6.1 Time-dependent perturbation theory

Time-dependent perturbation theory tackles quantum systems affected by external influences that change over time. It's like studying how a calm pond reacts when you toss in pebbles at different intervals. The theory helps us understand how these disturbances cause transitions between quantum states.

Fermi's Golden Rule is a key player here. It gives us a way to calculate how often these transitions happen. Think of it as a recipe for predicting the ripples in our quantum pond. This rule connects the strength of the disturbance to the likelihood of state changes.

Time-dependent perturbations in quantum systems

Understanding time-dependent perturbations

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• Time-dependent perturbations are external influences that vary with time and cause changes in a quantum system's behavior
• Described by a time-dependent potential energy term, V(t), added to the system's Hamiltonian
• The strength of the perturbation determines the degree to which the system's behavior deviates from its unperturbed state
• Examples of time-dependent perturbations include
  • Electromagnetic fields
  • Laser pulses
  • Time-varying electric or magnetic fields (oscillating fields)
• The goal of time-dependent perturbation theory is to calculate the probability of transitions between different quantum states due to the perturbation

Transition probabilities and Fermi's Golden Rule

• The transition probability between an initial state i and a final state f is given by $P_{i \to f}(t) = |\langle \phi_f | \Psi(t) \rangle|^2$, where $\Psi(t)$ is the time-dependent wavefunction and $\phi_f$ is the eigenstate of the final state
• In first-order perturbation theory, the transition probability can be approximated as $P_{i \to f}(t) \approx \frac{1}{\hbar^2} \left| \int_0^t dt' \langle \phi_f | V(t') | \phi_i \rangle \exp\left[\frac{i}{\hbar}(E_f - E_i)t'\right] \right|^2$, where $E_i$ and $E_f$ are the energies of the initial and final states, respectively
• The transition probability depends on the matrix element $\langle \phi_f | V(t') | \phi_i \rangle$, which represents the coupling between the initial and final states due to the perturbation
• Fermi's Golden Rule states that the transition rate, $\Gamma_{i \to f}$, is proportional to the square of the matrix element and the density of final states, $\rho(E_f)$: $\Gamma_{i \to f} = \frac{2\pi}{\hbar} |\langle \phi_f | V | \phi_i \rangle|^2 \rho(E_f)$
  • Provides a way to calculate the rate of transitions between quantum states
  • Widely used in various fields, such as atomic physics, condensed matter physics, and quantum optics

Time-dependent Schrödinger equation for perturbations

Deriving the time-dependent Schrödinger equation

• The time-dependent Schrödinger equation for a perturbed system is given by $i\hbar \frac{\partial \Psi(t)}{\partial t} = [H_0 + V(t)]\Psi(t)$, where $H_0$ is the unperturbed Hamiltonian and $V(t)$ is the time-dependent perturbation
• The solution to the time-dependent Schrödinger equation can be expressed as a linear combination of the unperturbed system's eigenstates, $\Psi(t) = \sum_n c_n(t) \phi_n$, where $c_n(t)$ are time-dependent coefficients and $\phi_n$ are the eigenstates of $H_0$
• Substituting the expansion of $\Psi(t)$ into the time-dependent Schrödinger equation leads to a set of coupled differential equations for the coefficients $c_n(t)$

Solving the time-dependent Schrödinger equation using perturbation theory

• The coupled differential equations can be solved using perturbation theory, which assumes that the perturbation is small compared to the unperturbed Hamiltonian
• In the interaction picture, the time-dependent Schrödinger equation becomes $i\hbar \frac{\partial c_n(t)}{\partial t} = \sum_m \langle \phi_n | V(t) | \phi_m \rangle \exp\left[\frac{i}{\hbar}(E_n - E_m)t\right] c_m(t)$
  • $c_n(t)$ are the time-dependent coefficients in the interaction picture
  • $\langle \phi_n | V(t) | \phi_m \rangle$ are the matrix elements of the perturbation between the unperturbed eigenstates
• The coupled differential equations can be solved iteratively, starting with the zeroth-order solution (the unperturbed system) and successively adding higher-order corrections

Transition probabilities with perturbation theory

First-order perturbation theory

• In first-order perturbation theory, the transition probability can be approximated as $P_{i \to f}(t) \approx \frac{1}{\hbar^2} \left| \int_0^t dt' \langle \phi_f | V(t') | \phi_i \rangle \exp\left[\frac{i}{\hbar}(E_f - E_i)t'\right] \right|^2$
• First-order perturbation theory considers only the linear terms in the perturbation expansion and is valid for weak perturbations
• Provides a good approximation for the transition probabilities and energy shifts in many cases
• Examples of applications:
  • Calculating the absorption and emission of light by atoms or molecules (electric dipole transitions)
  • Describing the interaction of spin systems with oscillating magnetic fields (magnetic resonance)

Higher-order perturbation theory

• Second-order perturbation theory includes quadratic terms in the perturbation expansion and is necessary when first-order corrections are insufficient or when dealing with degenerate states
• Higher-order perturbation theory (third-order and beyond) becomes increasingly complex and is used when dealing with stronger perturbations or when high accuracy is required
• The convergence of the perturbation series depends on the strength of the perturbation relative to the unperturbed system's energy scale
  • Stronger perturbations may require higher-order terms or non-perturbative methods
• Examples of applications:
  • Calculating the Stark effect (shifting and splitting of energy levels due to an external electric field)
  • Describing multi-photon processes, such as two-photon absorption or Raman scattering

Orders of perturbation theory

Validity and limitations of perturbation theory

• Perturbation theory is an approximation method that assumes the perturbation is small compared to the unperturbed system's Hamiltonian
• The validity of perturbation theory depends on the strength of the perturbation relative to the energy scale of the unperturbed system
• For weak perturbations, first-order perturbation theory often provides accurate results
  • Example: the Zeeman effect (splitting of energy levels in a weak magnetic field)
• As the perturbation strength increases, higher-order terms become more important, and the perturbation series may converge slowly or diverge
  • Example: the anharmonic oscillator (a system with a potential energy that deviates from the harmonic potential)

Non-perturbative methods

• When the perturbation is strong or the perturbation series does not converge, non-perturbative methods may be necessary
• Examples of non-perturbative methods:
  • Variational methods: Approximate the wavefunction by minimizing the energy expectation value with respect to a trial wavefunction
  • Numerical methods: Solve the time-dependent Schrödinger equation directly using computational techniques (finite difference, finite element, or spectral methods)
  • Resummation techniques: Rearrange the perturbation series to improve convergence (Padé approximants, Borel resummation)
• Non-perturbative methods can provide accurate results for strongly perturbed systems but often require more computational resources and may lack the analytical insight provided by perturbation theory