Quantum Mechanics Course

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← Part VI/Time-Independent Perturbation

1. Time-Independent Perturbation Theory

Reading time: ~50 minutes | Pages: 10

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Time-independent perturbation theory is one of the most powerful and widely-used approximation methods in quantum mechanics. When we encounter a system whose Hamiltonian can be written as a sum of a solvable "unperturbed" part and a small "perturbation," we can systematically calculate corrections to the energies and wave functions.

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Video Lecture

Introduction to Time-Independent Perturbation Theory

Comprehensive introduction to the fundamental concepts of perturbation theory, including the setup, first-order corrections, and physical interpretation.

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

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Development of Perturbation Theory

Schrödinger's Perturbation Method

1926

Erwin Schrödinger

Developed the first systematic perturbation theory for quantum mechanics in his series of papers on wave mechanics. Used it to calculate corrections to energy levels in atoms.

Rayleigh-Schrödinger Perturbation Theory

1930

Various physicists

Formalized the complete mathematical framework combining classical Rayleigh perturbation methods with quantum mechanics, establishing the systematic power series expansion.

Perturbative QED

1949

Feynman, Schwinger, Tomonaga

Extended perturbation theory to quantum field theory, calculating corrections to quantum electrodynamics with extraordinary precision (Nobel Prize 1965).

💡 These developments represent key milestones in the evolution of quantum mechanics.

The Setup

The fundamental idea is to split the Hamiltonian into two parts:

$$\hat{H} = \hat{H}_0 + \lambda\hat{H}'$$

where:

  • $\hat{H}_0$ is the "unperturbed" Hamiltonian with known exact solutions
  • $\hat{H}'$ is the "perturbation" (typically small)
  • $\lambda$ is a dimensionless parameter measuring the strength of the perturbation (often set to 1 at the end)

The strategy is to expand the true energies and wave functions as power series in $\lambda$, then solve order by order. This works remarkably well when $\hat{H}'$ introduces only small changes compared to the level spacings of $\hat{H}_0$.

Unperturbed Problem

We assume we know the complete solution to the unperturbed problem:

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

The superscript (0) denotes unperturbed quantities. The states $\{|\psi_n^{(0)}\rangle\}$ form a complete orthonormal basis.

Perturbation Series

We expand both the energy eigenvalues and eigenstates as power series in $\lambda$:

$$E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \lambda^3 E_n^{(3)} + \cdots$$
$$|\psi_n\rangle = |\psi_n^{(0)}\rangle + \lambda|\psi_n^{(1)}\rangle + \lambda^2|\psi_n^{(2)}\rangle + \lambda^3|\psi_n^{(3)}\rangle + \cdots$$

The corrections $E_n^{(k)}$ and $|\psi_n^{(k)}\rangle$ are called the k-th order energy and wave function corrections. We assume these series converge for $\lambda \ll 1$.

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Video Lecture

Derivation of First-Order Perturbation Theory

Step-by-step derivation of the first-order energy correction and wave function correction formulas.

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

First-Order Energy Correction

The first-order correction to the energy is simply the expectation value of the perturbation:

$$E_n^{(1)} = \langle\psi_n^{(0)}|\hat{H}'|\psi_n^{(0)}\rangle$$

This is remarkably simple: just calculate $\langle \hat{H}' \rangle$ in the unperturbed state! This first-order correction is often sufficient for many applications and gives excellent agreement with experiment.

First-Order Wave Function Correction

The perturbed wave function gains admixtures from all other unperturbed states:

$$|\psi_n^{(1)}\rangle = \sum_{m\neq n}\frac{\langle\psi_m^{(0)}|\hat{H}'|\psi_n^{(0)}\rangle}{E_n^{(0)} - E_m^{(0)}}|\psi_m^{(0)}\rangle$$

This formula tells us that the perturbation "mixes in" other unperturbed states. The mixing amplitude is proportional to the matrix element $\langle m|\hat{H}'|n\rangle$ and inversely proportional to the energy difference $(E_n^{(0)} - E_m^{(0)})$.

Key insight: States with energies close to $E_n^{(0)}$ contribute more strongly to the correction. If any $E_m^{(0)} \approx E_n^{(0)}$, the perturbation series fails (degeneracy).

Second-Order Energy Correction

The second-order correction involves a sum over all intermediate states:

$$E_n^{(2)} = \sum_{m\neq n}\frac{|\langle\psi_m^{(0)}|\hat{H}'|\psi_n^{(0)}\rangle|^2}{E_n^{(0)} - E_m^{(0)}}$$

Notice that this is always negative for the ground state (since $E_0^{(0)} < E_m^{(0)}$ for all $m \neq 0$), so perturbations always lower the ground state energy to second order.

📝 Worked Example: Anharmonic Oscillator Ground State

Problem: A quantum harmonic oscillator is perturbed by a small anharmonic (quartic) term. Calculate the first-order energy correction to the ground state.

The full Hamiltonian is:

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2 + \lambda\hat{x}^4$$

Step 1: Identify $\hat{H}_0$ and $\hat{H}'$

$$\hat{H}_0 = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2, \quad \hat{H}' = \hat{x}^4$$

Step 2: Use the first-order formula

$$E_0^{(1)} = \lambda\langle 0|\hat{x}^4|0\rangle$$

Step 3: Express $\hat{x}$ in terms of ladder operators

$$\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}(\hat{a} + \hat{a}^\dagger)$$

Step 4: Expand $\hat{x}^4$

$$\hat{x}^4 = \left(\frac{\hbar}{2m\omega}\right)^2(\hat{a} + \hat{a}^\dagger)^4$$

Step 5: Apply to ground state

Only terms with equal numbers of $\hat{a}$ and $\hat{a}^\dagger$ contribute. The non-zero contributions come from $\hat{a}^2(\hat{a}^\dagger)^2$, $\hat{a}\hat{a}^\dagger\hat{a}\hat{a}^\dagger$, $\hat{a}\hat{a}^\dagger\hat{a}^\dagger\hat{a}$, etc.

Final Result:

$$E_0^{(1)} = \frac{3\lambda\hbar^2}{4m^2\omega^2}$$

💡 The anharmonic term raises the ground state energy, as expected from the positive quartic potential.

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Video Lecture

Applications: Stark Effect and Zeeman Effect

Classic applications of perturbation theory to atoms in electric and magnetic fields.

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Example: Stark Effect (Linear)

The Stark effect describes the splitting of atomic energy levels in an external uniform electric field $\vec{E} = E_0\hat{z}$. The perturbation is:

$$\hat{H}' = eE_0 z = eE_0 r\cos\theta$$

For hydrogen, the first-order correction vanishes for all states with definite parity (since $z$ is odd under parity). Thus we must calculate the second-order correction:

$$E_n^{(2)} = (eE_0)^2\sum_{m\neq n}\frac{|\langle n\ell m|r\cos\theta|n'\ell' m'\rangle|^2}{E_n - E_{n'}}$$

Exception: For degenerate levels (like n=2 in hydrogen), there is a linear Stark effect due to mixing of $|2s\rangle$ and $|2p\rangle$ states. This requires degenerate perturbation theory.

Example: Zeeman Effect (Weak Field)

An atom placed in a weak magnetic field $\vec{B} = B\hat{z}$ experiences a perturbation:

$$\hat{H}' = -\vec{\mu}\cdot\vec{B} = \frac{eB}{2m}(\hat{L}_z + 2\hat{S}_z)$$

The first-order correction is:

$$E^{(1)} = \mu_B B(m_\ell + 2m_s) = \mu_B B(m_j + m_s)$$

where $\mu_B = e\hbar/(2m_e) \approx 5.79 \times 10^{-5}$ eV/T is the Bohr magneton. This splitting is directly observable in spectroscopy and is the basis for techniques like electron spin resonance (ESR).

📝 Worked Example: Second-Order Correction for Perturbed Infinite Well

Problem: A particle in an infinite square well (width $a$) is perturbed by $\hat{H}' = V_0\sin(\pi x/a)$. Calculate the second-order energy correction to the ground state.

Step 1: Unperturbed ground state

$$\psi_1^{(0)}(x) = \sqrt{\frac{2}{a}}\sin\left(\frac{\pi x}{a}\right), \quad E_1^{(0)} = \frac{\pi^2\hbar^2}{2ma^2}$$

Step 2: First-order energy correction

$$E_1^{(1)} = \langle\psi_1^{(0)}|\hat{H}'|\psi_1^{(0)}\rangle = \frac{2V_0}{a}\int_0^a \sin^2\left(\frac{\pi x}{a}\right)\sin\left(\frac{\pi x}{a}\right)dx = \frac{V_0}{a}\int_0^a \sin\left(\frac{\pi x}{a}\right)[1-\cos(2\pi x/a)]dx$$

The cosine term integrates to zero, leaving:

$$E_1^{(1)} = V_0$$

Step 3: Matrix elements for second order

We need $\langle n|\hat{H}'|1\rangle$ for all $n \neq 1$:

$$\langle n|\hat{H}'|1\rangle = \frac{2V_0}{a}\int_0^a \sin\left(\frac{n\pi x}{a}\right)\sin\left(\frac{\pi x}{a}\right)\sin\left(\frac{\pi x}{a}\right)dx$$

Using $\sin^2\theta = (1-\cos 2\theta)/2$ and orthogonality, only $n = 3$ contributes:

$$\langle 3|\hat{H}'|1\rangle = \frac{V_0}{2}$$

Step 4: Apply second-order formula

$$E_1^{(2)} = \frac{|\langle 3|\hat{H}'|1\rangle|^2}{E_1^{(0)} - E_3^{(0)}} = \frac{(V_0/2)^2}{\pi^2\hbar^2/(2ma^2) - 9\pi^2\hbar^2/(2ma^2)} = -\frac{V_0^2 ma^2}{32\pi^2\hbar^2}$$

💡 The second-order correction is negative (lowering energy) and scales as $V_0^2$, as expected from perturbation theory.

Validity Conditions

Perturbation theory is valid when the corrections are small compared to the unperturbed quantities:

$$|E_n^{(1)}| \ll |E_n^{(0)} - E_m^{(0)}| \quad \text{for all } m \neq n$$

Equivalently, the perturbation must be small compared to the level spacings. More precisely:

  • First-order correction should be much smaller than unperturbed energy: $|E_n^{(1)}| \ll |E_n^{(0)}|$
  • Series should appear to converge: $|E_n^{(2)}| \ll |E_n^{(1)}|$
  • No near-degeneracies in spectrum (requires degenerate perturbation theory)

🤔 Self-Check Question

Question: In second-order perturbation theory for the ground state $|0\rangle$, why is the energy correction $E_0^{(2)}$ always negative?

Show Answer

Answer: $E_0^{(2)}$ is always negative for the ground state.

Reason: The second-order formula is:

$$E_0^{(2)} = \sum_{m\neq 0}\frac{|\langle m|\hat{H}'|0\rangle|^2}{E_0^{(0)} - E_m^{(0)}}$$

Since $|0\rangle$ is the ground state, we have $E_0^{(0)} < E_m^{(0)}$ for all $m \neq 0$. Therefore $(E_0^{(0)} - E_m^{(0)}) < 0$ for all terms in the sum.

The numerator $|\langle m|\hat{H}'|0\rangle|^2$ is always positive (or zero). Thus each term in the sum is negative, making the total sum negative.

Physical interpretation: Any perturbation allows the ground state to "mix in" excited states, effectively lowering its energy through quantum fluctuations—this is the quantum mechanical version of the variational principle.

Key Limitations

⚠️ When Perturbation Theory Fails:

  • Degeneracy: If $E_n^{(0)} = E_m^{(0)}$, the denominators blow up. Must use degenerate perturbation theory.
  • Near-degeneracy: Even if levels aren't exactly degenerate, small energy differences lead to large corrections that violate convergence.
  • Strong perturbations: If $\hat{H}'$ is comparable to or larger than $\hat{H}_0$, the series doesn't converge.
  • Level crossings: Cannot describe situations where energy levels cross or exhibit avoided crossings.
  • Asymptotic series: Many perturbation series are asymptotic (diverge at high order) even when low orders are accurate.

💡 Application: Atomic and Molecular Physics

Time-independent perturbation theory is the workhorse of atomic, molecular, and solid-state physics:

  • Fine structure: Relativistic corrections and spin-orbit coupling treated as perturbations to hydrogen
  • Hyperfine structure: Nuclear magnetic moment interactions split atomic levels
  • Stark and Zeeman effects: External fields split degenerate levels
  • van der Waals forces: Second-order perturbation gives induced dipole interactions
  • Crystal field theory: Ligand fields in transition metal complexes
  • Band structure: Nearly-free electron model treats periodic potential as perturbation

Despite being an approximation method, perturbation theory predictions agree with experimental measurements to extraordinary precision (often parts per million or better).

💡 Application: Quantum Field Theory and QED

Perturbation theory extends far beyond non-relativistic quantum mechanics into quantum field theory:

  • Feynman diagrams: Systematic graphical representation of perturbative expansions
  • Scattering amplitudes: Cross sections calculated order by order in coupling constants
  • Renormalization: Technique to handle divergences in higher-order terms
  • Precision tests: Electron anomalous magnetic moment calculated to 5 loops, agrees with experiment to 12 decimal places

The perturbative methods you learn here form the foundation for understanding the Standard Model of particle physics—the most successful scientific theory ever developed.

🔗 Related Topics: Degenerate Perturbation Theory - Essential when unperturbed levels have the same energy | Time-Dependent Perturbation Theory - For time-varying perturbations and transitions

Part VI OverviewDegenerate Perturbation →