Table of Contents

1. Introduction
2. Why Degeneracy Requires Special Treatment
3. The Failure of Non-Degenerate Formulas
4. Basic Setup of Degenerate Perturbation Theory
5. Perturbation Within the Degenerate Subspace
6. Constructing the Effective Hamiltonian
7. Diagonalizing the Perturbation Matrix
8. First-Order Corrections to Energy and States
9. Physical Interpretation
10. Example: Zeeman Effect in Hydrogen
11. Example: Stark Effect in Degenerate Systems
12. Higher-Order Corrections
13. Applications and Importance
14. Limitations and Alternatives
15. Conclusion

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1. Introduction

In quantum mechanics, degenerate perturbation theory is used when the unperturbed Hamiltonian of a system has multiple eigenstates corresponding to the same energy. These degenerate states require a special treatment since standard perturbation theory produces undefined expressions due to division by zero. Degenerate perturbation theory helps correctly describe how these energy levels and states evolve under a small perturbation.

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2. Why Degeneracy Requires Special Treatment

Degenerate energy levels occur in systems with symmetries. Because the energy is the same for multiple eigenstates, any linear combination of these states is also an eigenstate. Perturbations typically lift this degeneracy and introduce energy splittings. However, standard perturbation theory fails in this scenario, necessitating a more careful diagonalization of the perturbation within the degenerate subspace.

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3. The Failure of Non-Degenerate Formulas

The standard first-order correction to the wavefunction is given by:

\[
|\psi_n^{(1)}\rangle = \sum_{k \ne n} \frac{\langle \psi_k^{(0)} | \hat{H}’ | \psi_n^{(0)} \rangle}{E_n^{(0)} – E_k^{(0)}} |\psi_k^{(0)}\rangle
\]

This expression diverges when \( E_k^{(0)} = E_n^{(0)} \), which occurs in degenerate systems. Moreover, choosing a specific \( |\psi_n^{(0)}\rangle \) from the degenerate set is ambiguous without further constraints.

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4. Basic Setup of Degenerate Perturbation Theory

Suppose \( \hat{H}0 \) has a degenerate eigenvalue \( E^{(0)} \) with \( d \) linearly independent eigenstates \( \{ |\phi_a^{(0)}\rangle \}{a=1}^d \). The goal is to find linear combinations of these states that remain eigenstates when the perturbation \( \hat{H}’ \) is added.

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5. Perturbation Within the Degenerate Subspace

We construct the matrix of the perturbation \( \hat{H}’ \) within the degenerate subspace:

\[
H’_{ab} = \langle \phi_a^{(0)} | \hat{H}’ | \phi_b^{(0)} \rangle
\]

This Hermitian matrix represents the perturbation’s action on the degenerate space.

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6. Constructing the Effective Hamiltonian

We define the effective Hamiltonian:

\[
\hat{H}_{\text{eff}} = P \hat{H}’ P
\]

Where \( P \) is the projector onto the degenerate subspace. The eigenvectors of \( \hat{H}_{\text{eff}} \) give the proper combinations of the original degenerate states that diagonalize \( \hat{H}’ \).

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7. Diagonalizing the Perturbation Matrix

Solving the eigenvalue problem:

\[
\sum_{b=1}^d H’_{ab} \chi_b^{(k)} = E_k^{(1)} \chi_a^{(k)}
\]

yields:

• \( E_k^{(1)} \): first-order energy corrections.
• \( \chi^{(k)} \): coefficients for forming new orthonormal states:
  \[
  |\psi_k^{(0)}\rangle = \sum_{a=1}^d \chi_a^{(k)} |\phi_a^{(0)}\rangle
  \]

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8. First-Order Corrections to Energy and States

These corrections depend on the eigenvalues and eigenvectors of the matrix \( H’ \) within the degenerate subspace. These become the new “unperturbed” states to which higher-order corrections can be applied if needed.

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9. Physical Interpretation

Degenerate perturbation theory explains:

• How symmetry breaking lifts degeneracy.
• Why degeneracies split into multiple nearby energy levels.
• How the “direction” of perturbation in Hilbert space determines the new eigenstates.

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10. Example: Zeeman Effect in Hydrogen

When an external magnetic field \( B \) is applied, the degenerate \( m_l \) levels of hydrogen split due to the interaction term:

\[
\hat{H}’ = -\mu_B B \hat{L}_z
\]

Constructing and diagonalizing \( H’ \) within the \( n = 2 \) degenerate manifold reveals the Zeeman energy shifts and the new magnetic quantum number eigenstates.

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11. Example: Stark Effect in Degenerate Systems

The hydrogen atom’s \( n=2 \) subspace includes 2s and 2p states. A uniform electric field in the \( z \)-direction leads to:

\[
\hat{H}’ = -e E z
\]

Only states with \( \Delta \ell = \pm 1 \) and \( \Delta m = 0 \) mix. Solving the eigenvalue problem for this restricted \( 2 \times 2 \) matrix gives new states and energies that predict the linear Stark effect.

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12. Higher-Order Corrections

Once degeneracy is resolved, standard (non-degenerate) second-order perturbation theory may be used. Corrections beyond first order require projecting out contributions from both inside and outside the degenerate space.

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13. Applications and Importance

• Atomic fine structure.
• Spectroscopy of atoms in magnetic/electric fields.
• Level splitting in solid-state systems.
• Symmetry breaking in molecular configurations.

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14. Limitations and Alternatives

• Complex for high-dimensional degenerate subspaces.
• Breakdown for strong perturbations.
• Requires precise knowledge of matrix elements.
• Alternatives include variational and numerical methods.

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15. Conclusion

Degenerate perturbation theory provides a systematic method for understanding how degenerate quantum systems behave under small perturbations. It resolves mathematical ambiguities and gives physical insight into symmetry breaking, energy level splitting, and new state formation. It is a cornerstone of quantum mechanics, with wide applications in atomic, molecular, and condensed matter physics.

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Table of Contents

1. Introduction
2. Motivation for Perturbation Theory
3. The General Setup
4. Unperturbed and Perturbed Hamiltonians
5. Expansion of Energy and Wavefunctions
6. First-Order Corrections to Energy
7. First-Order Corrections to Wavefunctions
8. Second-Order Corrections to Energy
9. Degenerate Perturbation Theory
10. Example: Non-Degenerate Perturbation in 1D Harmonic Oscillator
11. Example: Degenerate Perturbation in the Hydrogen Atom
12. Applications of Time-Independent Perturbation Theory
13. Limitations of the Method
14. Variants and Extensions
15. Conclusion

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1. Introduction

Perturbation theory is a powerful approximation technique in quantum mechanics that allows us to find approximate solutions to problems that cannot be solved exactly. In many physical systems, the Hamiltonian can be written as a sum of a solvable part and a small perturbing term. Time-independent perturbation theory applies when the perturbation does not vary with time.

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2. Motivation for Perturbation Theory

• Exact solutions are known only for a few systems (e.g., hydrogen atom, harmonic oscillator).
• Most realistic systems involve additional terms or interactions not present in the solvable version.
• Perturbation theory gives a way to expand around a known solution and include small corrections.

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3. The General Setup

Suppose the full Hamiltonian is:

\[
\hat{H} = \hat{H}_0 + \lambda \hat{H}’
\]

Where:

• \( \hat{H}_0 \): unperturbed Hamiltonian with known eigenstates and eigenvalues.
• \( \hat{H}’ \): perturbing Hamiltonian.
• \( \lambda \): bookkeeping parameter (set to 1 at the end).

We seek the eigenvalues and eigenstates of \( \hat{H} \) in a power series expansion.

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4. Unperturbed and Perturbed Hamiltonians

Let:

\[
\hat{H}_0 |\psi_n^{(0)}\rangle = E_n^{(0)} |\psi_n^{(0)}\rangle
\]

Then the true energy and wavefunction are expanded as:

\[
E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \dots
\]
\[
|\psi_n\rangle = |\psi_n^{(0)}\rangle + \lambda |\psi_n^{(1)}\rangle + \lambda^2 |\psi_n^{(2)}\rangle + \dots
\]

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5. Expansion of Energy and Wavefunctions

Substitute the expansions into the Schrödinger equation and match powers of \( \lambda \):

\[
(\hat{H}_0 + \lambda \hat{H}’)(|\psi_n^{(0)}\rangle + \lambda |\psi_n^{(1)}\rangle + \dots) = (E_n^{(0)} + \lambda E_n^{(1)} + \dots)(|\psi_n^{(0)}\rangle + \lambda |\psi_n^{(1)}\rangle + \dots)
\]

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6. First-Order Corrections to Energy

The first-order energy correction is:

\[
E_n^{(1)} = \langle \psi_n^{(0)} | \hat{H}’ | \psi_n^{(0)} \rangle
\]

This is the expectation value of the perturbation in the unperturbed state.

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7. First-Order Corrections to Wavefunctions

The first-order wavefunction correction is:

\[
|\psi_n^{(1)}\rangle = \sum_{k \ne n} \frac{\langle \psi_k^{(0)} | \hat{H}’ | \psi_n^{(0)} \rangle}{E_n^{(0)} – E_k^{(0)}} |\psi_k^{(0)}\rangle
\]

Only states different from \( n \) contribute.

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8. Second-Order Corrections to Energy

The second-order correction is:

\[
E_n^{(2)} = \sum_{k \ne n} \frac{|\langle \psi_k^{(0)} | \hat{H}’ | \psi_n^{(0)} \rangle|^2}{E_n^{(0)} – E_k^{(0)}}
\]

This gives information on how much the other states influence the perturbed energy level.

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9. Degenerate Perturbation Theory

When \( \hat{H}_0 \) has degenerate eigenstates, the standard method fails because denominators vanish. In this case:

• Diagonalize \( \hat{H}’ \) within the degenerate subspace.
• Construct new linear combinations of degenerate states that diagonalize the perturbation.
• Use these as the basis for applying perturbation theory.

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10. Example: Non-Degenerate Perturbation in 1D Harmonic Oscillator

Let the perturbation be:

\[
\hat{H}’ = \alpha x^4
\]

This leads to shifts in energy levels that can be computed using the known oscillator eigenstates and matrix elements of \( x^4 \).

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11. Example: Degenerate Perturbation in the Hydrogen Atom

In hydrogen:

• The degeneracy in \( \ell \) is lifted by a perturbation like the spin-orbit coupling.
• Degenerate perturbation theory helps to calculate fine structure corrections.

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12. Applications of Time-Independent Perturbation Theory

• Fine structure in atoms
• Stark effect (electric field perturbation)
• Zeeman effect (magnetic field perturbation)
• Anharmonic oscillators in molecular vibrations

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13. Limitations of the Method

• Assumes the perturbation is small.
• Diverges if energy levels are too close or perturbation is too strong.
• Not suitable for non-analytic or discontinuous perturbations.

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14. Variants and Extensions

• Time-dependent perturbation theory for dynamics and transitions.
• Rayleigh–Schrödinger perturbation theory
• Brillouin–Wigner perturbation theory

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15. Conclusion

Time-independent perturbation theory is a cornerstone of quantum approximation methods. It allows physicists to incorporate the effects of weak interactions and fields on known systems, and predict how energy levels and states shift in realistic environments. It bridges exact theory with experimental reality, enabling the study of a wide range of quantum systems.

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Table of Contents

1. Introduction
2. What Is Fine Structure?
3. Sources of Fine Structure
4. Spin-Orbit Coupling: The Physical Picture
5. Mathematical Formulation of Spin-Orbit Interaction
6. Derivation from Magnetic Interaction
7. Total Angular Momentum \( \vec{J} = \vec{L} + \vec{S} \)
8. Quantum Numbers and Term Symbols
9. Fine Structure Splitting in Hydrogen
10. Energy Corrections: Dirac and Pauli Theories
11. Selection Rules and Transition Lines
12. Fine Structure in Multi-Electron Atoms
13. Spectroscopic Observations
14. Relativistic Origin and Historical Significance
15. Applications and Experimental Techniques
16. Conclusion

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1. Introduction

In atomic physics, fine structure refers to small but significant corrections to atomic energy levels that arise due to relativistic effects and spin-orbit interaction. These corrections lead to the splitting of spectral lines that would otherwise be degenerate in the non-relativistic Schrödinger model. Understanding fine structure is essential to achieving high precision in spectroscopy and in understanding the full quantum mechanical behavior of atoms.

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2. What Is Fine Structure?

Fine structure is observed as a splitting of spectral lines due to corrections that include:

• Relativistic corrections to the electron’s kinetic energy.
• Interaction between the electron’s magnetic moment (due to spin) and the magnetic field created by its orbital motion — known as spin-orbit coupling.
• The Darwin term, which arises from the relativistic zitterbewegung (trembling motion) of the electron.

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3. Sources of Fine Structure

These corrections collectively modify the energy levels of atoms, especially noticeable in hydrogen and hydrogen-like systems. They are:

• \( \Delta E_{\text{rel}} \): The relativistic kinetic energy correction.
• \( \Delta E_{\text{SO}} \): The spin-orbit coupling correction.
• \( \Delta E_{\text{Darwin}} \): The Darwin term affecting s-orbitals.

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4. Spin-Orbit Coupling: The Physical Picture

From the electron’s perspective (its rest frame), the nucleus appears to orbit the electron, generating a magnetic field. This magnetic field interacts with the magnetic dipole moment of the electron caused by its intrinsic spin, leading to a coupling energy that depends on the relative orientation of the spin and orbital angular momentum.

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5. Mathematical Formulation of Spin-Orbit Interaction

The spin-orbit coupling term in the Hamiltonian is typically written as:

\[
\hat{H}_{\text{SO}} = \xi(r) \vec{L} \cdot \vec{S}
\]

Where:

• \( \vec{L} \) is the orbital angular momentum operator.
• \( \vec{S} \) is the spin angular momentum operator.
• \( \xi(r) \) is a function depending on the radial distance and the potential.

For hydrogen-like atoms:

\[
\xi(r) = \frac{1}{2 m_e^2 c^2 r} \frac{dV}{dr}
\]

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6. Derivation from Magnetic Interaction

In the electron’s rest frame:

• The magnetic field from the moving nucleus is:

\[
\vec{B} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{Ze}{m_e c^2 r^3} \vec{L}
\]

• The spin magnetic moment of the electron is:

\[
\vec{\mu}_s = -g_s \mu_B \frac{\vec{S}}{\hbar}
\]

• The spin-orbit energy shift becomes:

\[
\Delta E = -\vec{\mu}_s \cdot \vec{B}
\]

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7. Total Angular Momentum \( \vec{J} = \vec{L} + \vec{S} \)

Total angular momentum is given by:

\[
\vec{J} = \vec{L} + \vec{S}
\]

The value of \( \vec{L} \cdot \vec{S} \) can be expressed in terms of quantum numbers as:

\[
\vec{L} \cdot \vec{S} = \frac{1}{2} \left( J^2 – L^2 – S^2 \right)
\]

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8. Quantum Numbers and Term Symbols

Energy levels split into sub-levels labeled by term symbols:

\[
^{2S+1}L_J
\]

Where:

• \( S \) is the total spin quantum number.
• \( L \) is the total orbital angular momentum (S = 0, P = 1, D = 2, F = 3, etc.).
• \( J = L + S, L + S – 1, …, |L – S| \)

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9. Fine Structure Splitting in Hydrogen

Using the Dirac equation, the corrected energy levels are:

\[
E_{n,j} = -\frac{m_e c^2 \alpha^2}{2n^2} \left[ 1 + \frac{\alpha^2}{n^2} \left( \frac{n}{j + 1/2} – \frac{3}{4} \right) \right]
\]

Where:

• \( \alpha \) is the fine-structure constant.
• \( n \) is the principal quantum number.
• \( j \) is the total angular momentum quantum number.

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10. Energy Corrections: Dirac and Pauli Theories

The Dirac equation, which naturally includes relativistic and spin effects, gives the full fine structure without ad hoc corrections. In contrast, the Pauli equation introduces spin in a semi-classical way and adds correction terms to the Schrödinger Hamiltonian.

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11. Selection Rules and Transition Lines

Allowed transitions must obey selection rules:

• \( \Delta J = 0, \pm 1 \) (except \( J = 0 \rightarrow J = 0 \) is forbidden)
• \( \Delta L = \pm 1 \)
• \( \Delta S = 0 \)

Fine structure explains the splitting of spectral lines into multiple closely spaced components.

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12. Fine Structure in Multi-Electron Atoms

In multi-electron atoms:

• Electrons interact through LS coupling (Russell-Saunders coupling) in light atoms.
• jj-coupling is used for heavy atoms.
• Energy levels depend on both the total spin and total orbital angular momentum of the entire system.

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13. Spectroscopic Observations

Fine structure is visible in:

• Hydrogen spectral series.
• Sodium D-lines (\( 589.0 \) nm and \( 589.6 \) nm).
• Helium transitions.

Observed with high-resolution instruments like Fabry–Pérot interferometers.

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14. Relativistic Origin and Historical Significance

• Validated special relativity in atomic physics.
• Led to the development of the Dirac equation.
• Confirmed the existence and role of electron spin.

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15. Applications and Experimental Techniques

• Fine structure is used in atomic clocks for time standards.
• Influences laser cooling and trapping.
• Essential in quantum electrodynamics and testing the standard model via precision spectroscopy.

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16. Conclusion

Fine structure and spin-orbit coupling enrich our understanding of atomic structure by including relativistic and intrinsic spin effects. They provide the basis for interpreting high-resolution spectra and have critical applications in technology, fundamental physics, and quantum information science.

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