💫Intro to Quantum Mechanics II Unit 5 – Time-Independent Perturbation Theory

Key Concepts and Foundations

• Time-independent perturbation theory (TIPT) is a method for approximating the eigenstates and eigenvalues of a quantum system with a small perturbation
• TIPT assumes the perturbation is time-independent and much smaller than the unperturbed Hamiltonian
  • Allows the use of perturbative expansions to find approximate solutions
• Unperturbed system is described by a known Hamiltonian $H_0$ with known eigenstates $|n^{(0)}\rangle$ and eigenvalues $E_n^{(0)}$
• Perturbation is represented by an additional term $V$ in the Hamiltonian, such that the total Hamiltonian is $H = H_0 + V$
• Goal of TIPT is to find the corrected eigenstates $|n\rangle$ and eigenvalues $E_n$ of the perturbed system
• Perturbative corrections are calculated order by order, with each order providing a more accurate approximation
• TIPT is particularly useful when the exact solution to the perturbed system is difficult or impossible to obtain analytically

Mathematical Framework

• Perturbation theory relies on the expansion of the eigenstates and eigenvalues in powers of a small parameter $\lambda$
  • $|n\rangle = |n^{(0)}\rangle + \lambda|n^{(1)}\rangle + \lambda^2|n^{(2)}\rangle + ...$
  • $E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ...$
• The parameter $\lambda$ represents the strength of the perturbation, with $\lambda = 0$ corresponding to the unperturbed system
• Corrections to the eigenstates and eigenvalues are obtained by substituting the expansions into the Schrödinger equation and equating terms of the same order in $\lambda$
• First-order corrections to the energy are given by the expectation value of the perturbation in the unperturbed state: $E_n^{(1)} = \langle n^{(0)}|V|n^{(0)}\rangle$
• First-order corrections to the eigenstates involve a sum over all other unperturbed states: $|n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle m^{(0)}|V|n^{(0)}\rangle}{E_n^{(0)} - E_m^{(0)}} |m^{(0)}\rangle$
• Higher-order corrections become increasingly complex and involve nested sums over unperturbed states

Non-Degenerate Perturbation Theory

• Non-degenerate perturbation theory applies when the unperturbed energy levels are non-degenerate (i.e., no two states have the same energy)
• First-order correction to the energy is simply the expectation value of the perturbation in the unperturbed state: $E_n^{(1)} = \langle n^{(0)}|V|n^{(0)}\rangle$
• Second-order correction to the energy involves a sum over all other unperturbed states: $E_n^{(2)} = \sum_{m \neq n} \frac{|\langle m^{(0)}|V|n^{(0)}\rangle|^2}{E_n^{(0)} - E_m^{(0)}}$
  • Denominators in the sum represent the energy differences between the unperturbed states
• First-order correction to the eigenstate is a sum over all other unperturbed states weighted by the matrix elements of the perturbation and the energy differences: $|n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle m^{(0)}|V|n^{(0)}\rangle}{E_n^{(0)} - E_m^{(0)}} |m^{(0)}\rangle$
• Non-degenerate perturbation theory breaks down when the energy differences between unperturbed states become small compared to the perturbation strength
  • Leads to divergences in the perturbative corrections

Degenerate Perturbation Theory

• Degenerate perturbation theory is used when the unperturbed system has degenerate energy levels (i.e., multiple states with the same energy)
• In the presence of degeneracy, the unperturbed states are not unique, and the perturbation can mix the degenerate states
• First step in degenerate perturbation theory is to diagonalize the perturbation matrix within the degenerate subspace
  • Yields a new set of unperturbed states that are eigenstates of the perturbation within the degenerate subspace
• Perturbative corrections are then calculated using the new set of unperturbed states
• First-order correction to the energy is given by the eigenvalues of the perturbation matrix within the degenerate subspace
• First-order correction to the eigenstates involves a linear combination of the degenerate unperturbed states, with coefficients determined by the eigenvectors of the perturbation matrix
• Higher-order corrections in degenerate perturbation theory are more complex and involve coupling between different degenerate subspaces

Applications in Atomic Systems

• TIPT is widely used to calculate energy levels and transitions in atomic systems
• Fine structure of hydrogen-like atoms can be treated using TIPT, with the relativistic corrections and spin-orbit coupling acting as perturbations to the non-relativistic Hamiltonian
  • Explains the splitting of energy levels and the observed spectral lines
• Zeeman effect, the splitting of atomic energy levels in an external magnetic field, can be described using TIPT
  • Magnetic field acts as a perturbation, lifting the degeneracy of the magnetic sublevels
• Stark effect, the shifting and splitting of atomic energy levels in an external electric field, can also be treated using TIPT
  • Electric field acts as a perturbation, mixing the atomic states and leading to energy shifts and level splittings
• Hyperfine structure, arising from the interaction between the electron and nuclear spins, can be calculated using TIPT
  • Hyperfine interaction acts as a perturbation, leading to the splitting of energy levels and the observed hyperfine spectral lines

Limitations and Considerations

• TIPT is an approximate method and is only valid when the perturbation is small compared to the unperturbed Hamiltonian
  • Accuracy of the perturbative corrections decreases as the perturbation strength increases
• Perturbative expansions are asymptotic series, meaning they do not necessarily converge to the exact result even when carried out to infinite order
  • In some cases, the perturbative corrections may diverge, indicating a breakdown of the perturbative approach
• Degenerate perturbation theory can fail when the perturbation mixes degenerate states from different unperturbed energy levels
  • Requires the use of more advanced techniques, such as the Wigner-Brillouin perturbation theory or the Rayleigh-Schrödinger perturbation theory
• TIPT assumes that the unperturbed states form a complete basis, which may not always be the case in practice
  • Incomplete basis sets can lead to errors in the perturbative corrections
• Care must be taken when applying TIPT to systems with strong correlations or non-perturbative effects, such as in strongly coupled quantum systems or in the presence of quantum phase transitions

Problem-Solving Strategies

• Identify the unperturbed Hamiltonian $H_0$ and the perturbation $V$
  • Ensure that the perturbation is small compared to the unperturbed Hamiltonian
• Determine the unperturbed eigenstates $|n^{(0)}\rangle$ and eigenvalues $E_n^{(0)}$
  • If the unperturbed system has degenerate energy levels, identify the degenerate subspaces
• Calculate the matrix elements of the perturbation in the unperturbed basis: $\langle m^{(0)}|V|n^{(0)}\rangle$
  • For degenerate perturbation theory, diagonalize the perturbation matrix within each degenerate subspace
• Use the appropriate formulas for the perturbative corrections to the energies and eigenstates, depending on whether the system is non-degenerate or degenerate
  • For non-degenerate systems, use the formulas for $E_n^{(1)}$, $E_n^{(2)}$, and $|n^{(1)}\rangle$
  • For degenerate systems, use the formulas for the first-order corrections based on the diagonalized perturbation matrix
• Interpret the results and assess the validity of the perturbative approach
  • Check the convergence of the perturbative corrections and compare with experimental data, if available

Real-World Connections

• TIPT is essential for understanding the electronic structure and spectroscopic properties of atoms and molecules
  • Used to interpret and predict the results of spectroscopic experiments, such as absorption and emission spectra
• In quantum chemistry, TIPT is used to calculate the properties of molecules, such as bond lengths, vibrational frequencies, and electronic transition energies
  • Helps in the design and optimization of molecular materials and drugs
• TIPT plays a crucial role in the development of quantum technologies, such as quantum sensors and quantum computers
  • Used to model the behavior of quantum systems in the presence of external fields or perturbations
• In condensed matter physics, TIPT is used to study the effects of impurities, defects, and external fields on the electronic and magnetic properties of materials
  • Helps in the design and engineering of novel materials with desired properties, such as high-temperature superconductors or topological insulators
• TIPT is also applied in nuclear physics to calculate the properties of atomic nuclei and to understand the interactions between nucleons
  • Used in the study of nuclear reactions, nuclear structure, and the properties of exotic nuclei