Angular momentum: commutators

• Definition of angular momentum \(\hat{L} = \hat{r} \times \hat{p}\)
• position/momentum commutator relations in 3D: \([\hat{p}_i, r_j] = -i\hbar \delta_{ij}\)

\[ [\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z, \qquad [\hat{L}_y, \hat{L}_z] = i\hbar \hat{L}_x, \qquad [\hat{L}_z, \hat{L}_x] = i\hbar \hat{L}_y, \qquad \]

• Traditionally written using the Levi-Civita symbol \(\epsilon_{ijk}\):
  • \(\epsilon_{ijk} = -1\) for odd permutations \((1,2,3)\) = \(132\), \(213\), \(321\)
  • \(\epsilon_{ijk} = 1\) for even permutations \((1,2,3)\) = \(123\), \(312\), \(231\)
  • \(\epsilon_{ijk} = 0\) for any same indices appearing \((1,2,3)\) = \(223\), \(311\), \(333\), etc.

\[ [\hat{L}_i, \hat{L}_j] = i\hbar \sum_k \epsilon_{ijk} \hat{L}_k \]

Angular momentum: eigenvalues

• \(\hat{\bf L} = (\hat{L_x}, \hat{L_y}, \hat{L_z})\) is a vector with a magnitude \(\|\hat{L}\|\) and direction,
• \(\hat{L_x}\), \(\hat{L_y}\), \(\hat{L_z}\) don’t commute: no common eigenstate basis,
• Magnitude (squared) of the momentum is \(\hat{\bf L}^2\)
• Magnitude does commute with each vector component
  \(\longrightarrow\) Use eigenstates basis (\(| a, b \rangle\)) of \(\hat{\bf L}^2\) and one component \(\hat{L_z}\).

\[ \hat{\bf L}^2 | a,b \rangle = a | a,b \rangle \, \qquad \hat{L_z} | a,b \rangle = b | a,b \rangle \]

Algebraic solution: ladder operators

\[ \hat{\bf L}^2\, | a,\,b \rangle = a\, | a,\,b \rangle \, \qquad \hat{L_z}\, | a,\,b \rangle = b\, | a,\,b \rangle \]

• Define ladder operators \(\quad \hat{L_\pm} = \hat{L}_x \pm i\hat{L}_y\)
• Is \(\hat{L_\pm} | a,\,b \rangle\) an eigenstate?
• Yes, since \([\hat{\bf L}^2, \hat{L}_i] = 0\):

\[ \hat{\bf L}^2 \left(\hat{L}_\pm \, | a,\,b \rangle \right) = \hat{L}_\pm \hat{\bf L}^2 \, | a,\,b \rangle = a \left(\hat{L}_\pm \, | a,\,b \rangle \right) \]

• So \(\hat{L}_\pm \, | a,\,b \rangle\) is eigenstate of \(\hat{\bf L}^2\) (or the zero state) with eigenvalue \(a\).
• What about \(\hat{L}_z\)?

Algebraic solution: ladder operators

• The ladder operators \(\hat{L}_\pm\) do not commute with \(\hat{L}_z\):

\[ [\hat{L}_z, \hat{L}_\pm] = \pm \hbar \hat{L}_\pm \]

• We use this relation to extract the impact on the eigenstates

\[ \hat{L}_z \hat{L}_\pm \, | a,\,b \rangle = \hat{L}_\pm \hat{L}_z \, | a,\,b \rangle + [\hat{L}_z \hat{L}_\pm] \, | a,\,b \rangle = (b \pm \hbar) \hat{L}_\pm \, | a,\,b \rangle \]

• So \(\hat{L}_\pm \, | a,\,b \rangle\) is eigenstate of \(\hat{L}_z\) (or the zero state) with eigenvalue \(b \pm \hbar\):

\[ \hat{L}_\pm \, | a,\,b \rangle = C_\pm(a,b) | a,\,b \pm \hbar \rangle \]

With \(C_\pm(a,b)\) a normalization factor (still unknown)

Eigenvalue ranges

• Procedure is similar to the one for finding the zeroth eigenstate of the harmonic oscillator. An upper/lower bound leads to discrete ladder
• We use \(\langle a, b | a, b \rangle = 1\) and express that the norm

\[ \|\hat{L}_\pm | a, b \rangle\|^2 = \langle a, b | \hat{L}^\dagger_\pm \hat{L}_\pm | a, b \rangle \geq 0 \]

We can rewrite \(\hat{L}^\dagger_\pm \hat{L}_\pm = \hat{L}_\mp \hat{L}_\pm\) as:

\[ \hat{L}_\mp \hat{L}_\pm = (\hat{L}_x \mp i\hat{L}_y)(\hat{L}_x \pm i\hat{L}_y) = \hat{L}^2_x + \hat{L}^2_y \pm i[\hat{L}_x, \hat{L}_y] = \hat{\bf L}^2 - \hat{L}_z^2 \mp \hbar \hat{L}_z \]

This lead to a relation between the eigenvalues

\[ \langle a, b | \hat{L}^\dagger_\pm \hat{L}_\pm | a, b \rangle = \langle a, b | \hat{\bf L}^2 - \hat{L}_z^2 \mp \hbar \hat{L}_z | a, b \rangle = (a - b^2 \mp \hbar b) \langle a, b | a, b \rangle \geq 0 \]

We will use \(a \geq 0\) and \(b \in \Re\) to find \(b\) and \(a\) values

Eigenvalue range: m

• \(a \geq 0\) and \(b \in \Re\)
• Smallest eigenvalue \(a = 0\) gives limits on \(b\):

\[ \begin{aligned} \langle a, b_\textrm{max} | \hat{L}^\dagger_+ \hat{L}_+ | a, b_\textrm{max} \rangle &= a - b_\textrm{max}^2 - \hbar b_\textrm{max} = 0\\ \langle a, b_\textrm{min} | \hat{L}^\dagger_- \hat{L}_- | a, b_\textrm{min} \rangle &= a - b_\textrm{min}^2 + \hbar b_\textrm{min} = 0 \end{aligned} \]

This both determines \(a\):

\[ \begin{aligned} a &= b_\textrm{max}^2 + \hbar b_\textrm{max}\\ a &= b_\textrm{min}^2 - \hbar b_\textrm{min} \end{aligned} \]

• Resulting in \(b_\textrm{min} = - b_\textrm{max}\)

• For any \(a\): starting from \(| a, b_\textrm{min}\rangle\) and move up the ladder with \(\hat{L}_+\) until \(| a, b_\textrm{max} \rangle\):

\[ b_\textrm{min}, \quad b_\textrm{min} + \hbar, \quad b_\textrm{min} + 2\hbar, \quad\dots\quad, \quad b_\textrm{min} + n\hbar = b_\textrm{max} \]

Eigenvalue range: l and m

• Since also \(b_\textrm{max} = - b_\textrm{min}\), the eigenvalue range \(-(n - 1)\hbar/2, \dots , (n - 1)\hbar/2\) or \(-n\hbar/2, \dots, n\hbar/2\)
• Eigenvalues of \(\hat{L}_z\) are \(b = m\hbar\) with \(m_\textrm{max} = -m_\textrm{min} = l\)
• Eigenvalues \(a\) of \(\hat{\bf L}^2\) are given by

\[ a = m_\textrm{max}\hbar (m_\textrm{max}\hbar + \hbar) = l(l+1)\hbar^2 \]

• In summary:

\[ \boxed{\hat{\bf L}^2|l, m\rangle = l(l+1)\hbar^2|l, m\rangle} \qquad \qquad \boxed{\hat{L}_z |l, m\rangle = m \hbar|l, m\rangle} \]

• \(l\) and \(m\) either even integer or odd half-integers with integer steps

Normalization

\[ \boxed{\hat{\bf L}^2|l, m\rangle = l(l+1)\hbar^2|l, m\rangle, \qquad \qquad \hat{L}_z |l, m\rangle = m \hbar|l, m\rangle} \]

Using the previous definition of the ladder operator

\[ \begin{aligned} 1 = \langle l, m | \hat{L}^\dagger_\pm \hat{L}_\pm | l, m \rangle &= \langle l, m | \hat{\bf L}^2 - \hat{L}_z^2 \pm \hbar \hat{L}_z | l, m \rangle\\ \end{aligned} \]

Taking the square root of the norm results in:

\[ \boxed{\begin{aligned} \hat{L}_+ | l, m \rangle &= \sqrt{l(l+1) - m(m+1)} \hbar | l, m+1 \rangle\\ \hat{L}_- | l, m \rangle &= \sqrt{l(l+1) - m(m-1)} \hbar | l, m-1 \rangle\\ \end{aligned}} \]

Corresponding eigenfunctions

• Constructing the eigenstate functions: Spherical harmonics \(\quad Y_{lm}(\theta, \phi) = \langle \theta, \phi| l, m \rangle\)

Expressing the operators: \(\hat{\bf L}^2\), \(\hat{L}_z\), and \(\hat{L}_\pm\) with \(\hat{\bf L} = {\bf r} \times \hat{\bf p} = -i\hbar{\bf r} \times \nabla\)

The gradient \(\nabla\) in spherical coordinates:

\[ \nabla = \vec{e}_r\frac{\partial}{\partial r} + \vec{e}_\theta\frac{1}{r}\frac{\partial}{\partial \theta} + \vec{e}_\phi\frac{1}{r \sin\theta} \frac{\partial}{\partial \phi} \]

\[ \Longrightarrow\boxed{\begin{aligned} \hat{L}_z & = - i \hbar \frac{\partial}{\partial\phi}, \quad \hat{L}_\pm = \hbar e^{\pm i\phi} \left[\pm \frac{\partial}{\partial \theta} + i \cot \theta \frac{\partial}{\partial \phi}\right]\\ &\\ \hat{\bf L}^2 &= -\hbar^2 \left[ \frac{1}{\sin\theta} \frac{\partial \sin \theta \frac{\partial}{\partial \theta}}{\partial \theta} + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial \phi^2}\right] \end{aligned}} \]

The \(\phi\) - component

\[ \boxed{\begin{aligned} \hat{L}_z & = - i \hbar \frac{\partial}{\partial\phi}, \quad \hat{L}_\pm = \hbar e^{\pm i\phi} \left[\pm \frac{\partial}{\partial \theta} + i \cot \theta \frac{\partial}{\partial \phi}\right]\\ \end{aligned}} \]

• Apply \(\hat{L}_z\) to proposed solution \(Y_{lm}(\theta, \phi)\)

\[ \begin{aligned} \hat{L}_z Y_{lm}(\theta, \phi) = - i \hbar \frac{\partial}{\partial\phi}Y_{lm}(\theta, \phi) = m\hbar Y_{lm}(\theta, \phi) \end{aligned} \]

Separation of the variables:

\[ Y_{lm}(\theta, \phi) = F(\theta) \, e^{im\phi}, \quad \textrm{with} -l \leq m \leq l \]

• If \(l\) is an integer, the requirement \(e^{im\phi} = e^{im(\phi + 2\pi)}\) is satisfied

The \(\theta\) - component

\[ \boxed{\begin{aligned} \hat{L}_z & = - i \hbar \frac{\partial}{\partial\phi}, \quad \hat{L}_\pm = \hbar e^{\pm i\phi} \left[\pm \frac{\partial}{\partial \theta} + i \cot \theta \frac{\partial}{\partial \phi}\right]\\ \end{aligned}} \]

• Apply \(\hat{L}_\pm\) to proposed solution \(F(\theta)\)
• Use the fact that \(m_\textrm{max}\) is \(\,l\): \(\,\,\hat{L}_+ | l, l \rangle = 0\):

\[ \begin{aligned} 0 = \langle \theta, \phi | \hat{L}_+ | l,l\rangle &= \hbar e^{i\phi} \left[\frac{\partial}{\partial \theta} + i \cot \theta \frac{\partial}{\partial \phi}\right] e^{il\phi} F(\theta)\\ & = \hbar e^{i(l+1)\phi} \left[\frac{\partial}{\partial \theta} - l \cot \theta\right] F(\theta)\\ \end{aligned} \]

\[ \Rightarrow \frac{\partial }{\partial \theta}F(\theta) = l \cot \theta F(\theta)\quad \Rightarrow \quad F(\theta) = C \sin^l(\theta) \]

Spherical Harmonics

The lower states with \(m < l\) are generated by applying \(\hat{L}_-\)

\[ \begin{aligned} Y_{lm}(\theta, \phi) = C \hat{L}_-^{(l-m)} \left[\sin^l \theta e^{il\phi}\right] = C \left[\frac{\partial}{\partial \theta} + i \cot \theta \frac{\partial}{\partial \phi}\right] \left[\sin^l \theta e^{il\phi}\right] \end{aligned} \]

The solutions are the spherical harmonics:

\[ \begin{aligned} Y_{lm}(\theta, \phi) = (-1)^{m+|m|} \left[ \frac{2l + 1}{4\pi} \,\frac{(l - |m|)!}{(l + |m|)!} \right]^{1/2} P^{|m|}_l(\cos\theta) e^{im\phi} \end{aligned} \]

with the associated Legendre polynomials

\[ P^{m}_l(x) = \frac{(1-x^2)^m/2}{2^l l!} \frac{d^{m+l}}{dx^{m+l}}(x^2 - 1)^l \]

Summary angular momentum

Common basis of eigenstates of angular momentum:

\[ \boxed{\hat{\bf L}^2|l, m\rangle = l(l+1)\hbar^2|l, m\rangle} \qquad \qquad \boxed{\hat{L}_z |l, m\rangle = m \hbar|l, m\rangle} \]

• \(l\) and \(m = -l, \dots, -1, 0, 1,\dots, l\) integers to get single-valued \(Y(\theta, \phi) = \Theta(\theta)\Phi(\phi) = \Theta(\theta) e^{im\phi}\)
• The solutions are the spherical harmonics \(Y(\theta, \phi)\):

\[ \begin{aligned} Y_{lm}(\theta, \phi) = (-1)^{m+|m|} \left[ \frac{2l + 1}{4\pi} \,\frac{(l - |m|)!}{(l + |m|)!} \right]^{1/2} P^{|m|}_l(\cos\theta) e^{im\phi} \end{aligned} \]

with the associated Legendre polynomials

\[ P^{m}_l(x) = \frac{(1-x^2)^m/2}{2^l l!} \frac{d^{m+l}}{dx^{m+l}}(x^2 - 1)^l \]

Summary angular momentum

\[ \begin{aligned} Y_{lm}(\theta, \phi) = (-1)^{m+|m|} \left[ \frac{2l + 1}{4\pi} \,\frac{(l - |m|)!}{(l + |m|)!} \right]^{1/2} P^{|m|}_l(\cos\theta) e^{im\phi} \end{aligned} \]

\[ P^{m}_l(x) = \frac{(1-x^2)^m/2}{2^l l!} \frac{d^{m+l}}{dx^{m+l}}(x^2 - 1)^l \]

Lower order \(P^{m}_l(z)\) polynomials:

Lower order eigenfunctions

\[ \begin{aligned} Y_{lm}(\theta, \phi) = (-1)^{m+|m|} \left[ \frac{2l + 1}{4\pi} \,\frac{(l - |m|)!}{(l + |m|)!} \right]^{1/2} P^{|m|}_l(\cos\theta) e^{im\phi} \end{aligned} \]

Angular momentum eigenfunction in spherical and carthesian coordinates

\[ \begin{aligned} Y_{l,m} & \qquad Y_{l,m}(\theta, \phi) & Y_{l,m}(x, y, z) \\ Y_{0,0} & \qquad \frac{1}{\sqrt{4\pi}} \qquad & \frac{1}{\sqrt{4\pi}}\\ & & \\ Y_{1,-1} & \qquad \sqrt{\frac{3}{8\pi}}\, \sin\theta \, e^{-i\phi} & \sqrt{\frac{3}{8\pi}}\,\frac{x - iy}{r}\\ Y_{1,0} & \qquad \frac{1}{\sqrt{4\pi}}\, \cos\theta \qquad & \frac{1}{\sqrt{4\pi}}\frac{z}{r}\\ Y_{1,1} & \qquad \sqrt{\frac{3}{8\pi}}\, \sin\theta \, e^{i\phi} & -\sqrt{\frac{3}{8\pi}}\,\frac{x + iy}{r}\\ \end{aligned} \]

Visual representation

• The \(l = 1\) and \(\,\,m = \pm 1\) eigenstates have a torus (donut) shape
• Specific basis connected to z-axis via \(\hat{L}_z\),
• Other choices \(\hat{L}_x\) or \(\hat{L}_y\) possible, correspond to other bases
• The magnetic quantum numbers \(\,m_l\,\) are eigenvalues of \(\hat{L}_z\)

Spherical Harmonics: real basis

• Rotating the basis using spherical symmetry
• Superposition of eigenstates (\(\hat{L}_z\)), start from the solutions:

\[ \begin{aligned} Y_{lm}(\theta, \phi) = (-1)^{m+|m|} \left[ \frac{2l + 1}{4\pi} \,\frac{(l - |m|)!}{(l + |m|)!} \right]^{1/2} P^{|m|}_l(\cos\theta) e^{im\phi} \end{aligned} \]

Use the identities \(\,\,\cos(m\phi) = \frac{e^{im\phi} + e^{-im\phi}}{2}\,\,\) and \(\,\,\sin(m\phi) = \frac{e^{im\phi} - e^{-im\phi}}{2i}\,\):

\[ \left\{ \begin{aligned} Y_x(\theta, \phi) &= \Re\{Y_{lm}\} = (-1)^{m+|m|} \left[ \frac{2l + 1}{4\pi} \,\frac{(l - |m|)!}{(l + |m|)!} \right]^{1/2} P^{|m|}_l(\cos\theta)\, \cos(m\phi)\\ Y_y(\theta, \phi) &= \Im\{Y_{lm}\} = (-1)^{m+|m|} \left[ \frac{2l + 1}{4\pi} \,\frac{(l - |m|)!}{(l + |m|)!} \right]^{1/2} P^{|m|}_l(\cos\theta)\, \sin(m\phi) \end{aligned} \right. \]

Spherical Harmonics: real basis

• Rotating the basis using spherical symmetry
• Superposition of eigenstates (\(\hat{L}_z\)), start from the solutions
• For \(l = 1\) the carthesian formula shows this very clear:

\[ \begin{aligned} Y_{l,m} & \qquad Y_{l,m}(\theta, \phi) & Y_{l,m}(x, y, z) \\ Y_{0,0} & \qquad \frac{1}{\sqrt{4\pi}} \qquad & \frac{1}{\sqrt{4\pi}}\\ & & \\ Y_{1,-1} & \qquad \sqrt{\frac{3}{8\pi}}\, \sin\theta \, e^{-i\phi} & \sqrt{\frac{3}{8\pi}}\,\frac{x - iy}{r}\\ Y_{1,0} & \qquad \frac{1}{\sqrt{4\pi}}\, \cos\theta \qquad & \frac{1}{\sqrt{4\pi}}\frac{z}{r}\\ Y_{1,1} & \qquad \sqrt{\frac{3}{8\pi}}\, \sin\theta \, e^{i\phi} & -\sqrt{\frac{3}{8\pi}}\,\frac{x + iy}{r}\\ \end{aligned} \]

Different basis for Y

• Different eigenstate basis: Symmetric dumbells vs. donuts in xy-plane
• Superposition of a \(Y_x\) and a \(Y_y\) (p-orbitals) gives a torus(donut) and vice versa.

Spherical Harmonics wave function

• Appropriate real-valued orthonormal basis, superposition of orbitals

Spherical Harmonics Probability

• Appropriate real-valued orthonormal basis, superposition of orbitals

Superposition

• Electrons can be in a superposition of orbital states.
• We used a specific basis, other bases are possible
• The electron magnetic quantum number \(m_l\) is an eigenvalue of the donut basis of eigenstates

Evolution in time

• Phase factor \(\,\,e^{i m \phi + i E_{nl} t/\hbar}\) over time
  • \(Y_{10}\) (\(p_0\)-orbital): Phase changes in time, phase difference of \(\pi\)
  • \(Y_{11}\) (\(p_1\)-orbital): Phase rotates

Diatomic molecule: Hamiltonian

• Two atoms with rigid bond (approximation, ignore stretching)
• Center of mass:

\[ \vec{R} = \frac{m_1 \vec{r_1} + m_2 \vec{r}_2}{m_1 + m_2}, \qquad \vec{r} = \vec{r}_2 - \vec{r}_1, \qquad M = m_1 + m_2, \qquad \mu = \frac{m_1 m_2}{m_1 + m_2} \]

• Hamiltonian contains linear momentum: center of mass moving
• and internal angular momentum with inertia \(I = \mu r^2\)

\[ \hat{H} = \frac{\hat{P}^2}{2M} + \frac{\hat{L}^2}{2I} \]

Diatomic molecule: Eigenenergies

\[ \hat{H} = \frac{\hat{P}^2}{2M} + \frac{\hat{L}^2}{2I} \]

• Center of mass \(\vec{R}\), and orientation \(\vec{r}\) are independent
• Separation of variables: \(\quad\psi(\vec{R}, \vec{r}) = e^{i\vec{K}\cdot\vec{R}} \, Y(\theta, \phi)\)
• Eigenenergies:

\[ E_{\vec{K}, l} = \frac{\hbar K^2}{2m} + \frac{\hbar^2 l(l+1)}{2I}. \]

• (2l + 1) Degeneracy in magnetic quantum number \(m \equiv m_s\)

Central potential: Hamiltonian

\[ \hat{H} = \frac{\hat{p}^2}{2m} + V(r) \]

• Central potential \(\longrightarrow\) go to spherical coordinates
• We need \(\hat{p}^2 = -\hbar^2 \nabla^2\) in spherical coordinates with:

\[ \nabla = \vec{e}_r\frac{\partial}{\partial r} + \vec{e}_\theta\frac{1}{r}\frac{\partial}{\partial \theta} + \vec{e}_\phi\frac{1}{r \sin\theta} \frac{\partial}{\partial \phi} \]

\[ \nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r} r^2 \frac{\partial}{\partial r} + \frac{1}{r^2} \left[\frac{1}{\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial}{\partial \theta}\right) + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial \phi^2}\right] \]

\[ \nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r} r^2 \frac{\partial}{\partial r} - \frac{\hat{L}^2}{r^2\hbar^2} \]

Central potential: Hamiltonian

\[ \hat{H} = \frac{\hat{p}^2}{2m} + V(r) \]

• Central potential \(\longrightarrow\) go to spherical coordinates
• We need \(\hat{p}^2 = -\hbar^2 \nabla^2\) in spherical coordinates with:

\[ \boxed{\quad\hat{H} = \left[-\frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r}\right) + \frac{\hat{L}^2}{2m r^2} + V(r)\right] \psi(\vec{r}) = E \psi(\vec{r})\quad} \]

Central potential: solutions

\[ \boxed{\left[-\frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r}\right) + \frac{\hat{L}^2}{2m r^2} + V(r)\right] \psi(\vec{r}) = E \psi(\vec{r})\quad} \]

Separation of variables \(\psi(\vec{r}) = R(r) Y(\theta, \phi)\)

\[ \left[-\frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r}\right) + \frac{\hat{L}^2}{2m r^2} + V(r)\right] R(r) Y(\theta, \phi) = E R(r) Y(\theta, \phi) \]

\[ \Longrightarrow \quad \left[-\frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r}\right) + \frac{l(l+1)\hbar^2}{2m r^2} + V(r)\right] R(r) = E R(r) \]

Simplifying by substituting \(R(r) = \chi(r)/r\) and defining a new effective potential \(V_e(r)\):

\[ \left[-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial r^2} + V_{e}(r)\right] \chi(r) = E\chi(r) \quad\textrm{ with}\quad V_e(r) = \frac{l(l+1)\hbar^2}{2m r^2} + V(r) \]

Boundary conditions

\[ \left[-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial r^2} + V_{e}(r)\right] \chi(r) = E\chi(r) \quad\textrm{ with}\quad V_e(r) = \frac{l(l+1)\hbar^2}{2m r^2} + V(r) \]

• To find \(\psi(\vec{r}) = \chi(r)/r Y(\theta, \phi)\) we need to find \(\chi(r)\)
• Because \(R(r) = \chi(r)/r\) we require \(\chi(0) = 0\)
• Normalization

\[ \int_\Omega |\psi(\vec{r})|^2 d\vec{r} = \int_0^\infty |R(r)|^2 r^2 dr = \int_0^\infty |\chi(r)|^2 dr = 1 \]

A valid bound state requires \(\chi(r)\) to decrease fast enough: \(\quad\lim_{r\longrightarrow\infty}\chi(r) \le \frac{1}{\sqrt{r}}\)

For a solution we need to know the potential \(V(r)\)

The Hydrogen atom: Hamiltonian

• Hamiltonian for two particles, electron and proton
• Electron and proton interact via Coulomb potential

\[ \hat{H} = -\frac{\hbar^2}{2m_e}\nabla_e^2 -\frac{\hbar^2}{2m_p}\nabla_p^2 + V(|\vec{r}_e - \vec{r}_p|), \quad\textrm{ with}\quad V(r) = -\frac{e^2}{4\pi \varepsilon_0 |\vec{r}_e - \vec{r}_p|}. \]

• Schrodinger equation with \(\psi(\vec{r}_e, \vec{r}_p)\) having 6 degrees of freedom:

\[ \left[-\frac{\hbar^2}{2m_e}\nabla_e^2 -\frac{\hbar^2}{2m_p}\nabla_p^2 + V(|\vec{r}_e - \vec{r}_p|)\right] \psi(\vec{r}_e, \vec{r}_p)= \psi(\vec{r}_e, \vec{r}_p). \]

• Go over to relative coordinates and center of mass

Hydrogen atom: Relative coordinates

• Two particles, electron and proton, use center of mass \(\vec{R}\) and relative coordinates:

\[ \vec{R} = \frac{m_e \vec{r_e} + m_p \vec{r}_p}{m_e + m_p}, \qquad \vec{r} = \vec{r}_e - \vec{r}_p, \qquad M = m_e + m_p, \qquad \mu = \frac{m_e m_p}{m_e + m_p} \]

• The Schrodinger equation becomes:

\[ \left[-\frac{\hbar^2}{2M} \nabla_R^2 - \frac{\hbar^2}{2\mu} \nabla_r^2 + V(r)\right] \psi(\vec{R}, \vec{r}) = E \psi(\vec{R}, \vec{r}) \]

• Separation of variable \(-\frac{\hbar^2}{2M} \nabla_R^2\) leads to a factor \(e^{i\vec{K}\cdot\vec{R}}\) with \(E_R = \frac{K^2\hbar^2}{2M}\)

\[ \left[- \frac{\hbar^2}{2\mu} \nabla^2 + V(r)\right] \psi(\vec{R}, \vec{r}) = E \psi(\vec{R}, \vec{r}) \]

Hydrogen atom: TISE

\[ \left[- \frac{\hbar^2}{2\mu} \nabla^2 + V(r)\right] \psi(\vec{R}, \vec{r}) = E \psi(\vec{R}, \vec{r}) \]

Similar: Separation of variables \(\psi(\vec{r}) = R(r) Y(\theta, \phi)\) and using \(Y_{lm}(\theta, \phi)\) as solutions for the angular part:

\[ \left[-\frac{\hbar^2}{2\mu} \left(\frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r}\right) + \frac{\hat{L}^2}{2\mu r^2} + V(r)\right] R(r) Y(\theta, \phi) = E R(r) Y(\theta, \phi) \]

\[ \Longrightarrow \quad \left[-\frac{\hbar^2}{2\mu} \left(\frac{d^2}{d r^2} + \frac{2}{r}\frac{d}{d r}\right) + \frac{l(l+1)\hbar^2}{2\mu r^2} + V(r)\right] R(r) = E R(r) \]

Simplifying by substituting \(R(r) = \chi(r)/r\) and defining a new effective potential \(V_e(r)\):

\[ \left[-\frac{\hbar^2}{2\mu} \frac{d^2}{d r^2} + V_{e}(r)\right] \chi(r) = E\chi(r) \quad\textrm{ with}\quad V_e(r) = \frac{l(l+1)\hbar^2}{2\mu r^2} + V(r) \]

Solution radial equation

\[ \left[-\frac{\hbar^2}{2\mu} \frac{d^2}{d r^2} + V_{e}(r)\right] \chi(r) = E\chi(r) \quad\textrm{ with}\quad V_e(r) = \frac{l(l+1)\hbar^2}{2\mu r^2} + V(r) \]

Define energy \(\,\,E = -\frac{\mathrm{Ry}}{n^2}\,\) with \(\,n \in \mathbb{R}\,\) and radial coordinate \(\,\,s = \frac{2\,r}{n\,a_0}\,\), where:

• Bohr radius: radius of Hydrogen atom: \(\,\,a_0 = \frac{4\pi\varepsilon_0\hbar^2}{e^2\mu} = 0.529\, \overset{\lower.5em\circ}{\mathrm{A}}\)
• Rydberg energy: \(\mathrm{Ry} = \frac{\hbar^2}{2\mu a_0^2} = \frac{\mu}{2}\left(\frac{e^2}{4\pi\varepsilon_0\hbar}\right)^2 = 13.6\) eV

This simplifies the radial equation:

\[ \Longrightarrow \quad \frac{d^2 \chi(s)}{d s^2} - \left[\frac{l(l+1)}{s^2} - \frac{n}{s} + \frac{1}{4}\right] \chi(s) = 0 \]

Solution radial equation CTU’d

\[ \Longrightarrow \quad \frac{d^2 \chi(s)}{d s^2} - \left[\frac{l(l+1)}{s^2} - \frac{n}{s} + \frac{1}{4}\right] \chi(s) = 0 \]

• Propose a solution \(\chi(s) \propto \exp(-s/2)\) to remove the \(\frac{1}{4}\) (and decaying)
• To remove the \(\frac{l(l+1)}{s^2}\) term we further require \(\chi(s) \propto s^{l+1}\)
• We obtain proposed solution of the form (with \(\,f(s)\,\) unknown):

\[ \chi(s) = f(s) s^{l+1} \exp(-s/2) \]

• Substitution leads to a differential equation for \(f(s)\):

\[ \quad s \, \frac{d^2 f(s)}{d s^2} - [s - 2\,(l+1)] \frac{d f(s)}{d s} + [n - (l+1)]\, f(s) = 0 \]

Solution radial equation CTU’d

\[ \quad s \, \frac{d^2 f(s)}{d s^2} - [s - 2\,(l+1)] \frac{d f(s)}{d s} + [n - (l+1)]\, f(s) = 0 \]

• Solve by expanding \(f(s)\) as a power series \(f(s) = \sum_{j=0}^\infty a_j s^j\)
• Solution demands that the power series is finite:
  • \(n\) must be an integer, and
  • \(n > l+1\)
• \(f(s)\) is then given by the associated Laguerre polynomials:

\[ L^{2l+1}_{n-l-1} = \sum_{q=0}^{n-l-1}(-1)^q \frac{(n+l)!}{(n-l-1-q)!\,(2l+1+q)!} \, s^q \]

Solutions

Total solutions together with the associated Laguerre polynomials

\[ \chi(s) = s^{l+1} L_{n-l-1}^{2l+1}(s) e^{-s/2}, \qquad s = \frac{2r}{na_0} \]

Normalization factor for \(R(r) = \chi(s)/r\):

\[ 1 = \int^\infty_0 R^2(r) \, r^2\, dr = \int^\infty_0 s^{2l} \left( L^{2l+1}_{n-l-1}(s) \right) \, e^{-s} \, s^2\, ds = \frac{2\,n\,(n+l)!}{(n - l - 1)!} \]

The normalized radial solutions:

\[ \boxed{\quad R_{nl}(s) = \left[ \frac{(n - l - 1)!}{2\,n\,(n+l)!} \left(\frac{2}{n\,a_0}\right)^3 \right]^{1/2} s^l \, L^{2l+1}_{n-l-1}(s) \, e^{-s/2}\quad} \]

And the eigenenergies are given by \(E_n = -\frac{\mathrm{Ry}}{n^2} = -\frac{13.6}{n^2}\) eV

Radial part: wave function

• Radial wave function \(R(r)\) has \(\,\,n - l - 1\) zeros

Radial part: probability

• Electron localization:
  • S-orbitals large probability in zero
  • P-orbital zero probability in zero

Orbitals: Radial + Angular parts

The wave function \(\,\,\psi(\vec{r}) = R_{nl}(r) Y_{lm}(\theta, \phi)\)

\[ \boxed{\begin{aligned} & \\ \quad Y_{lm}(\theta, \phi) &= (-1)^{m+|m|} \left[ \frac{2l + 1}{4\pi} \,\frac{(l - |m|)!}{(l + |m|)!} \right]^{1/2} P^{|m|}_l(\cos\theta) e^{im\phi} \quad\\ & \\ R_{nl}(s) &= \left[ \frac{(n - l - 1)!}{2\,n\,(n+l)!} \left(\frac{2}{n\,a_0}\right)^3 \right]^{1/2} s^l \, L^{2l+1}_{n-l-1}(s) \, e^{-s/2}\\ & \\ \end{aligned}} \]

with \(s = \frac{2\,r}{n\,a_0}\,\), associated Legendre and associated Laguerre polynomials

\[ P^{m}_l(z) = \frac{(1-z^2)^m/2}{2^l l!} \frac{d^{m+l}}{dz^{m+l}}(z^2 - 1)^l, \qquad L_k^{\alpha}(z) = \sum_{j = 0}^{k}(-1)^j \begin{pmatrix}k+\alpha\\ k - j\end{pmatrix} \frac{x^j}{j!} \]

Comparison of s-p-d-f orbitals

• Overview of all orbitals:

Adapted from Wikipedia

Orbitals of the Hydrogen atom

• Orbital eigenstates are \(\,\,| n, l, m_l, m_s \rangle\)
• Electron eigenstates in the Hydrogen atom are structured in:
  • Shells: Different principal quantum number \(n\)
  • Sub-shells: Different azimuthal quantum number \(l\)
    • Named s, p, d, f orbitals
    • Historical names: Sharp, Principal, Diffuse, and Fundamental coming from appearance of atomic spectral lines
  • Specific orbitals: magnetic quantum number \(m\) (also called \(m_l\))
  • Spin of the electron: spin quantum number \(m_s\) (we will see spin afterwards)

Energy diagram of s-p-d-f orbitals

• Hydrogen has only 1 electron: energy depends on \(n\) only (shell): \(E_n = \frac{\mathrm{Ry}}{n^2}\)
• Other atoms: Multiple electrons and different sub-shells have different energies
  • electron-electron interactions
  • different sub-shell result in different electron distances (interaction-energies)