Comprehensive University Study Notes: Quantum Mechanics and Atomic Physics

Principles and Quantum Nature of Matter

• Classical Physics vs. Quantum Mechanics

  • Classical Description: In a potential landscape $E<em>{pot}(\vec{r})$, if the position $\vec{r}(t</em>0)$ and momentum $\vec{p}(t_0)$ are known, all future states are predictable using equations of motion.

  • Quantum Nature in Macroscopic Devices: An electric motor operates on $\vec{F} = \ell\vec{I} \times \vec{B}$. The current consists of electrons with a quantized elementary charge. However, the quantum nature remains hidden in macroscopic systems due to the vast number of particles.

  • Nanoscopic Systems: In a 10 nm nano-island separated by gaps, quantum effects become dominant.

    • Tunneling Effect: Electrons can traverse classically forbidden potential barriers across gaps.

    • Coulomb Blockade: Due to tiny capacitance ($9 \cdot 10^{-20} F$), a single electron increases the island potential by $1.8 V$ ($U = Q/C$), preventing further electrons from entering until the first one tunnels out.

Criteria for Quantum Treatment

• Physical Constants as Benchmarks:

  • In Relativity, the speed of light $c$ is the limit. For electrons in an electron microscope, $v \approx 0.7c$, requiring relativistic treatment.

  • In Quantum Mechanics, the benchmark is Planck’s constant: $h = 6.62607015 \cdot 10^{-34} Js$.

• The Action Dimension:

  • Unit of $h$: $[Energy] \cdot [Time] = [Length] \cdot [Momentum] = [Angular \, Momentum]$.

  • Criterion: If a dynamic variable of action $X \gg h$, classical physics applies. If $X \approx h$, quantum treatment is necessary.

• Example: Sand Grain on a Carousel:

  • $m = 1 mg$, $r = 1 m$, $T = 10 s$.

  • Angular momentum $L = r \, m \frac{2\pi r}{T} = m \omega r^2$.

  • Calculation shows $L \approx 10^{27} \cdot h$, meaning the grain behaves classically.

Blackbody Radiation and the Birth of Quantum Theory

• Experimental Observation: Heated objects emit electromagnetic radiation. Intensity $I(\nu)$ shifts from red to orange/white as temperature increases.

• Cavity Radiation (Hohlraumstrahlung): A metal cavity with a small hole acts as an ideal blackbody (absorption = 1).

  • Spectral Energy Density $E(\nu)$: Energy per volume per frequency interval $\Delta\nu$.

  • Wien's Displacement Law: The frequency of the maximum $\nu_{max} = bT$, where $b = 5.879 \cdot 10^{10} \, (Ks)^{-1}$.

• Rayleigh-Jeans Law (1900):

  • Theory: Light forms standing wave modes in the cavity. Wave numbers are discrete: $k<em>x = n</em>x \frac{\pi}{L}$.

  • Mode Density $n(\nu) = \frac{8\pi\nu^2}{c^3}$.

  • Equipartition Theorem Error: They assumed each mode has energy $\epsilon = k_B T$ (like a harmonic oscillator).

  • Result: $E(\nu) = \frac{8\pi\nu^2}{c^3} k_B T$. This leads to the Ultraviolet Catastrophe where energy density goes to infinity at high frequencies.

• Planck’s Breakthrough (Dec 14, 1900):

  • Ad-hoc Assumption: Energy is exchanged only in discrete packets ("quanta") of $h\nu$.

  • Energy of a mode: $\epsilon(n, \nu) = nh\nu$.

  • Boltzmann Statistics: Probability $P(n, \nu) = \frac{e^{-\epsilon/(k<em>B T)}}{\sum e^{-\epsilon/(k</em>B T)}}$.

  • Mean Energy: $\bar{\epsilon}(\nu) = \frac{h\nu}{e^{\frac{h\nu}{k_B T}} - 1}$.

  • Planck’s Radiation Law: $E(\nu) = \frac{8\pi h}{c^3} \frac{\nu^3}{e^{\frac{h\nu}{k_B T}} - 1}$.

Photoelectric Effect and Einstein's Hypothesis

• Experiment: UV light hits a zinc plate. If negatively charged, it discharges (electrons are ejected). If positively charged, no change occurs.

• Observations contradicting classical theory:

  • Kinetic energy is independent of intensity (classically, higher intensity = stronger $E$-field = more energy).

  • Kinetic energy depends linearly on frequency.

• Einstein’s Explanation (1905):

  • Light consists of energy quanta (Photons) with $E = h\nu$.

  • A photon transfers its entire energy to an electron.

  • Energy Equation: $E<em>{kin} = h\nu - W</em>A$, where $W_A$ is the work function (exit work).

• Millikan’s Verification (1916):

  • Used a vacuum tube and stopping potential $U$. $eU = E_{kin}$.

  • Determining $h$ from the slope of $U$ vs. $\nu$ graph.

• Photon Momentum: Derived from $E^2 = p^2 c^2 + m^2 c^4$ with $m=0$.

  • $p = \frac{h\nu}{c} = \frac{h}{\lambda}$.

The Compton Effect and Wave-Particle Dualism

• Compton Scattering (1922): X-rays ($\lambda \approx 0.07 nm$) scattered by graphite show a wavelength shift $\lambda' > \lambda$.

• Mechanism: Elastic collision between a photon and a stationary electron.

• Compton Equation: $\lambda' - \lambda = \lambda_c (1 - \cos \theta)$.

  • Compton Wavelength: $\lambda<em>c = \frac{h}{m</em>e c} \approx 2.43 pm$.

• Conclusion: X-rays, which show wave properties in Laue diffraction, show particle properties in scattering.

Matter Waves (De Broglie)

• Hypothesis (1924): Particles with momentum $p$ have a wavelength $\lambda$.

• De Broglie Relation: $p = \frac{h}{\lambda} = \hbar k$.

• Experimental Evidence: Electrons in an electron microscope show interference patterns at a biprism (Möllenstedt biprism).

• Single Particle Interference: Even when only one electron is in the microscope at a time, an interference pattern builds up over time.

  • Interpretation: Propagation is wave-like; detection is particle-like. Probability of detection $\propto |\text{Amplitude}|^2$.

Atomic Models and Quantization

• Rutherford’s Scattering: Atom has a tiny nucleus ($< 10^{-13} m$) containing almost all mass.

• Bohr Model (1913):

  • Assumes circular orbits where $F<em>{centrifugal} = F</em>{Coulomb}$.

  • Quantization of Angular Momentum: $L = n\hbar$.

  • Bohr Radius: $r<em>n = \frac{4\pi \epsilon</em>0 \hbar^2}{m<em>e e^2} n^2$. $a</em>0 = 0.53 \, \text{\AA}$.

  • Energy States: $E_n = -Ry \frac{1}{n^2}$, with $Ry \approx 13.6 eV$.

• Atomic Spectra:

  • Emission/Absorption: Photons emitted/absorbed when jumping between states: $h\nu = E<em>j - E</em>k$.

  • Series: Lyman ($n \to 1$, UV), Balmer ($n \to 2$, visible), Paschen ($n \to 3$, IR).

Heisenberg's Uncertainty Principles

• Position-Momentum Uncertainty:

  • Arises from diffraction at a slit of width $\Delta y$. Narrowing the slit (better position) causes wider diffraction (higher $\Delta p_y$).

  • Relation: $\Delta y \cdot \Delta p_y \gtrsim \frac{\hbar}{2}$.

  • Consequence: Classical trajectories are invalid in QM.

• Energy-Time Uncertainty:

  • If a wave exists for only a duration $\Delta t$, it is a superposition of frequencies.

  • Relation: $\Delta E \cdot \Delta t \approx \hbar$.

  • Natural Line Width: Atomic states with lifetime $\Delta t$ have an energy spread $\Delta E$.

The Schrödinger Equation

• Wave Function $\psi(\vec{r}, t)$: Describes the probability amplitude.

  • Probability Density: $|\psi(\vec{r}, t)|^2$.

  • Normalization: $\int |\psi|^2 d^3r = 1$.

• Evolution:

  • Time-Dependent Schrödinger Equation: $i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r}, t) \right] \psi$.

• Stationary States: If $V$ is time-independent, use separation of variables $\psi(\vec{r}, t) = \psi(\vec{r})e^{-iEt/\hbar}$.

  • Time-Independent Schrödinger Equation: $-\frac{\hbar^2}{2m}\nabla^2 \psi(\vec{r}) + V(\vec{r})\psi(\vec{r}) = E\psi(\vec{r})$.

Fundamental Potential Problems

• 1D Infinite Potential Well (Size $\ell$):

  • Boundaries: $\psi(0) = \psi(\ell) = 0$.

  • Energies: $E_n = \frac{\hbar^2 \pi^2}{2m\ell^2} n^2$.

• Potential Step ($E > V_0$):

  • Unlike classical balls, electrons can be reflected even if $E > V_0$.

  • Reflectivity $R = \left( \frac{k<em>1 - k</em>2}{k<em>1 + k</em>2} \right)^2$.

• Tunneling Effect ($E < V_0$):

  • The wave function decays exponentially inside the barrier but has finished value on the other side.

  • Scanning Tunneling Microscopy (STM): Uses exponential dependence of tunneling current on distance to image atoms.

• Harmonic Oscillator ($V = \frac{1}{2}m\omega^2 x^2$):

  • Solutions involve Hermite Polynomials.

  • Energies: $E_n = \hbar\omega\left( n + \frac{1}{2} \right)$.

  • Includes Zero-Point Energy ($E_0 = \frac{1}{2}\hbar\omega$) at $T = 0$.

Quantum Mechanics Axioms

• Axiom I: The state is fully described by $\psi(\vec{r}, t)$.

• Axiom II: Observables $A(\vec{r}, \vec{p})$ correspond to Hermitian operators $\hat{A}$ (e.g., $\hat{p} = -i\hbar\nabla$).

• Axiom III: Possible measurement results are eigenvalues $a<em>n$ of $\hat{A}\psi</em>n = a<em>n\psi</em>n$.

• Axiom IV: Probability of measuring $a<em>n$ for state $\phi$ is $W(a</em>n) = |(\psi_n, \phi)|^2$.

  • Expectation Value: $\langle A \rangle = (\phi, \hat{A}\phi)$.

• Axiom V: Time evolution follows the Schrödinger equation.

Uncertainty and Commutation

• Commutator: $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$.

• Simultaneous Measurement: Two observables can be precisely measured together if and only if their operators commute ($[\hat{A}, \hat{B}] = 0$).

• Canonical Commutation: $[\hat{x}, \hat{p}_x] = i\hbar$.

Perturbation Theory

• Stationary Perturbation: For $\hat{H} = \hat{H}_0 + \hat{V}'$, where $\hat{V}'$ is small.

  • First-order energy shift: $\Delta E<em>n = (\psi</em>n^0, \hat{V}'\psi_n^0)$.

• Time-Dependent Perturbation:

  • Used to calculate transition probabilities (e.g., light absorption).

  • Transition Rate: $P<em>{km}(t) \propto |(\psi</em>k, \hat{H}'\psi_m)|^2$. Leads to Fermi's Golden Rule.

  • Dipole Matrix Element: $D<em>{km} = (\psi</em>k, e\vec{r}\psi_m)$. Determines selection rules.

The Hydrogen Atom

• Schrödinger Equation in Spherical Coordinates: $\psi(r, \theta, \phi) = R(r)Y_{\ell}^m(\theta, \phi)$.

• Angular Solution (Spherical Harmonics):

  • Angular Momentum Squared: $\hat{L}^2 Y = \hbar^2 \ell(\ell+1) Y$.

  • Z-Component: $\hat{L}_z Y = m\hbar Y$.

• Radial Solution:

  • Energy depends only on Hauptquantenzahl $n$: $E_n = -Ry \frac{Z^2}{n^2}$.

  • Constraints: $\ell = 0, \dots, n-1$ and $m = -\ell, \dots, \ell$.

• Fine Structure:

  • Relativistic correction and Spin-Orbit Coupling cause energy levels to depend on $n$ and $j$ (total angular momentum).

  • Selection Rules for photons: $\Delta \ell = \pm 1, \Delta m = 0, \pm 1$.

Electron Spin and Multi-Electron Atoms

• Stern-Gerlach Experiment (1921): Silver atoms in a magnetic gradient split into two beams, proving a non-classical angular momentum.

• Spin $\vec{S}$: $s = 1/2$. $S_z = \pm \frac{1}{2} \hbar$.

• Pauli Principle: No two electrons in an atom can have the same four quantum numbers $(n, \ell, m, m_s)$.

  • Total wave function must be antisymmetric upon exchange of two electrons.

• He-Atom and Exchange Interaction:

  • Parahelium ($S=0$, singlet): Symmetric spatial, antisymmetric spin.

  • Orthohelium ($S=1$, triplet): Antisymmetric spatial, symmetric spin. Lower energy due to the Exchange Integral $K_{ij}$.

• Hund's Rules:

  1. Maximize $S$.

  2. Maximize $L$.

  3. $J$ orientation depends on shell being more or less than half-full.

• Shell Model and Periodic Table:

  • Orbitals fill in order: $1s, 2s, 2p, 3s, 3p, 4s, 3d \dots$.

  • Exceptions (e.g., $4s$ before $3d$) caused by screening effects and effective nuclear charge $Z_{eff} = Z - S$.

X-Ray Physics

• Generation: Electrons hit a metal anode.

• Bremsstrahlung: Continuous spectrum caused by electron deceleration in nuclear fields. Has a short-wavelength limit $\lambda<em>G = \frac{hc}{eU</em>B}$.

• Characteristic X-Rays: Electrons ejected from inner shells ($K, L, M \dots$) are replaced by outer electrons.

  • Moseley's Law: $\sqrt{\nu} \propto Z$.

  • Notation: $K<em>\alpha$ is $n=2 \to n=1$; $K</em>\beta$ is $n=3 \to n=1$.

• Absorption: Dominated by Photoelectric effect, Compton scattering, and at high energies ($> 1.022 MeV$), Pair production.