1.6: Angular Momentum

Angular momentum operators

To describe quantum mechanical rotation or orbital motion, one must quantize angular momentum. The orbital angular momentum operator is defined as \[\hat{L}=\hat{r} \times \hat{p}=i \hbar(\hat{r} \times \nabla)\]

It has three components \((\hat{L}_{x}, \hat{L}_{y}, \hat{L}_{z})\) that generate rotation about the \(x, y\), or \(z\) axis, and whose magnitude is given by \(\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}\). The angular momentum operators follow the commutation relationships \[\begin{align}
\left[H, L^{2}\right] &= 0 \\[3pt]
\left[L^{2}, L_{z}\right] &= 0 \\[3pt]
\left[L_{x}, L_{y}\right] &= i \hbar L_{z} \end{align}\]

Note that in eq. (1.6.4) the \(x, y,\) and \(z\) indices can be cyclically permuted. There is an eigenbasis common to \(H\) and \(L^{2}\) and one of the three \(L_{i}\), which is \(L_{z}\) by convention. The eigenvalues for the orbital angular momentum operator \(L\) and \(z\)-projection of the angular momentum \(L_{z}\) are \[\begin{align}
L^{2}|\ell m\rangle &=\hbar^{2} \ell(\ell+1)|\ell m\rangle & \ell &=0,1,2 \ldots \\[3pt]
L_{z}|\ell m\rangle &=\hbar m|\ell m\rangle &m &=0, \pm 1, \pm 2 \ldots \pm \ell \end{align}\]

where the eigenstates \(|\ell m\rangle\) are labeled by the orbital angular momentum quantum number \(\ell\), and the magnetic quantum number \(m\). The eigenstates are found to be spherical harmonics \(|\ell m\rangle=Y_{\ell}^{m}(\theta, \phi)\) as discussed below.

Similar to the strategy used for the harmonic oscillator, we can also define raising and lowering operators for the total angular momentum, \[\hat{L}_{\pm}=\hat{L}_{i} \pm \mathrm{i} \hat{L}_{y}\]

which follow the commutation relations \([\hat{L}^{2}, \hat{L}_{\pm}]=0\) and \([\hat{L}_{z}, \hat{L}_{\pm}]=\pm \, \hbar \hat{L}_{\pm}\), and satisfy the eigenvalue equation \[\begin{align} &\hat{L}_{\pm}|\ell m\rangle=C^{\pm}_{\ell, m}|\ell(m \pm 1)\rangle \\ &C^{\pm}_{\ell, m}=\hbar[\ell(\ell+1)-m(m \pm 1)]^{1 / 2} \end{align}\]

Thus, \(\hat{L}_{+}\)and \(\hat{L}_{-}\)respectively raise and lower the magnetic quantum number by 1, changing the projection of the angular momentum on \(z\) by \(\pm \hbar\).

Spherically Symmetric Potential

Let’s examine the role of angular momentum for the case of a particle experiencing a spherically symmetric potential \(V(r)\) such as the hydrogen atom and a 3D isotropic harmonic oscillator. For a particle with mass \(m_{_{R}}\), the Hamiltonian is \[\hat{H}=-\frac{\hbar^{2}}{2 m_{_{R}}} \nabla^{2}+V(r)\]

Writing the kinetic energy operator in spherical coordinates, \[-\frac{\hbar^{2}}{2 m_{_{R}}} \nabla^{2}=-\frac{\hbar^{2}}{2 m_{_{R}}}\left(\frac{1}{r^{2}} \frac{\partial}{\partial r} r^{2} \frac{\partial}{\partial r}-\frac{1}{r^{2}} L^{2}\right)\]

where the square of the total angular momentum is \[L^{2}=-\frac{1}{\sin \theta}\left(\frac{1}{\sin \theta} \frac{\partial^{2}}{\partial \phi^{2}}+\frac{\partial}{\partial \theta} \sin \theta \frac{\partial}{\partial \theta}\right)\]

We note that this representation separates the radial dependence in the Hamiltonian from the angular part. We therefore expect that the overall wavefunction can be written as a product of a radial and an angular part in the form \[\psi(r, \theta, \phi)=R(r) Y(\theta, \phi)\]

Substituting eq. (1.6.13) into the TISE, we find that we solve for the orientational and radial wavefunctions separately. Considering solutions first to the angular part, we note that the potential is only a function of \(r\), and only need to consider the angular momentum. This leads to the identities in eq. (1.6.5) and eq. (1.6.6), revealing that the \(|\ell m\rangle\) wavefunctions projected onto spherical coordinates are represented by the spherical harmonics: \[Y_{\ell}^{m}(\theta, \phi)=N_{\ell m}^{(Y)} P_{\ell}^{|m|}(\cos \theta) \mathrm{e}^{i m \phi}\] where \(\ell=0,1,2 \ldots\) and \(m=0, \pm 1, \pm 2 \ldots P_{\ell}^{m}\) are the associated Legendre polynomials, and the normalization factor is \[N_{\ell m}^{(Y)}=(-1)^{(m+|m|) / 2} i^{\ell}\left[\frac{2 \ell+1}{4 \pi} \frac{(\ell-|m|) !}{(\ell+|m|) !}\right]^{1 / 2}\]

The angular components of the wavefunction are common to all eigenstates of spherically symmetric potentials. In chemistry, it is common to use real angular wavefunctions instead of the complex form in eq. (1.6.14). These are constructed from the linear combinations \(Y_{\ell, m} \pm Y_{\ell,-m}\).

Substituting eq. (1.6.11) and eq. (1.6.5) into eq. (1.6.10) leads to a new Hamiltonian that can be inserted into the Schrödinger equation. This can be solved as a purely radial problem for a given \[\left(-\frac{\hbar^{2}}{2 m_{_{R}}} \frac{\partial^{2}}{\partial r^{2}}+U(r, \ell)\right) \chi=E \chi\]

\(U\) plays the role of an effective potential: \[U(r, \ell)=V(r)+\frac{\hbar^{2}}{2 m_{_{R}}  r^{2}} \ell(\ell+1)\]

Equation (1.6.16) is known as the radial wave equation. It looks like the TISE for a one-dimensional problem in \(r\), where we could solve this equation for each value of \(\ell\). Note \(U\) has a repulsive wall as \(r \rightarrow 0\) due to the centrifugal kinetic energy that scales as \(r^{-2}\) for \(\ell>0\).

The wavefunctions defined in eq. (1.6.13) are normalized such that \[\int|\psi|^{2} d \Omega=1\]

where \[\int d \Omega \equiv \int_{0}^{\infty} r^{2} d r \int_{0}^{\pi} \sin \theta d \theta \int_{0}^{2 \pi} d \phi\]

If we restrict the integration to be over all angles, we find that the probability of finding a particle between a distance \(r\) and \(r+d r\) is \(P(r)=4 \pi r^{2}|R(r)|^{2}=4 \pi|\chi(r)|^{2}\).

To this point the treatment of orbital angular momentum is identical for any spherically symmetric potential. Now we must consider the specific form of the potential. For instance, in the case of the isotropic harmonic oscillator, \(U(r)=\frac{1}{2} \kappa r^{2}\). In the case of a free particle, we substitute \(V(r)=0\) in eq. (1.6.17) and find that the radial solutions can be written in terms of spherical Bessel functions, \(j_{\ell}\). Then the solutions to the full wavefunction for the free particle can be expanded in the product states: \[\Psi_{\ell, m}(r, \theta, \phi)=j_{\ell}(k r) Y_{\ell}^{m}(\theta, \phi)\]

where the wavevector \(k\) is defined as in eq. (1.4.4). Of course, these spherical waves form a poor substitute for describing linear motion of a particle with a plane wave basis set.

Hydrogen Atom

For a hydrogen-like atom, a single electron of charge \(e\) and rest mass \(m_{e}\) interacts with a nucleus of charge \(Z e\) under the influence of a Coulomb potential \[V_{H}(r)=-\frac{Z e^{2}}{4 \pi \dot{o}_{0}} \frac{1}{r}\]

We can simplify the expression by defining atomic units for distance and energy. The Bohr radius is defined as \[a_{0}=4 \pi \varepsilon_{0} \frac{\hbar^{2}}{m_{e} e^{2}}=5.2918 \times 10^{-11} m\]

and the Hartree is \[\mathcal{E}_{H}=\frac{1}{4 \pi \varepsilon_{0}} \frac{e^{2}}{a_{0}}=4.3598 \times 10^{-18} \mathrm{~J}=27.2 \mathrm{eV}\]

Written in terms of atomic units, we can see from eq. (1.6.23) that eq. (1.6.21) becomes \(\left(V / \mathcal{E}_{H}\right)=-Z /\left(r / a_{0}\right)\). Thus, the conversion effectively sets the SI variables \(m_{\mathrm{e}}=e=\left(4 \pi \varepsilon_{0}\right)^{-1}=\hbar\) \(=1\). Then the radial wave equation is \[\frac{\partial^{2} \chi}{\partial r^{2}}+\left(\frac{2 Z}{r}-\frac{\ell(\ell+1)}{r^{2}}\right) \chi=2 E \chi\]

The effective potential within the parentheses in eq. (1.6.24) is shown in Figure 1 for varying \(\ell\). Solutions to the radial wavefunction for the hydrogen atom take the form \[R_{n \ell}(r)=N_{n \ell}^{(H)} \rho^{\ell} \mathcal{L}_{n+\ell}^{2 \ell+1}(\rho) e^{-\rho / 2}\]

where the reduced radius \(\rho=2 r / n a_{0}\) and \(\mathcal{L}_{k}^{\alpha}(z)\) are the associated Laguerre polynomials. The primary quantum number takes on integer values \(n=1,2,3 \ldots\), and \(\ell\) is constrained such that \(\ell=\) \(0,1,2 \ldots n-1\). The radial normalization factor in eq. (1.6.25) is \[N_{n \ell}^{(H)}=-\frac{2}{n^{3} a_{0}^{3 / 2}}\left[\frac{(n-\ell-1) !}{[(n+1) !]^{3}}\right]^{1 / 2}\]

The energy eigenvalues are \[E_{n}=-\frac{Z^{2}}{2 n^{2}} \mathcal{E}_{H}\]

Figure \(\PageIndex{1}\). The radial effective potential for the hydrogen atom, \(U_{\text {eff}}(\rho)\).

Electron Spin

In describing electronic wavefunctions, we also include the contribution from electron spin. The electron spin angular momentum \(S\) and its \(z\)-projection are quantized as \[\begin{align} S^{2}\left|s m_{s}\right\rangle &=\hbar^{2} s(s+1)\left|s m_{s}\right\rangle & s & =0, \tfrac{1}{2},1, \tfrac{3}{2},2 \ldots \\[3pt] S_{z}\left|s m_{s}\right\rangle &=\hbar m_{s}\left|s m_{s}\right\rangle & m_{s} &=-s,-s+1, \ldots, s \end{align}\]

where the electron spin eigenstates \(\left|s m_{s}\right\rangle\), are labeled by the electron spin angular momentum quantum number \(s\), and the spin magnetic quantum number \(m_{s}\). The number of values of \(S_{z}\) is \(2 s+1\) and is referred to as the spin multiplicity. As fermions, electrons have half-integer spin, and each unpaired electron contributes \(\frac{1}{2}\) to the electron spin quantum number \(s\). A single unpaired electron has \(s=\frac{1}{2}\), for which \(m_{s}=\pm \frac{1}{2}\) corresponding to spin-up and spin-down configurations. For multielectron systems, the spin is calculated as the vector sum of spins, essentially \(\frac{1}{2}\) times the number of unpaired electrons.

The resulting total angular momentum for an electron is \(J=L+S . J\) has associated with it the total angular momentum quantum number \(j\), which takes on values of \(j=|\ell-s|,|\ell-s|+1, \ldots, \ell+s\). The additive nature of the orbital and spin contributions to the angular momentum leads to a total electronic wavefunction that is a product of spatial and spin wavefunctions. \[\Psi_{t o t}=\Psi(r, \theta, \phi)\left|s m_{s}\right\rangle\]

Thus, the state of an electron can be specified by four quantum numbers \(\Psi_{\text {tot }}=\left|n \ \ell \ m_{\ell} \ m_{s}\right\rangle\).

Rigid Rotor

In the case of an isolated molecule spinning freely about its center of mass, the total angular momentum \(J\) is obtained from the sum of the orbital angular momentum \(L\) and spin angular momentum \(S\) for the molecular constituents: \(J=L+S\), where \(L=\sum_{\mathrm{i}} L_{\mathrm{i}}\) and \(S=\sum_{\mathrm{i}} S_{\mathrm{i}}\). The case of the rigid rotor refers to the minimal model for the rotational quantum states of a freely spinning object with one principle axis of rotation (cylindrical symmetry) and no magnetic spin. Then, the Hamiltonian is given by the rotational kinetic energy \[H_{r o t}=\frac{\hat{J}^{2}}{2 I}\]

\(I\) is the moment of inertia about the principle axis of rotation. The eigenfunctions for this Hamiltonian \(|J M\rangle\) can be expressed as the spherical harmonics \(Y_{J, M}(\theta, \phi)\) in eq. (1.6.14) \[\begin{align} \hat{J}^{2}|J M\rangle & =\hbar^{2} J(J+1)|J M\rangle & J &=0,1,2 \ldots \\[3pt]  \hat{J}_{z}|J M\rangle &=M \hbar|J M\rangle & M &=-J,-J+1, \ldots, J \end{align}\]

\(J\) is the rotational quantum number. \(M\) is its projection onto the \(z\) axis. The energy eigenvalues for \(H_{r o t}\) are \[E_{J, M}=\bar{B} J(J+1)\]

where the rotational constant \(\bar{B}\) is \[\bar{B}=\frac{\hbar^{2}}{2 I}\]

More commonly, \(\bar{B}\) is given in spectroscopic units of \(\mathrm{cm}^{-1}\) using \(\bar{B}=h / 8 \pi^{2} I c\).

 

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[1].    N. Zettili, Quantum Mechanics: Concepts and Applications, 2nd ed. (Wiley, Chichester, New York, 2009).