\gammaγ is gyromanatric ratio
μ⃗=−gee2meS⃗,γ=gee2me,Sz=ℏ2σz,Ω≈eBme\vec{\mu} = -\frac{g_e e}{2m_e}\vec{S},\quad \gamma = \frac{g_e e}{2m_e},\quad S_z = \frac{\hbar}{2}\sigma_z,\quad \Omega\approx\frac{eB}{m_e} μ=−2megeeS,γ=2megee,Sz=2ℏσz,Ω≈meeB

第三章角动量

D(n^,ϕ)=exp⁡(−J⃗⋅n⃗ϕ/ℏ)D(\hat{n},\phi)=\exp(-\vec{J}\cdot\vec{n}\phi/\hbar)D(n^,ϕ)=exp(−J⋅nϕ/ℏ), [Ji,Jj]=iϵijkℏJz[J_i, J_j]=i\epsilon_{ijk}\hbar J_z[Ji,Jj]=iϵijkℏJz has no restriction on spins
for spin-1/2 exp⁡(−S⃗⋅n^ϕ/ℏ)=exp⁡(−σ⃗⋅n^ϕ/2)=1cos⁡(ϕ/2)−iσ⃗⋅n^sin⁡(ϕ/2)\exp(-\vec{S}\cdot\hat{n}\phi/\hbar)=\exp(-\vec{\sigma}\cdot\hat{n}\phi/2)=1\cos(\phi/2)-i\vec{\sigma}\cdot\hat{n}\sin(\phi/2)exp(−S⋅n^ϕ/ℏ)=exp(−σ⋅n^ϕ/2)=1cos(ϕ/2)−iσ⋅n^sin(ϕ/2)
exp⁡(−iσ⃗⋅n^ϕ/2)∣ϕ=2π=−1\exp(-i\vec{\sigma}\cdot\hat{n}\phi/2)|_{\phi=2\pi}=-1 exp(−iσ⋅n^ϕ/2)∣ϕ=2π=−1
i.e. rotate 4π4\pi4π back to itself per se.
D(α,β,γ)=Dz(α)Dy(β)Dz(γ)D(\alpha,\beta,\gamma)=D_z(\alpha)D_y(\beta)D_z(\gamma)D(α,β,γ)=Dz(α)Dy(β)Dz(γ)
D(α,β,γ)=e(−iJzα/ℏ)e(−iJyβ/ℏ)e(−iJzγ/ℏ)Dm′,mj(α,β,γ)=⟨j,m′∣D∣j,m⟩=e−i(m′α+mγ)⟨j,m′∣e(−iJyβ/ℏ)∣j,m⟩dm′,m(j)=⟨j,m′∣e(−iJyβ/ℏ)∣j,m⟩,Jy=(J++J_)/2id1/2=(cos⁡(β/2)−sin⁡(β/2)sin⁡(β/2)cos⁡(β/2))\begin{aligned} D(\alpha,\beta,\gamma)&=e^{(-iJ_z\alpha/\hbar)}e^{(-iJ_y\beta/\hbar)}e^{(-iJ_z\gamma/\hbar)}\\ D_{m',m}^{j}(\alpha,\beta,\gamma) &= \langle j,m'|D|j,m\rangle\\ & = e^{-i(m'\alpha+m\gamma)}\langle j,m'|e^{(-iJ_y\beta/\hbar)}|j,m\rangle \\ d_{m',m}^{(j)} &=\langle j,m'|e^{(-iJ_y\beta/\hbar)}|j,m\rangle,\quad J_y = (J_++J_\_)/ 2i \\ d^{1/2} &= \begin{pmatrix} \cos(\beta/2)& -\sin(\beta/2)\\ \sin(\beta/2)& \cos(\beta/2)\\ \end{pmatrix}\\ \end{aligned} D(α,β,γ)Dm′,mj(α,β,γ)dm′,m(j)d1/2=e(−iJzα/ℏ)e(−iJyβ/ℏ)e(−iJzγ/ℏ)=⟨j,m′∣D∣j,m⟩=e−i(m′α+mγ)⟨j,m′∣e(−iJyβ/ℏ)∣j,m⟩=⟨j,m′∣e(−iJyβ/ℏ)∣j,m⟩,Jy=(J++J_)/2i=(cos(β/2)sin(β/2)−sin(β/2)cos(β/2))
orbital angular momentum L=x×pL=x\times pL=x×p, [Li,Lj]=iℏϵijkLk[L_i, L_j]=i\hbar \epsilon_{ijk}L_k[Li,Lj]=iℏϵijkLk
L2∣l,m⟩=l(l+1)ℏ2∣l,m⟩Lz∣l,m⟩=mℏ∣l,m⟩L±∣l,m⟩=l(l+1)−m(m±1)∣l,m±1⟩⟨θ,ϕ∣l,m⟩=Yl,m(θ,ϕ)\begin{aligned} L^2 |l,m\rangle &= l(l+1)\hbar^2|l,m\rangle\\ L_z |l,m\rangle &= m\hbar |l,m\rangle \\ L_{\pm}|l,m\rangle &= \sqrt{l(l+1)-m(m\pm 1)}|l,m\pm 1\rangle \\ \langle \theta,\phi|l,m\rangle &= Y_{l,m}(\theta, \phi) \end{aligned} L2∣l,m⟩Lz∣l,m⟩L±∣l,m⟩⟨θ,ϕ∣l,m⟩=l(l+1)ℏ2∣l,m⟩=mℏ∣l,m⟩=l(l+1)−m(m±1)∣l,m±1⟩=Yl,m(θ,ϕ)
relation between D-matrix and Yl,mY_{l,m}Yl,m and ∣j,m⟩|j,m\rangle∣j,m⟩:
Dm,0(l)(α,β,γ=0)=4π(2l+1)Yl,m∗∣θ=β,ϕ=αD^{(l)}_{m,0}(\alpha,\beta,\gamma=0) = \left.\sqrt{\frac{4\pi}{(2l+1)}} Y_{l,m^*}\right|_{\theta=\beta,\phi=\alpha} Dm,0(l)(α,β,γ=0)=(2l+1)4πYl,m∗∣∣∣∣∣θ=β,ϕ=α
d0,0l(θ)=Pl(cos⁡θ)d^{l}_{0,0}(\theta) = P_l(\cos\theta) d0,0l(θ)=Pl(cosθ)
J2D(R)∣j,m⟩=D(R)J2∣j,m⟩=j(j+1)ℏ2[D(R)∣j,m⟩]D(R)∣j,m⟩=∑m′∣j,m′⟩Dm,m′j(R)\begin{aligned} J^2 D(R)|j,m\rangle &= D(R) J^2|j,m\rangle = j(j+1)\hbar^2 [D(R)|j,m\rangle]\\ D(R)|j,m\rangle &= \sum_{m'} |j,m'\rangle D_{m,m'}^j(R) \end{aligned} J2D(R)∣j,m⟩D(R)∣j,m⟩=D(R)J2∣j,m⟩=j(j+1)ℏ2[D(R)∣j,m⟩]=m′∑∣j,m′⟩Dm,m′j(R)
Addition of angular momentum J=J1+J2J=J_1+J_2J=J1+J2. Degeneracy:
Degeneracy=∑j=∣j1−j2∣j1+j2(2j+1)=(2j1+1)(2j2+1)\text{Degeneracy} = \sum_{j=|j_1-j_2|}^{j_1+j_2} (2j+1)=(2j_1+1)(2j_2+1) Degeneracy=j=∣j1−j2∣∑j1+j2(2j+1)=(2j1+1)(2j2+1)
CG-coefficient:
⟨j1j2;m1m2∣j1,j2;j,m⟩=(−1)j1−j2+m2j+1(j1j2jm1m2−m)\langle j_1j_2;m_1m_2|j_1,j_2;j,m\rangle = (-1)^{j_1-j_2+m}\sqrt{2j+1}\begin{pmatrix}j_1&j_2&j\\m_1&m_2&-m\\ \end{pmatrix} ⟨j1j2;m1m2∣j1,j2;j,m⟩=(−1)j1−j2+m2j+1(j1m1j2m2j−m)
recursion relationshio:
⟨j1j2;m1m2∣J±∣j1,j2;j,m⟩=⟨j1j2;m1m2∣J1±+J2±∣j1,j2;j,m⟩j(j+1)−m(m±1)⟨j1j2;m1m2∣j1,j2;j,m±1⟩=j1(j1+1)−m1(m1±1)⟨j1j2;(m1±1)m2∣j1,j2;j,m⟩+j2(j2+1)−m2(m2±1)⟨j1j2;m1(m2±1)∣j1,j2;j,m⟩\begin{aligned} \langle j_1j_2;m_1m_2|J_\pm|j_1,j_2;j,m\rangle = \langle j_1j_2;m_1m_2|J_{1\pm}+J_{2\pm}|j_1,j_2;j,m\rangle\\ \sqrt{j(j+1)-m(m\pm 1)}\langle j_1j_2;m_1m_2|j_1,j_2;j,m\pm 1\rangle = \\ \sqrt{j_1(j_1+1)-m_1(m_1\pm 1)}\langle j_1j_2;(m_1\pm1)m_2|j_1,j_2;j,m\rangle\\ +\sqrt{j_2(j_2+1)-m_2(m_2\pm 1)}\langle j_1j_2;m_1(m_2\pm 1)|j_1,j_2;j,m\rangle \end{aligned} ⟨j1j2;m1m2∣J±∣j1,j2;j,m⟩=⟨j1j2;m1m2∣J1±+J2±∣j1,j2;j,m⟩j(j+1)−m(m±1)⟨j1j2;m1m2∣j1,j2;j,m±1⟩=j1(j1+1)−m1(m1±1)⟨j1j2;(m1±1)m2∣j1,j2;j,m⟩+j2(j2+1)−m2(m2±1)⟨j1j2;m1(m2±1)∣j1,j2;j,m⟩
Rotation Matrices:
Dj1⊗Dj2=Dj1+j2⊕Dj1+j2−1⊕⋯⊕D∣j1−j2∣D^{j_1}\otimes D^{j_2} = D^{j_1+j_2}\oplus D^{j_1+j_2-1}\oplus\dots\oplus D^{|j_1-j_2|} Dj1⊗Dj2=Dj1+j2⊕Dj1+j2−1⊕⋯⊕D∣j1−j2∣
∫dΩYlm∗Yl1m1Yl2m2=(2l1+1)(2l2+1)4π(2l+1)⟨l1l2;00∣l1l2;l0⟩⟨l1l2;m1m2∣l1l2;lm⟩\int d\Omega Y_{l}^{m^*}Y_{l_1}^{m_1}Y_{l_2}^{m_2} = \sqrt{\frac{(2l_1+1)(2l_2+1)}{4\pi(2l+1)}}\langle l_1l_2;00|l_1l_2;l0\rangle\langle l_1l_2;m_1m_2|l_1l_2;lm\rangle ∫dΩYlm∗Yl1m1Yl2m2=4π(2l+1)(2l1+1)(2l2+1)⟨l1l2;00∣l1l2;l0⟩⟨l1l2;m1m2∣l1l2;lm⟩
Schwinger: two-mode-osscilator = angular momentum, ∣n+,n−⟩|n_+,n_-\rangle∣n+,n−⟩
J+=ℏa+†a−,J−=ℏa−†a+N=N1+N2=a+†a++a−†a−Jz=ℏ2(a+†a+−a−†a−)J2=ℏ22N(N2+1)\begin{aligned} &J_+ = \hbar a^\dag_+a_-, \quad J_- = \hbar a^\dag_-a_+\\ &N = N_1+N_2 = a^\dag_+a_++a^\dag_-a_-\\ &J_z = \frac{\hbar}{2}(a^\dag_+a_+-a^\dag_-a_-)\\ &J^2 = \frac{\hbar^2}{2}N(\frac{N}{2}+1) \end{aligned} J+=ℏa+†a−,J−=ℏa−†a+N=N1+N2=a+†a++a−†a−Jz=2ℏ(a+†a+−a−†a−)J2=2ℏ2N(2N+1)
n+=j+m,n−=j−m,∣j,m⟩∣j,m⟩=(a+†)j+m(a−†)j−m(j+m)!(j−m)!∣0⟩n_+ = j+m, \quad n_- = j-m, \quad |j,m\rangle\\ |j,m\rangle = \frac{(a^\dag_+)^{j+m}(a^\dag_-)^{j-m}}{\sqrt{(j+m)!(j-m)!}}|0\rangle n+=j+m,n−=j−m,∣j,m⟩∣j,m⟩=(j+m)!(j−m)!(a+†)j+m(a−†)j−m∣0⟩
Spherical Harmonic and Vector:
Y1,0=34πcos⁡θ=34πzrY1,+1=−34πx+iyrY1,−1=34πx−iyr\begin{aligned} Y_{1,0} = \sqrt{\frac{3}{4\pi}}\cos\theta = \sqrt{\frac{3}{4\pi}}\frac{z}{r} \\ Y_{1,+1} = -\sqrt{\frac{3}{4\pi}}\frac{x+iy}{r}\\ Y_{1,-1} = \sqrt{\frac{3}{4\pi}}\frac{x-iy}{r} \end{aligned} Y1,0=4π3cosθ=4π3rzY1,+1=−4π3rx+iyY1,−1=4π3rx−iy
the Winger-Echart Theorem:
⟨α′,j′m∣Tq(k)∣α,jm⟩=⟨jk;mq∣jk;j′m′⟩⟨α′j′∣∣T(k)∣∣αj⟩2j+1\langle \alpha',j'm|T^{(k)}_q|\alpha,jm\rangle = \langle jk;mq|jk;j'm'\rangle\frac{\langle \alpha'j'||T^{(k)}||\alpha j\rangle}{\sqrt{2j+1}} ⟨α′,j′m∣Tq(k)∣α,jm⟩=⟨jk;mq∣jk;j′m′⟩2j+1⟨α′j′∣∣T(k)∣∣αj⟩
⟨j′m′∣Tq(k)∣j,m⟩∝⟨jk;mq∣jk;j′m′⟩\langle j'm'|T^{(k)}_q|j,m\rangle \propto \langle jk;mq|jk;j'm'\rangle ⟨j′m′∣Tq(k)∣j,m⟩∝⟨jk;mq∣jk;j′m′⟩

第五章微扰理论

1. time-independent perturbation theory: non-degeneracy case

(H0+ϵV)(∣n(0)⟩+ϵ∣n(1)⟩)+ϵ2∣n(2⟩+⋯)=(En(0)+ϵEn(1)+ϵEn(2)+⋯)(∣n(0)⟩+ϵ∣n(1)⟩)+ϵ2∣n(2⟩+⋯)\begin{aligned} &(H_0+ \epsilon V)(|n^{(0)}\rangle+\epsilon|n^{(1)}\rangle)+\epsilon^2|n^{(2}\rangle+\cdots) = \\ &(E_n^{(0)}+\epsilon E_n^{(1)}+\epsilon E_n^{(2)}+\cdots)(|n^{(0)}\rangle+\epsilon |n^{(1)}\rangle)+\epsilon^2|n^{(2}\rangle+\cdots) \end{aligned} (H0+ϵV)(∣n(0)⟩+ϵ∣n(1)⟩)+ϵ2∣n(2⟩+⋯)=(En(0)+ϵEn(1)+ϵEn(2)+⋯)(∣n(0)⟩+ϵ∣n(1)⟩)+ϵ2∣n(2⟩+⋯)
0 order of ϵ:H0∣n(0)⟩=En(0)∣n(0)⟩1 order of ϵ:H0∣n(1)⟩+V∣n(0)⟩=En(0)∣n(1)⟩+En(1)∣n(0)⟩2 order of ϵ:H0∣n(2)⟩+V∣n(1)⟩=En(0)∣n(2)⟩+En(1)∣n(2)⟩+En(2)∣n(0)⟩…\begin{aligned} \text{0 order of }\epsilon: \quad & H_0 |n^{(0)}\rangle = E_n^{(0)}|n^{(0)}\rangle \\ \text{1 order of }\epsilon: \quad & H_0 |n^{(1)}\rangle +V|n^{(0)}\rangle = E_n^{(0)}|n^{(1)}\rangle + E_n^{(1)}|n^{(0)}\rangle \\ \text{2 order of }\epsilon: \quad & H_0 |n^{(2)}\rangle +V|n^{(1)}\rangle = E_n^{(0)}|n^{(2)}\rangle + E_n^{(1)}|n^{(2)}\rangle + E_n^{(2)}|n^{(0)}\rangle\\ \dots & \end{aligned} 0 order of ϵ:1 order of ϵ:2 order of ϵ:…H0∣n(0)⟩=En(0)∣n(0)⟩H0∣n(1)⟩+V∣n(0)⟩=En(0)∣n(1)⟩+En(1)∣n(0)⟩H0∣n(2)⟩+V∣n(1)⟩=En(0)∣n(2)⟩+En(1)∣n(2)⟩+En(2)∣n(0)⟩
for 1st order:
left multiply ⟨n(0)∣:⟨n(0)∣V∣n(0)⟩=En(1)=Vn,nleft multiply ⟨m(0)∣(m≠n):Em(0)⟨m(0)∣n(1)⟩+⟨m(0)∣V∣n(0)⟩=En(0)⟨m(0)∣n(1)⟩Vm,n=(En0−Em(0))⟨m(0)∣n(1)⟩∣n(1)⟩=∑m∣m(0)⟩⟨m(0)∣n(1)⟩=∑m≠nVm,nEn(0)−Em(0)∣m(0)⟩\begin{aligned} &\text{left multiply }\langle n^{(0)}|: &\langle n^{(0)}|V|n^{(0)}\rangle &= E_n^{(1)} = V_{n,n} \\ &\text{left multiply }\langle m^{(0)}| (m\ne n): &E_m^{(0)}\langle m^{(0)}|n^{(1)}\rangle + \langle m^{(0)}|V|n^{(0)}\rangle &= E_n^{(0)}\langle m^{(0)}|n^{(1)}\rangle \\ & &V_{m,n} &= (E_n^{0}-E_m^{(0)}) \langle m^{(0)}|n^{(1)}\rangle \\ & &|n^{(1)}\rangle = \sum_{m} |m^{(0)}\rangle \langle m^{(0)}|n^{(1)}\rangle &= \sum_{m\ne n} \frac{V_{m,n}}{E_n^{(0)}-E_m^{(0)}} |m^{(0)}\rangle \end{aligned} left multiply ⟨n(0)∣:left multiply ⟨m(0)∣(m=n):⟨n(0)∣V∣n(0)⟩Em(0)⟨m(0)∣n(1)⟩+⟨m(0)∣V∣n(0)⟩Vm,n∣n(1)⟩=m∑∣m(0)⟩⟨m(0)∣n(1)⟩=En(1)=Vn,n=En(0)⟨m(0)∣n(1)⟩=(En0−Em(0))⟨m(0)∣n(1)⟩=m=n∑En(0)−Em(0)Vm,n∣m(0)⟩
for 2nd order:
left multiply ⟨n(0)∣:En(2)=⟨n(0)∣V∣n(1)⟩=∑m≠n∣Vm,n∣2En(0)−Em(0)\begin{aligned} &\text{left multiply }\langle n^{(0)}|: & E_n^{(2)} = \langle n^{(0)}|V|n^{(1)}\rangle = \sum_{m\ne n} \frac{|V_{m,n}|^2}{E_n^{(0)}-E_m^{(0)}} \end{aligned} left multiply ⟨n(0)∣:En(2)=⟨n(0)∣V∣n(1)⟩=m=n∑En(0)−Em(0)∣Vm,n∣2
for k-th order:
En(k)=⟨n(0)∣V∣n(k)⟩E_n^{(k)} = \langle n^{(0)}|V|n^{(k)}\rangle En(k)=⟨n(0)∣V∣n(k)⟩

2. time-independent perturbation theory: degeneracy case

why degeneracy different: over disturbation, past-degenerate state split off. we need to choose correct state followed disturbation.

the most easiest case: E0=E1=EDE_0 = E_1=E_DE0=E1=ED, d=2d=2d=2, ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle=\alpha|0\rangle+\beta|1\rangle∣ψ⟩=α∣0⟩+β∣1⟩
(H+ϵV)(∣ψ(0)⟩+ϵ∣ψ(1)⟩+ϵ2∣ψ(2)⟩⋯)=(ED(0)+ϵED(1)+ϵ2ED(2)⋯)(∣ψ(0)⟩+ϵ∣ψ(1)⟩+ϵ2∣ψ(2)⟩⋯)\begin{aligned} &(H+\epsilon V)(|\psi^{(0)}\rangle+\epsilon|\psi^{(1)}\rangle+\epsilon^2|\psi^{(2)}\rangle\cdots) \\ =& (E_D^{(0)}+\epsilon E_D^{(1)}+\epsilon^2E_D^{(2)}\cdots)(|\psi^{(0)}\rangle+\epsilon|\psi^{(1)}\rangle+\epsilon^2|\psi^{(2)}\rangle\cdots) \end{aligned} =(H+ϵV)(∣ψ(0)⟩+ϵ∣ψ(1)⟩+ϵ2∣ψ(2)⟩⋯)(ED(0)+ϵED(1)+ϵ2ED(2)⋯)(∣ψ(0)⟩+ϵ∣ψ(1)⟩+ϵ2∣ψ(2)⟩⋯)

for the 0th order of ϵ:H∣ψ(0)⟩=ED∣ψ(0)⟩for the 1st order of ϵ:H0∣ψ(1)⟩+V∣ψ(0)⟩=ED(0)∣ψ(1)⟩+ED(1)∣ψ(0)⟩for the 2nd order of ϵ:H0∣ψ(2)⟩+V∣ψ(1)⟩=ED(0)∣ψ(2)⟩+ED(1)∣ψ(1)⟩+ED(2)∣ψ(0)⟩…\begin{aligned} &\text{for the 0th order of }\epsilon: & \quad H|\psi^{(0)}\rangle &= E_D |\psi^{(0)}\rangle \\ &\text{for the 1st order of }\epsilon: & H_0 |\psi^{(1)}\rangle +V|\psi^{(0)}\rangle &= E_D^{(0)}|\psi^{(1)}\rangle + E_D^{(1)}|\psi^{(0)}\rangle \\ &\text{for the 2nd order of }\epsilon: \quad & H_0 |\psi^{(2)}\rangle +V|\psi^{(1)}\rangle &= E_D^{(0)}|\psi^{(2)}\rangle + E_D^{(1)}|\psi^{(1)}\rangle + E_D^{(2)}|\psi^{(0)}\rangle\\ &\dots \end{aligned} for the 0th order of ϵ:for the 1st order of ϵ:for the 2nd order of ϵ:…H∣ψ(0)⟩H0∣ψ(1)⟩+V∣ψ(0)⟩H0∣ψ(2)⟩+V∣ψ(1)⟩=ED∣ψ(0)⟩=ED(0)∣ψ(1)⟩+ED(1)∣ψ(0)⟩=ED(0)∣ψ(2)⟩+ED(1)∣ψ(1)⟩+ED(2)∣ψ(0)⟩

for the 1st order:
left multiply ⟨n(0)∣,n=0,1:⟨n(0)∣V∣ψ(0)⟩=ED(1)⟨n0∣ψ(0)⟩∑mcm⟨n(0)∣V∣m(0)⟩=ED(1)cnδn,0(V00V01V10V11)(αβ)=ED(1)(αβ)left multiply ⟨k(0)∣,k≠0,1:⟨k(0)∣V∣ψ(0)⟩=(ED−Ek(0))⟨k∣ψ(1)⟩∣ψ(1)⟩=∑k∉D⟨k(0)∣V∣ψ(0)⟩ED−Ek∣k(0)⟩\begin{aligned} &\text{left multiply }\langle n^{(0)}|, n=0,1: & \quad \langle n^{(0)}|V|\psi^{(0)}\rangle = E_D^{(1)} \langle n^{0}|\psi^{(0)}\rangle \\ & & \sum_m c_m \langle n^{(0)}|V|m^{(0)}\rangle = E_D^{(1)} c_n\delta_{n,0}\\ & & \begin{pmatrix}V_{00}&V_{01}\\V_{10}&V_{11} \end{pmatrix}\begin{pmatrix}\alpha\\ \beta \end{pmatrix}=E_D^{(1)}\begin{pmatrix}\alpha\\ \beta \end{pmatrix} \\ &\text{left multiply }\langle k^{(0)}|, k\ne 0,1: &\langle k^{(0)}|V|\psi^{(0)}\rangle = (E_D-E_k^{(0)})\langle k|\psi^{(1)}\rangle \\ & & |\psi^{(1)}\rangle = \sum_{k\notin D} \frac{\langle k^{(0)}|V|\psi^{(0)}\rangle}{E_D-E_k}|k^{(0)}\rangle\\ \end{aligned} left multiply ⟨n(0)∣,n=0,1:left multiply ⟨k(0)∣,k=0,1:⟨n(0)∣V∣ψ(0)⟩=ED(1)⟨n0∣ψ(0)⟩m∑cm⟨n(0)∣V∣m(0)⟩=ED(1)cnδn,0(V00V10V01V11)(αβ)=ED(1)(αβ)⟨k(0)∣V∣ψ(0)⟩=(ED−Ek(0))⟨k∣ψ(1)⟩∣ψ(1)⟩=k∈/D∑ED−Ek⟨k(0)∣V∣ψ(0)⟩∣k(0)⟩
for the 2nd order:
left multiply ⟨n(0)∣,n=0,1:⟨n(0)∣V∣ψ(1)⟩=ED(2)⟨n(0)∣ψ(0)⟩ED(2)∣⟨n(0)∣ψ(0)⟩∣2=∑k∉D⟨k(0)∣V∣ψ(0)⟩ED−Ek∑n∈D⟨ψ(0)∣n(0)⟩⟨n(0)∣V∣k(0)⟩ED(2)=∑k∉D∣⟨k(0)∣V∣ψ(0)⟩∣2ED−Ek\begin{aligned} &\text{left multiply }\langle n^{(0)}|, n=0,1: &\quad \langle n^{(0)}|V|\psi^{(1)}\rangle &= E_D^{(2)} \langle n^{(0)}|\psi^{(0)}\rangle \\ & & E_D^{(2)} |\langle n^{(0)}|\psi^{(0)}\rangle|^2 &= \sum_{k\notin D} \frac{\langle k^{(0)}|V|\psi^{(0)}\rangle}{E_D-E_k}\sum_{n\in D}\langle \psi^{(0)}|n^{(0)}\rangle \langle n^{(0)}|V|k^{(0)}\rangle\\ & & E_D^{(2)} &= \sum_{k\notin D} \frac{|\langle k^{(0)}|V|\psi^{(0)}\rangle|^2 }{E_D-E_k} \end{aligned} left multiply ⟨n(0)∣,n=0,1:⟨n(0)∣V∣ψ(1)⟩ED(2)∣⟨n(0)∣ψ(0)⟩∣2ED(2)=ED(2)⟨n(0)∣ψ(0)⟩=k∈/D∑ED−Ek⟨k(0)∣V∣ψ(0)⟩n∈D∑⟨ψ(0)∣n(0)⟩⟨n(0)∣V∣k(0)⟩=k∈/D∑ED−Ek∣⟨k(0)∣V∣ψ(0)⟩∣2

if it’s d-degeneracy, then there is d values of ED(1)E_D^{(1)}ED(1), there is d states split.

More General Expression:

3. Time-dependent Purturbation Theory:

from Dyson theory:
for the 0th order: U(t,t0)(0)=1for the 1st order: U(t,t0)(1)=−iℏ∫t0tdt1H(t1)for the 2nd order: U(t,t0)(2)=(−iℏ)2∫t0tdt1H(t1)∫t0t1dt2H(t2)…\begin{aligned} \text{for the 0th order: }& \quad U(t,t_0)^{(0)} = 1 \\ \text{for the 1st order: }& \quad U(t,t_0)^{(1)} = \frac{-i}{\hbar}\int_{t_0}^t dt_1\ H(t_1) \\ \text{for the 2nd order: }& \quad U(t,t_0)^{(2)} = (\frac{-i}{\hbar})^2\int_{t_0}^t dt_1H(t_1)\int_{t_0}^{t_1}dt_2\ H(t_2)\\ \dots \end{aligned} for the 0th order: for the 1st order: for the 2nd order: …U(t,t0)(0)=1U(t,t0)(1)=ℏ−i∫t0tdt1 H(t1)U(t,t0)(2)=(ℏ−i)2∫t0tdt1H(t1)∫t0t1dt2 H(t2)

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量子力学常用知识汇总 (部分)

第一章算符

第二章动力学

第三章角动量

第五章微扰理论

1. time-independent perturbation theory: non-degeneracy case

2. time-independent perturbation theory: degeneracy case

3. Time-dependent Purturbation Theory:

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量子力学常用知识汇总 (部分)

第一章 算符

第二章 动力学

第三章 角动量

第五章 微扰理论

1. time-independent perturbation theory: non-degeneracy case

2. time-independent perturbation theory: degeneracy case

3. Time-dependent Purturbation Theory:

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第一章算符

第二章动力学

第三章角动量

第五章微扰理论