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

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

Summary angular momentum

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

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

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

Visual representation

• The \(\,\,m = \pm 1\) 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 \(\,\,\sin(m\phi) = \frac{e^{im\phi} - e^{-im\phi}}{2i}\,\):

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

Different basis P-states

• Different eigenstate basis: Symmetric dumbells vs. donuts in xy-plane
• Superposition of a \(p_x\) and a \(p_y\)-orbital gives a torus(donut)-orbital 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
  • \(p_0\)-orbital: Phase changes in time, phase difference of \(\pi\)
  • \(p_1\)-orbital: Phase rotates

Radial solutions

Solutions are given by Laguerre polynomials

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

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

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:

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

Radial solutions

\[ \boxed{\begin{aligned} \quad R_{nl}(s) &= \left[ \frac{(n - l - 1)!}{2\,n\,(n+l)!} \left(\frac{2}{n\,a_0}\right)^3 \right]^{1/2} s^l \, L^{2l+1}_{n-l-1}(s) \, e^{-s/2}\quad\\ &\\ \qquad s &= \frac{2r}{na_0}, \qquad\,\,\,\, a_0 = \frac{4\pi\varepsilon_0\hbar^2}{e^2 \mu} = 0.529\,\textrm{A}\\ \qquad E_n &= -\frac{Ry}{n^2},\qquad Ry = \frac{\hbar^2}{2\mu a_0^2} = \frac{\mu}{2} \left(\frac{e^2}{4\pi\varepsilon_0\hbar}\right)^2 = 13.6\,\textrm{eV}\\ \end{aligned}} \]

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:

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)
• Other atoms: Multiple electrons and different sub-shells have different energies
  • electron-electron interactions
  • different sub-shell result in different electron distances (interaction-energies)

Magnetic field: Spin interaction

• Extension Schrodinger equation
• Includes the spin magnetic moment: \(\vec{\mu_e} = g \mu_B \vec{\sigma}\)
• Energy of spin magnetic moment in magnetic field \(\vec{B}\)

\[ E_s = \vec{\mu_e}\cdot\vec{B} = g \mu_B \vec{\sigma}\cdot\vec{B} \]

Quantum Mechanics: Corresponding Hamiltonian (convert to operators)

\[ \hat{H}_s \,=\, \frac{g \mu_B}{2} \hat{\vec{\sigma}}\cdot\vec{B} \,=\, \frac{g\mu_B}{2} \left[ B_x \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} + B_y \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix} + B_z \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix} \right] \]

Factor 1/2 comes from \(\hat{\vec{\sigma}}\) definition

Magnetic field: Moving Charges

Classically: Lorentz force:

\[ \vec{F} = q\vec{E} + q (\vec{v} \times \vec{B}) \]

• Write the electric and magnetic fields (\(\vec{E}\) and \(\vec{B}\)) as potentials:

\[ \begin{aligned} \textrm{Electric field:} \qquad &\vec{E} = -e\nabla V - \frac{\partial \vec{A}}{\partial t}\\ \textrm{Magnetic field:} \qquad &\vec{B} = \nabla \times \vec{A} \end{aligned} \]

• These potentials are derived from Maxwell’s equations
• The particle’s momentum \(\,\,\vec{p} \longrightarrow \vec{p} - e\vec{A}\)

Quantum Mechanics: use corresponding momentum operator \(\,\,\hat{\vec{p}} \longrightarrow \hat{\vec{p}} - e\vec{A}\)

Pauli equation

Add both spin \(\frac{g \mu_B}{2} \hat{\vec{\sigma}}\cdot\vec{B}\) and vector potential \(\hat{\vec{p}} - e\vec{A}\).

\[ \left[ \frac{1}{2m_0} (\vec{\hat{p}} - e \vec{A})^2 \mathbb{1} + V \mathbb{1} + \frac{g \mu_B}{2} \hat{\vec{\sigma}}\cdot\vec{B} \right] \Psi(\vec{r}, t) = i\hbar \frac{\partial}{\partial t} \Psi(\vec{r}, t) \]

where \(\,\,\Psi(\vec{r}, t) = \begin{pmatrix}\Psi_\uparrow(\vec{r}, t) \\ \Psi_\downarrow(\vec{r}, t) \end{pmatrix}\,\,\) is a spinor

The part \((\vec{\hat{p}} - e \vec{A})^2\) can be written as (operator precedence):

\[ (\vec{\hat{p}} - e \vec{A})\cdot(\vec{\hat{p}} - e \vec{A}) = \hat{p}^2 -e\vec{A}\cdot \hat{\vec{p}} + i\hbar e \nabla\cdot\vec{A} + e^2 A^2 \]

• Last term \(\,\,e^2 A^2 \approx 0\,\) for most magnetic fields

Zeeman effect

• The Zeeman effect: splitting of spectral lines under magnetic field
• Energy level splitting of atomic orbital levels
• Level-splitting depends on spin and magnetic quantum number
• Additional selection rules for observed emission/spectral lines

Zeeman effect for Hydrogen

• Hydrogen as a model system (only 1 electron)
• Assume magnetic field \(\,\vec{B} = (0, 0, B_z)\) along the z-direction (spherical symmetry)
• Start from the Pauli-equation

\[ \left[ \frac{1}{2m_0} (\vec{\hat{p}} - e \vec{A})^2 \mathbb{1} + V \mathbb{1} + \frac{g \mu_B}{2} \hat{\vec{\sigma}}\cdot\vec{B} \right] \Psi(\vec{r}, t) = i\hbar \frac{\partial}{\partial t} \Psi(\vec{r}, t) \]

where a we choose vector potential \(\vec{A} = \frac{1}{2}\,B\,(-y, x, 0)\)

\[ \begin{aligned} (\vec{\hat{p}} - e \vec{A})^2 &= \hat{p}^2 -e\vec{A}\cdot \hat{\vec{p}} + i\hbar e \nabla\cdot\vec{A}\\ \end{aligned} \]

where we approximated \({e^2 A^2} \approx 0\)

Derivation Zeeman effect

\[ \begin{aligned} (\vec{\hat{p}} - e \vec{A})^2 &= \hat{p}^2 + i\hbar e \vec{A}\cdot \nabla + i\hbar e \nabla\cdot\vec{A}\\ &= \hat{p}^2 + \frac{i\hbar e B}{2}(-y, x, 0) \cdot (\hat{p}_x, \hat{p}_y, \hat{p}_z) + \frac{i\hbar e B}{2} (\hat{p}_x, \hat{p}_y, \hat{p}_z) \cdot (-y, x, 0)\\ &= \hat{p}^2 + \frac{\hbar^2 e B}{2}(-y \partial_x + x\partial_y) + \frac{\hbar^2 e B}{2}\, 0 \\ &= \hat{p}^2 + \frac{\hbar e B}{2} \hat{L}_z \end{aligned} \]

where the z-component of angular momentum: \(\hat{L}_z = -i\hbar (x\partial_x - y\partial_x)\)

Filling this in the Pauli equation:

\[ \left[ \left(\frac{\vec{\hat{p}}^2}{2m_0} + V\right) + \frac{\hbar e B}{4 m_0} \hat{L}_z + \frac{g \mu_B}{2} \hat{\vec{\sigma}}\cdot\vec{B} \right] \Psi(\vec{r}, t) = i\hbar \frac{\partial}{\partial t} \Psi(\vec{r}, t) \]

Derivation Zeeman effect CTU’d

Approximating \(g/2 \approx 1\) and using the Bohr magneton \(\mu_B = \frac{e\hbar}{m_0}\)

\[ \left[ \left(\frac{\vec{\hat{p}}^2}{2m_0} + V\right) + \frac{\mu_B\,B}{2\hbar} \left(\hat{L}_z + B \hat{\sigma}_z \right) \right] \Psi(\vec{r}, t) = i\hbar \frac{\partial}{\partial t} \Psi(\vec{r}, t) \]

The first term is just the Hydrogen Hamiltonian \(\hat{H}_0\)

\[ \left[ \hat{H}_0 + \frac{\mu_B\,B}{2\hbar} \left(\hat{L}_z + \hat{\sigma}_z \right) \right] \Psi(\vec{r}, t) = i\hbar \frac{\partial}{\partial t} \Psi(\vec{r}, t) \]

For stationary solutions we assume \(\Psi(\vec{r}, t) = \psi(\vec{r}) \, e^{-iEt/\hbar}\)

\[ \left[ \hat{H}_0 + \frac{\mu_B\,B}{2\hbar} \left(\hat{L}_z + \hat{\sigma}_z \right) \right] \psi(\vec{r}) = E \psi(\vec{r}) \]

Derivation Zeeman effect CTU’d

Remember that \(\psi(\vec{r}) = \begin{pmatrix} \psi_\uparrow \\ \psi_\downarrow \end{pmatrix}\) and we have a matrix equation:

\[ \left[ \hat{H}_0\, \mathbb{1} + \frac{\mu_B\,B}{2\hbar} \left(\hat{L}_z \, \mathbb{1} + \hbar\, \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \right) \right] \begin{pmatrix} \psi_\uparrow(\vec{r}) \\ \psi_\downarrow(\vec{r}) \end{pmatrix}(\vec{r}) = E \begin{pmatrix} \psi_\uparrow(\vec{r}) \\ \psi_\downarrow(\vec{r}) \end{pmatrix} \]

This corresponds to two uncoupled equations for spin-up and spin-down:

\[ \begin{aligned} \left[ \hat{H}_0 + \frac{\mu_B\,B}{\hbar} \left(\hat{L}_z + \hbar \right)\right] \psi_\uparrow(\vec{r}) &= E \, \psi_\uparrow(\vec{r}) \\ \left[ \hat{H}_0 + \frac{\mu_B\,B}{\hbar} \left(\hat{L}_z - \hbar \right)\right] \psi_\downarrow(\vec{r}) &= E \, \psi_\downarrow(\vec{r}) \\ \end{aligned} \]

Fill in solutions of the Hydrogen atom (eigenstates of \(\hat{H}_0\)):

\[ |n,l,m,\uparrow \rangle = \psi_\uparrow, \qquad \qquad |n,l,m,\downarrow \rangle = \psi_\downarrow \]

Eigenenergies and eigenstates

\[ \begin{aligned} \hat{H}_0 \psi_\uparrow(\vec{r}) + \frac{\mu_B\,B}{\hbar} \left(\hat{L}_z + \hbar \right) \psi_\uparrow(\vec{r}) &= E \, \psi_\uparrow(\vec{r}) \\ \hat{H}_0 \psi_\downarrow(\vec{r}) + \frac{\mu_B\,B}{\hbar} \left(\hat{L}_z - \hbar \right) \psi_\downarrow(\vec{r}) &= E \, \psi_\downarrow(\vec{r}) \\ \end{aligned} \]

The solutions are eigenstates of the Hydrogen atom, and \(\hat{L}_z\)

\[ \begin{aligned} \hat{H}_0 |n,l,m,\uparrow (\downarrow) \rangle &= E_{n} |n,l,m,\uparrow (\downarrow)\rangle\\ \hat{L}_z |n,l,m,\uparrow \rangle &= m\hbar |n,l,m,\uparrow \rangle\\ \hat{L}_z |n,l,m,\downarrow \rangle &= -m\hbar |n,l,m,\downarrow \rangle\\ \end{aligned} \]

Fill in solutions of the Hydrogen atom (eigenstates of \(\hat{H}_0\)):

\[ \begin{aligned} E_{nl} \psi_\uparrow(\vec{r}) + \mu_B\,B (m + 1) \psi_\uparrow(\vec{r}) &= E \, \psi_\uparrow(\vec{r}) \\ E_{nl} \psi_\uparrow(\vec{r}) + \mu_B\,B (m - 1) \psi_\uparrow(\vec{r}) &= E \, \psi_\uparrow(\vec{r}) \\ \end{aligned} \]

Eigenenergies and eigenstates

Fill in solutions of the Hydrogen atom (eigenstates of \(\hat{H}_0\)):

\[ \begin{aligned} E_{nl} \psi_\uparrow(\vec{r}) + \mu_B\,B (m + 1) \psi_\uparrow(\vec{r}) &= E \, \psi_\uparrow(\vec{r}) \\ E_{nl} \psi_\uparrow(\vec{r}) + \mu_B\,B (m - 1) \psi_\uparrow(\vec{r}) &= E \, \psi_\uparrow(\vec{r}) \\ \end{aligned} \]

The eigenenergies \(E = E_{nlm\uparrow}\) depend now also on the magnetic quantum number \(m\) and the spin \(\uparrow (\downarrow)\):

\[\boxed{\begin{aligned} E_{nlm\uparrow} = E_{nl} + \mu_B\,B (m + 1) \qquad \textrm{with}\quad\psi_\uparrow\\ E_{nlm\downarrow} = E_{nl} + \mu_B\,B (m - 1) \qquad \textrm{with}\quad\psi_\downarrow \end{aligned}} \]

• The splitting of energy levels depends on magnetic field strength \(B\)
• Selection rules determine between which energy levels emission is allowed

Energy-level splitting

\[\boxed{\begin{aligned} E_{nlm\uparrow} = E_{nl} + \mu_B\,B (m + 1) \qquad \textrm{with}\quad\psi_\uparrow\\ E_{nlm\downarrow} = E_{nl} + \mu_B\,B (m - 1) \qquad \textrm{with}\quad\psi_\downarrow \end{aligned}} \]

• The splitting of energy levels depends on magnetic field strength \(B\)
• Magnetic field for p-orbital level splitting (right plot) \(B = 15\) T

Energy-level splitting

• Bohr magneton \(\mu_B \approx 5.8 \times 10^{−5}\) eV/T \(\,\,\longrightarrow\,\,\) splitting small for normal magnetic field strengths \(\,B\). Record laboratory magnetic fields are e.g.:
  • Stable magnetic field of \(45.5\) T
  • Peak, i.e. only a few \({\mu}s\), magnetic fields as a pulsed magnetic field: \(1200\) T
• Selection rules determine between which energy levels emission is allowed

Energy-level splitting

• Bohr magneton \(\mu_B \approx 5.8 \times 10^{−5}\) eV/T \(\,\,\longrightarrow\,\,\) splitting small for normal magnetic field strengths \(\,B\). Record laboratory magnetic fields are e.g.:
  • Stable magnetic field of \(45.5\) T
  • Peak, i.e. only a few \({\mu}s\), magnetic fields as a pulsed magnetic field: \(1200\) T
• The wavelength or frequency differences in spectral lines is given by:

\[ \frac{2\pi}{\Delta \lambda} = \Delta f = \frac{\Delta\omega}{2\pi} = \Delta E/h \]

• The Hydrogen emission spectral line at \(122\) nm will split in three lines.
• Selection rules: Spin is preserved, \(l \rightarrow l - \pm 1\) while different polarizations of light, therefore