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Quantum Mechanics – Chapter 9: Angular Momentum

Chapter 9

Angular Momentum

9.1  Orbital Angular Momentum

Classical definition

  • \(\vec L=\vec r\times\vec p\), with \(\vec r=x\hat{\imath}+y\hat{\jmath}+z\hat{k}\), \(\vec p=p_x\hat{\imath}+p_y\hat{\jmath}+p_z\hat{k}\): \[L_x=yp_z-zp_y,\qquad L_y=zp_x-xp_z,\qquad L_z=xp_y-yp_x.\]

  • Classically there is no uncertainty principle: \(L_x,L_y,L_z\) can all be measured simultaneously.

Quantum operators

  • \(\hat L_x=\hat Y\hat P_z-\hat Z\hat P_y=-i\hbar\Big(y\dfrac{\partial}{\partial z}-z\dfrac{\partial}{\partial y}\Big)\),   \(\hat L_y=\hat Z\hat P_x-\hat X\hat P_z=-i\hbar\Big(z\dfrac{\partial}{\partial x}-x\dfrac{\partial}{\partial z}\Big)\),

  • \(\hat L_z=\hat X\hat P_y-\hat Y\hat P_x=-i\hbar\Big(x\dfrac{\partial}{\partial y}-y\dfrac{\partial}{\partial x}\Big)\).

  • Commutators: \[[\hat L_x,\hat L_y]=[\hat Y\hat P_z,\hat Z\hat P_x]+[\hat Z\hat P_y,\hat X\hat P_z]=\hat Y(-i\hbar)\hat P_x+\hat X(i\hbar)\hat P_y=i\hbar\hat L_z.\]

  • Cyclically: \([\hat L_x,\hat L_y]=i\hbar\hat L_z\),   \([\hat L_y,\hat L_z]=i\hbar\hat L_x\),   \([\hat L_z,\hat L_x]=i\hbar\hat L_y\); compactly \(\vec L\times\vec L=i\hbar\vec L\).

  • The components do not commute \(\Rightarrow\) they cannot be measured simultaneously.

Complete basis: \(\hat L^2\) and \(\hat L_z\)

  • Build \(\hat L^2=\hat L_x^2+\hat L_y^2+\hat L_z^2\) (the analogue of \(|\vec L|^2\)).

  • \([\hat L^2,\hat L_x]=\hat L_y[\hat L_y,\hat L_x]+[\hat L_y,\hat L_x]\hat L_y+\hat L_z[\hat L_z,\hat L_x]+[\hat L_z,\hat L_x]\hat L_z=0\).

  • Likewise \([\hat L^2,\hat L_y]=[\hat L^2,\hat L_z]=0\).

  • \(\hat L^2\) and one component form a commuting pair; by convention we take \(\{\hat L^2,\hat L_z\}\) — two quantum numbers for the two angular variables \(\theta,\phi\).

Figure
\(x=r\sin\theta\cos\phi,\ y=r\sin\theta\sin\phi,\ z=r\cos\theta\)

Spherical polar form

  • \(\vec L=-i\hbar\,\vec r\times\nabla\) with \(\nabla=\hat r\,\partial_r+\hat\theta\,\frac1r\partial_\theta+\hat\phi\,\frac{1}{r\sin\theta}\partial_\phi\), and \(\hat r\times\hat r=0\), \(\hat r\times\hat\theta=\hat\phi\), \(\hat r\times\hat\phi=-\hat\theta\): \[\vec L=-i\hbar\Big(\hat\phi\frac{\partial}{\partial\theta}-\hat\theta\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}\Big).\]

  • Using \(\hat\theta=\cos\theta\cos\phi\,\hat{\imath}+\cos\theta\sin\phi\,\hat{\jmath}-\sin\theta\,\hat{k}\) and \(\hat\phi=-\sin\phi\,\hat{\imath}+\cos\phi\,\hat{\jmath}\): \[\begin{gathered} \hat L_x=i\hbar\Big(\sin\phi\frac{\partial}{\partial\theta}+\cot\theta\cos\phi\frac{\partial}{\partial\phi}\Big),\qquad \hat L_y=-i\hbar\Big(\cos\phi\frac{\partial}{\partial\theta}-\cot\theta\sin\phi\frac{\partial}{\partial\phi}\Big),\\ \hat L_z=-i\hbar\frac{\partial}{\partial\phi}. \end{gathered}\]

9.2  Eigenvalues and Eigenfunctions

Eigenfunctions of \(\hat L_z\)

  • \(-i\hbar\dfrac{d\Phi}{d\phi}=\mu\Phi\Rightarrow\Phi=A\,e^{i\mu\phi/\hbar}\), \(0\le\phi\le2\pi\); normalization \(\int_0^{2\pi}|\Phi|^2d\phi=1\Rightarrow A=\dfrac{1}{\sqrt{2\pi}}\).

  • Single-valuedness \(\Phi(\phi+2\pi)=\Phi(\phi)\Rightarrow e^{2\pi i\mu/\hbar}=1\Rightarrow\mu=m\hbar\): \[\Phi_m(\phi)=\frac{1}{\sqrt{2\pi}}e^{im\phi},\qquad \hat L_z\Phi_m=m\hbar\,\Phi_m,\qquad m=0,\pm1,\pm2,\ldots\]

Example 9.1   Measuring \(L_z\) (1)

\(\psi(\phi)\propto\cos2\phi\). What does a measurement of \(L_z\) give?

Solution

  • \(\cos2\phi\propto e^{2i\phi}+e^{-2i\phi}\) \(\Rightarrow m=\pm2\):   \(L_z=+2\hbar\) or \(-2\hbar\) (equal probability).

Example 9.2   Measuring \(L_z\) (2)

\(\psi(x,y)\propto\dfrac{1}{x+iy}\). What does a measurement of \(L_z\) give?

Solution

  • \(x+iy=r\sin\theta\,e^{i\phi}\) \(\Rightarrow\psi\propto\dfrac{e^{-i\phi}}{r\sin\theta}\) \(\Rightarrow m=-1\):   \(L_z=-\hbar\).

Eigenfunctions of \(\hat L^2\): Spherical Harmonics

  • \(\hat L^2=-\hbar^2\Big[\dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta}\Big(\sin\theta\dfrac{\partial}{\partial\theta}\Big)+\dfrac{1}{\sin^2\theta}\dfrac{\partial^2}{\partial\phi^2}\Big]\).

  • \([\hat L^2,\hat L_z]=0\) \(\Rightarrow\) common eigenfunctions \(\psi(\theta,\phi)=f(\theta)e^{im\phi}\); with \(\hat L^2\psi=\lambda\psi\): \[\frac{1}{\sin\theta}\frac{d}{d\theta}\Big(\sin\theta\frac{df}{d\theta}\Big)+\Big(\frac{\lambda}{\hbar^2}-\frac{m^2}{\sin^2\theta}\Big)f=0.\]

  • Put \(x=\cos\theta\):   \((1-x^2)f''-2xf'+\Big[\dfrac{\lambda}{\hbar^2}-\dfrac{m^2}{1-x^2}\Big]f=0\) — the associated Legendre equation.

  • Regular at \(x=\pm1\) only if \(\lambda=l(l+1)\hbar^2\), \(l=0,1,2,\ldots\), and \(-l\le m\le l\); solutions \(f=P_l^m(\cos\theta)\).

  • Spherical harmonics \(Y_l^m(\theta,\phi)\propto P_l^m(\cos\theta)e^{im\phi}\equiv\left|l,m\right\rangle\): \[\hat L^2\left|l,m\right\rangle=l(l+1)\hbar^2\left|l,m\right\rangle,\qquad \hat L_z\left|l,m\right\rangle=m\hbar\left|l,m\right\rangle.\]

  • Orthonormality: \(\displaystyle\int_0^{2\pi}\!\!\int_0^\pi Y_{l'}^{m'*}Y_l^m\sin\theta\,d\theta\,d\phi=\delta_{l'l}\delta_{m'm}\), i.e. \(\left\langle l',m'\middle|l,m\right\rangle =\delta_{l'l}\delta_{m'm}\).

Important spherical harmonics

  • \(Y_0^0=\dfrac{1}{2\sqrt\pi}\);   \(Y_1^0=\dfrac12\sqrt{\dfrac3\pi}\cos\theta\);   \(Y_1^{\pm1}=\mp\dfrac12\sqrt{\dfrac{3}{2\pi}}\sin\theta\,e^{\pm i\phi}\)

  • \(Y_2^0=\dfrac14\sqrt{\dfrac5\pi}(3\cos^2\theta-1)\);   \(Y_2^{\pm1}=\mp\dfrac12\sqrt{\dfrac{15}{2\pi}}\sin\theta\cos\theta\,e^{\pm i\phi}\);   \(Y_2^{\pm2}=\dfrac14\sqrt{\dfrac{15}{2\pi}}\sin^2\theta\,e^{\pm2i\phi}\)

  • Cartesian forms: \(rY_1^0\propto z\), \(rY_1^{\pm1}\propto\mp(x\pm iy)\);   \(r^2Y_2^0\propto3z^2-r^2\), \(r^2Y_2^{\pm1}\propto\mp(x\pm iy)z\), \(r^2Y_2^{\pm2}\propto(x\pm iy)^2\).

9.3  Angular Momentum Algebra and Ladder Operators

  • Ladder operators: \(\hat L_\pm=\hat L_x\pm i\hat L_y\);   \([\hat L^2,\hat L_\pm]=0\).

  • \([\hat L_z,\hat L_\pm]=\pm\hbar\hat L_\pm\),   \([\hat L_+,\hat L_-]=2\hbar\hat L_z\).

  • \(\hat L_-\hat L_+=\hat L^2-\hat L_z^2-\hbar\hat L_z\),   \(\hat L_+\hat L_-=\hat L^2-\hat L_z^2+\hbar\hat L_z\).

Raising and lowering

  • \(\hat L_z(\hat L_+\left|l,m\right\rangle)=(\hat L_+\hat L_z+\hbar\hat L_+)\left|l,m\right\rangle=(m+1)\hbar\,\hat L_+\left|l,m\right\rangle\) \(\Rightarrow\hat L_+\left|l,m\right\rangle=c_+\left|l,m+1\right\rangle\).

  • \(|c_+|^2=\left\langle l,m\middle|\hat L_-\hat L_+\middle|l,m\right\rangle =\big[l(l+1)-m(m+1)\big]\hbar^2\ge0\) \(\Rightarrow m\le l\).

  • Similarly \(|c_-|^2=\big[l(l+1)-m(m-1)\big]\hbar^2\ge0\Rightarrow m\ge-l\).

Ladder relations

  • \(\hat L_+\left|l,m\right\rangle=\sqrt{l(l+1)-m(m+1)}\,\hbar\left|l,m+1\right\rangle\),   \(\hat L_+\left|l,l\right\rangle=0\)

  • \(\hat L_-\left|l,m\right\rangle=\sqrt{l(l+1)-m(m-1)}\,\hbar\left|l,m-1\right\rangle\),   \(\hat L_-\left|l,-l\right\rangle=0\)

  • For fixed \(l\): \(2l+1\) states, \(m=-l,\ldots,l\).

Figure

9.4  Space Quantization

  • \(|\vec L|=\sqrt{l(l+1)}\,\hbar\) while \(L_z=m\hbar\), so the angle with \(\hat z\) obeys \[\cos\theta=\frac{m}{\sqrt{l(l+1)}}.\]

  • For fixed \(l\) only \(2l+1\) orientations are allowed — \(\theta\) is quantized, not continuous.

  • \(|m|<\sqrt{l(l+1)}\) always: \(\vec L\) can never point exactly along \(z\) (\(L_x,L_y\) cannot both vanish).

Figure

Example 9.3   Allowed angles for \(l=2\)

Find the possible values of \(\cos\theta\) for \(l=2\).

Solution

  • \(|\vec L|=\sqrt6\hbar\), \(m=2,1,0,-1,-2\) \(\Rightarrow\cos\theta=\dfrac{2}{\sqrt6},\ \dfrac{1}{\sqrt6},\ 0,\ -\dfrac{1}{\sqrt6},\ -\dfrac{2}{\sqrt6}\).

Example 9.4   \(L_x\) statistics in \(\left|l,m\right\rangle\)

Find (a) \(\left\langle \hat L_x\right\rangle\), \(\left\langle \hat L_x^2\right\rangle\); (b) \(\Delta L_x\); (c) the range of \(\Delta L_x\) in \(\left|l,m\right\rangle\).

Solution

  • (a)  \(\hat L_x=\tfrac12(\hat L_++\hat L_-)\) changes \(m\) \(\Rightarrow\left\langle \hat L_x\right\rangle=0\).

  • \(\hat L_x^2=\tfrac14\big(\hat L_+^2+\hat L_-^2+\hat L_+\hat L_-+\hat L_-\hat L_+\big)\) and \(\hat L_+\hat L_-+\hat L_-\hat L_+=2(\hat L^2-\hat L_z^2)\): \[\left\langle \hat L_x^2\right\rangle=\left\langle \hat L_y^2\right\rangle=\tfrac12\big[l(l+1)-m^2\big]\hbar^2.\]

  • (b)  \(\Delta L_x=\sqrt{\dfrac{l(l+1)-m^2}{2}}\,\hbar\).

  • (c)  Largest at \(m=0\), smallest at \(m=\pm l\):   \(\sqrt{\dfrac l2}\,\hbar\le\Delta L_x\le\sqrt{\dfrac{l(l+1)}{2}}\,\hbar\).

Example 9.5   Eigenfunction of \(L^2\) but not of \(L_z\)

\(\psi(x,y,z)\propto f(r)\,y\,r\). (a) Result of measuring \(\hat L^2\)? (b) Of \(\hat L_z\)? (c) Show \(\psi\) is an eigenfunction of \(\hat L^2\) but not of \(\hat L_z\).

Solution

  • \(y=r\sin\theta\sin\phi=\dfrac{r\sin\theta}{2i}(e^{i\phi}-e^{-i\phi})\propto Y_1^1+Y_1^{-1}\) \(\Rightarrow l=1\), \(m=\pm1\).

  • (a)  \(\hat L^2\): \(2\hbar^2\) with certainty.   (b) \(\hat L_z\): \(+\hbar\) or \(-\hbar\).

  • (c)  \(\hat L^2\psi=2\hbar^2\psi\); but \(\hat L_z\psi\propto\hbar(Y_1^1-Y_1^{-1})\ne m\hbar\,\psi\).

Example 9.6   Particle on a ring

A particle of mass \(\mu\) moves on a horizontal circle of radius \(a\). Find (i) \(H\), (ii) the quantum operator, (iii) the eigenfunctions and degeneracies.

Solution

  • (i)  \(L=\tfrac12\mu a^2\dot\phi^2\), \(p_\phi=\mu a^2\dot\phi\) \(\Rightarrow H=\dfrac{p_\phi^2}{2\mu a^2}\).

  • (ii)  \(p_\phi\) is \(L_z\):   \(\hat H=\dfrac{\hat L_z^2}{2\mu a^2}\).

  • (iii)  \([\hat H,\hat L_z]=0\):   \(\psi_m=\dfrac{e^{im\phi}}{\sqrt{2\pi}}\), \(E_m=\dfrac{m^2\hbar^2}{2\mu a^2}\), \(m=0,\pm1,\pm2,\ldots\)

  • Ground state (\(m=0\)

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