动量算符

动量算符 p⃗^=−iℏ∇\hat{\vec{p}}=-i\hbar \nabla，分量形式 p^x=−iℏ∂∂x,p^y=−iℏ∂∂y,p^z=−iℏ∂∂z\hat{p}_{x}=-i\hbar \frac{\partial }{\partial x},\hat{p}_{y}=-i\hbar \frac{\partial }{\partial y},\hat{p}_{z}=-i\hbar \frac{\partial }{\partial z}。

动量算符各分量和坐标算符各分量之间的对易关系；

[x^i,p^j]=iℏδij={0,i≠jiℏ,i=j[\hat{x}_i,\hat{p}_j]=i\hbar \delta_{ij}=\begin{cases} 0,& i\neq j \\ i \hbar ,&i=j \end{cases}

动量平方算符

p⃗^2=p^x2+p^y2+p^z2=−ℏ2∇2\hat{\vec{p}}^{2}=\hat{p}_{x}^{2}+\hat{p}_{y}^{2}+\hat{p}_{z}^{2}=-\hbar^{2}\nabla ^{2}

角动量算符

定义

L^=r^×p^=−iℏ(r×∇)\hat{\bm{L}}=\hat{\bm{r}}\times \hat{\bm{p}}=-i \hbar (\bm{r}\times \nabla )

分量形式

L^x=yp^z−zp^y=−iℏ(y∂∂z−z∂∂y)L^y=zp^x−xpz=−iℏ(z∂∂x−x∂∂z)L^z=xp^y−yp^x=−iℏ(x∂∂y−y∂∂x)\begin{aligned} \hat{L}_{x}&=y \hat{p}_{z} - z \hat{p}_{y}=-i \hbar (y\frac{\partial }{\partial z}-z\frac{\partial }{\partial y}) \\ \hat{L}_{y}&=z \hat{p}_{x}-x p_{z}=-i \hbar (z \frac{\partial }{\partial x}-x \frac{\partial }{\partial z}) \\ \hat{L}_{z}&=x \hat{p}_{y}-y \hat{p}_{x}=-i \hbar (x \frac{\partial }{\partial y}-y \frac{\partial }{\partial x}) \end{aligned}

角动量平方算符

L^2=L^x2+L^y2+L^z2\hat{\bm{L}}^{2}=\hat{L}_{x}^{2}+\hat{L}_y^{2}+\hat{L}_z^{2}

角动量算符各分量对易式

角动量各分量之间不对易

[L^x,L^y]=L^xL^y−L^yL^x=(yp^z−zp^y)(zp^x−xp^z)−(zp^x−xp^z)(yp^z−zp^y)=(p^zz−zp^z)yp^x+(zp^z−p^zz)xp^y=−iℏyp^x+iℏxp^y=iℏ(xp^y−yp^x)=iℏL^z\begin{aligned} [\hat{L}_{x},\hat{L}_y]&=\hat{L}_{x}\hat{L}_y-\hat{L}_y \hat{L}_{x} \\ &=(y \hat{p}_z-z \hat{p}_y)(z \hat{p}_{x}-x \hat{p}_z)-(z \hat{p}_{x}-x \hat{p}_z)(y \hat{p}_z- z \hat{p}_y)\\ &=(\hat{p}_z z- z\hat{p}_z)y \hat{p}_{x} +(z \hat{p}_z-\hat{p}_z z)x \hat{p}_y\\ &=-i\hbar y \hat{p}_{x}+i \hbar x \hat{p}_y=i \hbar (x \hat{p}_y-y \hat{p}_{x} )=i \hbar \hat{L}_z \end{aligned}

同理 [L^y,L^z]=iℏL^x[\hat{L}_y,\hat{L}_z]=i\hbar \hat{L}_{x}, [L^z,L^x]=iℏL^y[\hat{L}_z,\hat{L}_{x}]=i\hbar \hat{L}_y ⇒\Rightarrow L^×L^=iℏL^\hat{\bm{L}}\times \hat{\bm{L}}=i \hbar \hat{\bm{L}}

角动量平方算符与其各分量之间是对易的

[L^2,L^x]=[L^x2+L^y2+L^z2,L^x]=[L^x2,L^x=0][L^y2,L^x]+[L^z2,L^x]=L^y[L^y,L^x]+[L^y,L^x]L^y+L^z[L^z,L^x]+[L^z,L^x]L^z=iℏ(−L^yL^z−L^zL^y+L^zL^y+L^yL^z)=0\begin{aligned} [\hat{L}^{2},\hat{L}_{x}]&=[\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_z^{2},\hat{L}_{x}]\overset{[\hat{L}_{x}^{2},\hat{L}_{x}=0]}{=} [\hat{L}_y^{2},\hat{L}_{x}]+[\hat{L}_z^{2},\hat{L}_{x}] \\ &=\hat{L}_y[\hat{L}_y,\hat{L}_{x}]+[\hat{L}_y,\hat{L}_{x}]\hat{L}_y+\hat{L}_z[\hat{L}_z,\hat{L}_{x}]+[\hat{L}_z,\hat{L}_{x}]\hat{L}_z\\ &=i \hbar (-\hat{L}_y \hat{L}_z-\hat{L}_z \hat{L}_y + \hat{L}_z \hat{L}_y+ \hat{L}_y \hat{L}_z)=0 \end{aligned}

同理 [L^2,L^y]=0[\hat{L}^{2},\hat{L}_y]=0,[L^2,L^z]=0[\hat{L}^{2},\hat{L}_z]=0

球坐标系中的角动量(不要求记忆)

L^x=iℏ(sin⁡φ∂∂θ+ctg⁡θcos⁡φ∂∂φ)L^y=−iℏ(cos⁡φ∂∂θ−ctg⁡θsin⁡φ∂∂φ)L^z=−iℏ∂∂φ\begin{aligned} \hat{L}_{x}&=i \hbar\left(\sin \varphi \frac{\partial}{\partial \theta}+\operatorname{ctg} \theta \cos \varphi \frac{\partial}{\partial \varphi}\right) \\ \hat{L}_{y}&=-i \hbar\left(\cos \varphi \frac{\partial}{\partial \theta}-\operatorname{ctg} \theta \sin \varphi \frac{\partial}{\partial \varphi}\right) \\ \hat{L}_{z}&=-i \hbar \frac{\partial}{\partial \varphi} \end{aligned}

L^2=L^x2+L^y2+L^z2=−ℏ2[1sin⁡θ∂∂θ(sin⁡θ∂∂θ)+1sin⁡2θ∂2∂φ2]\hat{\mathbf{L}}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}=-\hbar^{2}\left[\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{\sin ^{2} \theta} \frac{\partial^{2}}{\partial \varphi^{2}}\right]

∇2=1r2[∂∂r(r2∂∂r)+1sin⁡θ∂∂θ(sin⁡θ∂∂θ)+1sin⁡2θ∂2∂φ2]=1r2[∂∂r(r2∂∂r)−L^2ℏ2]\begin{aligned} \nabla^{2}&=\frac{1}{r^{2}}\left[\frac{\partial}{\partial r}\left(r^{2} \frac{\partial}{\partial r}\right)+\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{\sin ^{2} \theta} \frac{\partial^{2}}{\partial \varphi^{2}}\right] \\ &=\frac{1}{r^{2}}\left[\frac{\partial}{\partial r}\left(r^{2} \frac{\partial}{\partial r}\right)-\frac{\hat{\mathbf{L}}^{2}}{\hbar^{2}}\right] \end{aligned}

角动量算符的本征函数和本征值

L^z\hat{L}_z 算符的本征值和本征函数

球坐标系中，L^z=−iℏ∂∂φ\displaystyle \hat{L}_z=-i\hbar \frac{\partial }{\partial \varphi}

L^zψ=lzψ⇒−iℏ∂∂φψ=lzψ\hat{L}_z \psi =l_z \psi \Rightarrow -i \hbar \frac{\partial }{\partial \varphi} \psi=l_z \psi

解得

ψ(φ)=Cexp⁡(iℏlzφ)\psi(\varphi)=C \exp (\frac{i}{\hbar}l_z \varphi)

同时考虑到 φ\varphi 和 φ+2π\varphi+2\pi 时系统处于同一状态，得到

lz=mℏ,m∈Zl_z=m \hbar , \quad m \in \mathbb{Z}

lz=mℏl_z=m \hbar 为算符 lzl_z 的本征值，对应的本征函数为

ψm(φ)=Ceimφ→归一化ψm(φ)=12πeimφ\psi_{m}(\varphi)=C e^{im \varphi} \xrightarrow{\text{归一化}} \psi_{m}(\varphi)=\frac{1}{\sqrt{2\pi}} e^{im \varphi}

球谐函数

球谐函数 Ylm(θ,φ)Y_{lm}(\theta, \varphi) 是 L^2\hat{L}^{2} 和 L^z\hat{L}_z 的共同本征波函数(下式要记)

{L^2Ylm(θ,φ)=l(l+1)ℏ2Ylm(θ,φ)L^zYlm(θ,φ)=mℏYlm(θ,φ)l=0,1,2,…m=−l,−l+1,…,0,…l−1,l\begin{cases} \hat{L}^{2}Y_{l m}(\theta, \varphi)=l(l+1)\hbar^{2}Y_{l m}(\theta,\varphi) \\ \hat{L}_zY_{l m}(\theta,\varphi)=m \hbar Y_{l m}(\theta,\varphi)\\ l=0,1,2, \ldots \quad m=-l,-l+1, \ldots ,0, \ldots l-1,l \end{cases}

ll 表征了角动量的大小，称为角量子数；mm 称为磁量子数，与原子光谱在外磁场中发生分裂有关，每一个 ll 值对应 2l+12l+1 个 mm。在无外磁场时，角量子数为 ll 的量子态是 2l+12l+1 重简并的。L^2,L^z\hat{L}^{2},\hat{L}_z 的本征值谱都是分立的，量子数为 ll 和 mm。

在 Ylm(θ,φ)Y_{l m}(\theta,\varphi) 的状态下，角动量大小为 L=l(l+1)ℏL=\sqrt{l(l+1)\hbar}，角动量在 zz 方向上的投影 Lz=mℏL_z=m\hbar。因此可以看出，角动量的空间取向是量子化的。

正交归一化条件：

∫02πdφ∫02πsin⁡θdθYlm∗(θ,φ)∗Yl′m′(θ,φ)=δll′δmm′\int_{0}^{2\pi}\mathrm{d}\varphi \int_{0}^{2\pi}\sin \theta \mathrm{d} \theta Y_{l m}^{*}(\theta,\varphi)^{*} Y_{l'm'}(\theta,\varphi)=\delta_{ll'}\delta_{mm'}

补充：量子力学五大假设

  • 粒子的态可以用波函数描述
  • 力学量对应算符一定是线性的厄密算符
  • 测量假设：对力学量进行测量，得到的一定是本征值中的一个
  • 态的演化可以用薛定谔方程描述
  • 多粒子系统中粒子不可分辨