4.1 Time-Independent Perturbation Theory
📚 Non-Degenerate Perturbation Theory
Hamiltonian: \(H = H_0 + \lambda V\)
First-Order Energy Correction:
\[ E_n^{(1)} = \langle n^{(0)} | V | n^{(0)} \rangle \]
Second-Order Energy Correction:
\[ E_n^{(2)} = \sum_{m \neq n} \frac{|\langle m^{(0)} | V | n^{(0)} \rangle|^2}{E_n^{(0)} - E_m^{(0)}} \]
First-Order Wavefunction Correction:
\[ |n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle m^{(0)} | V | n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} |m^{(0)}\rangle \]
Application Example: Stark Effect (Linear Stark Effect in Hydrogen Atom)
Hydrogen atom in electric field: \(V = eEz\)
The first-order energy correction is zero (parity conservation). Polarization appears at second order.
4.2 Time-Dependent Perturbation Theory
📚 Fermi's Golden Rule
Transition probability due to time-dependent perturbation \(V(t) = Ve^{-i\omega t}\):
\[ w_{i \to f} = \frac{2\pi}{\hbar} |\langle f | V | i \rangle|^2 \rho(E_f) \]
where \(\rho(E_f)\) is the density of final states.
Selection Rules: Dipole transitions
\[ \Delta l = \pm 1, \quad \Delta m = 0, \pm 1 \]
Application: Light Absorption and Emission
This describes the process of photon absorption and emission by atoms. It is related to Einstein's A and B coefficients.
4.3 Scattering Theory
📚 Scattering Cross Section
Differential Scattering Cross Section:
\[ \frac{d\sigma}{d\Omega} = |f(\theta)|^2 \]
where \(f(\theta)\) is the scattering amplitude.
Born Approximation (weak potential):
\[ f(\theta) = -\frac{m}{2\pi\hbar^2} \int e^{i\mathbf{q}\cdot\mathbf{r}} V(r) d^3r \]
where \(\mathbf{q} = \mathbf{k}_i - \mathbf{k}_f\) is the momentum transfer.
Partial Wave Expansion
Scattering in a central force field:
\[ f(\theta) = \frac{1}{k} \sum_{l=0}^\infty (2l+1) e^{i\delta_l} \sin\delta_l P_l(\cos\theta) \]
where \(\delta_l\) is the phase shift.
🎯 Exercise Problems
- Harmonic Oscillator Perturbation: Calculate the energy correction up to second order for a harmonic oscillator subjected to perturbation \(V = \alpha x^4\).
- Zeeman Effect: Calculate the energy splitting of a hydrogen atom in a magnetic field.
- Born Approximation: Find the scattering cross section for the Yukawa potential \(V(r) = \frac{g^2 e^{-\mu r}}{r}\).
- Resonance Scattering: Determine the resonance condition where the phase shift becomes \(\delta_0 = \pi/2\) for s-wave scattering (l=0).
Summary
In this chapter, we have learned perturbation theory and scattering theory. Time-independent perturbation theory provides systematic methods for calculating energy corrections, with applications to the Stark and Zeeman effects. Time-dependent perturbation theory yields Fermi's golden rule, selection rules, and descriptions of light absorption processes. Scattering theory introduces the scattering cross section, Born approximation for weak potentials, partial wave expansion, and the concept of phase shifts.
In the next chapter, we will study Relativistic Quantum Mechanics.