Perturbation Theory for GATE 2026 – Complete Notes, Solved PYQs & Quick Tricks

Fundamentals of Time-Independent Perturbation Theory

Time-independent perturbation theory allows physicists to estimate the eigenstates and eigenvalues of a Hamiltonian that differs slightly from a known solvable Hamiltonian. It operates on the principle that the Hamiltonian can be split into a dominant unperturbed part ($H_0$) and a small perturbation term ($H’$), enabling iterative corrections to energy and wavefunctions.

In quantum mechanics, exact solutions to the Schrödinger equation are rare. Most physical systems, such as atoms in electric fields or anharmonic oscillators, are too complex to solve analytically. Perturbation theory notes often start by expressing the total Hamiltonian $H$ as:

Here, $H_0$ represents the unperturbed Hamiltonian (the known system), $H’$ is the perturbation (the small disturbance), and $\lambda$ is a dimensionless parameter ranging from 0 to 1 that tracks the strength of the perturbation.

For perturbation theory GATE preparation, you must assume that the eigenenergies ($E_n^{(0)}$) and eigenfunctions ($\psi_n^{(0)}$) of the unperturbed system are fully known:

The goal is to find the corrections to these energies and wavefunctions as a power series in $\lambda$. When $\lambda$ is small, higher-order terms become negligible, allowing us to truncate the series after the first order correction or second order correction. This method is effective only when the perturbation is weak compared to the unperturbed energy level spacing.

Non-Degenerate Perturbation Theory: First & Second Order Corrections

Non-degenerate perturbation theory applies when each energy eigenvalue of the unperturbed system corresponds to a unique eigenstate. The corrections are derived by expanding the Schrödinger equation in powers of $\lambda$, providing a systematic way to calculate energy shifts and wavefunction modifications using matrix elements of the perturbation.

Deriving the First Order Correction

The first order correction to the energy is simply the expectation value of the perturbation Hamiltonian calculated using the unperturbed states. This is often the most frequently asked concept in perturbation theory GATE questions because it provides a quick estimate of the energy shift without complex summation.

Mathematically, the first-order energy correction $E_n^{(1)}$ is given by:

This equation states that the energy shift is the average value of the perturbing potential over the unperturbed state.

• If the perturbation $H’$ is symmetric (even) and the state $\psi_n^{(0)}$ has definite parity, the integral may vanish.
• If $H’$ is a constant $V_0$ added to the potential, the correction is simply $V_0$.

The first order correction to the wavefunction, $\psi_n^{(1)}$, involves a linear superposition of all other unperturbed states $\psi_m^{(0)}$:

This formula introduces the concept of matrix elements mixing states. The term $m \neq n$ ensures we do not divide by zero.

Calculating the Second Order Correction

The second order correction to the energy is required when the first-order term vanishes or when higher precision is needed. It involves a sum over all intermediate states and is always negative for the ground state, a crucial property for verifying numerical answers in exams.

The formula for the second-order energy correction $E_n^{(2)}$ is:

Key observations for perturbation theory notes:

1. Ground State Lowering: For the ground state ($n=0$), $E_0^{(0)} < E_m^{(0)}$ for all $m$. Thus, the denominator is always negative, making $E_0^{(2)}$ negative.
2. Energy Gaps: The magnitude of the correction is inversely proportional to the energy gap ($E_n^{(0)} – E_m^{(0)}$). Nearby energy levels influence the perturbation more strongly than distant ones.
3. Matrix Elements: The numerator depends on the square of the matrix elements $|\langle \psi_m^{(0)} | H’ | \psi_n^{(0)} \rangle|^2$, meaning the transition probability between states $n$ and $m$ dictates the strength of the interaction.

Degenerate Perturbation Theory for GATE Physics

Degenerate perturbation theory is used when multiple distinct quantum states share the same energy eigenvalue, causing the standard non-degenerate formulas to fail due to division by zero. This method resolves the ambiguity by diagonalizing the perturbation matrix within the degenerate subspace, effectively “lifting” the degeneracy and splitting the energy levels.

In degenerate perturbation theory, the standard formula for wavefunction correction fails because $E_n^{(0)} – E_m^{(0)} = 0$ in the denominator. To solve this, we must select a specific linear combination of the degenerate states that diagonalizes the perturbation $H’$.

For a generic case with two-fold degeneracy (states $\psi_a$ and $\psi_b$ with energy $E^0$), we form the perturbation matrix $W$ in the subspace of these degenerate states:

Where the matrix elements are $W_{ij} = \langle \psi_i | H’ | \psi_j \rangle$. To find the first order correction to the energies ($E^{(1)}$), we solve the secular equation:

Solving this quadratic equation yields two roots, $E_+^{(1)}$ and $E_-^{(1)}$. These are the energy shifts. If the roots are distinct, the degeneracy is lifted.

• Trace Trick: The sum of the perturbed energies equals the trace of the perturbation matrix ($E_+ + E_- = W_{aa} + W_{bb}$).
• Symmetry Trick: If an operator commutes with both $H_0$ and $H’$, we can use its simultaneous eigenstates to simplify the matrix $W$ to diagonal form immediately.

Standard Cases & Tricks: Infinite Square Well & Harmonic Oscillator

Applying perturbation theory to standard quantum potentials like the infinite square well and harmonic oscillator requires recognizing symmetry arguments and integral properties. These systems appear frequently in GATE, where identifying parity or using operator methods can shortcut lengthy integration processes.

Infinite Square Well Perturbations

The infinite square well is the most common testbed for time-independent perturbation problems.
Unperturbed system ($0 < x < a$):

Common Perturbation Types:

1. Constant Potential ($V_0$): The shift is exactly $V_0$ for all levels.
2. Delta Function $\delta(x – a/2)$:
   • For odd $n$ (states like $\psi_1, \psi_3$), the probability density at the center is maximum. The shift is large: $E_n^{(1)} = \frac{2}{a} V_0$.
   • For even $n$ (states like $\psi_2, \psi_4$), the wavefunction has a node at $a/2$. The shift is zero.
3. Step Perturbation: If the potential is raised by $V_0$ only from $0$ to $a/2$, the correction is $V_0/2$ for all states due to uniform probability distribution averaged over the well.

Harmonic Oscillator Tricks

For the harmonic oscillator, operator methods are superior to integration.

Using ladder operators $a$ and $a^\dagger$, where $x = \sqrt{\frac{\hbar}{2m\omega}}(a + a^\dagger)$:

• Odd Powers ($x, x^3, x^5$): The first order correction is always ZERO. The expectation value of an odd parity operator in a definite parity state (like harmonic oscillator eigenstates) vanishes.
• Even Powers ($x^4$): The first order correction is non-zero. For $H’ = \lambda x^4$, calculating $\langle n | x^4 | n \rangle$ involves terms like $a^\dagger a^\dagger a a$ and requires counting number operator pairs.

Critical Perspective: When Perturbation Theory Fails

Perturbation theory is not universally valid; it fails when the perturbation is strong or when the energy correction approaches the magnitude of the spacing between unperturbed levels. Understanding these limitations is essential for determining when to abandon approximation methods in favor of variational methods or exact numerical solutions.

A common misconception in perturbation theory notes is that a “small” $\lambda$ guarantees convergence. However, the true criterion for validity is not just the smallness of the perturbation Hamiltonian $H’$, but the ratio of the matrix element to the energy gap.

The perturbative series converges only if:

Why this matters:

• Near-Degeneracy: If two energy levels are very close (quasi-degenerate), the denominator becomes tiny, causing the correction term to blow up even for a weak perturbation. In such cases, one must switch to degenerate perturbation theory or a variational approach.
• Singular Potentials: Perturbations like $1/x$ or $\delta(x)$ in specific dimensions can sometimes lead to divergences that standard Rayleigh-Schrödinger perturbation theory cannot handle without renormalization techniques.
• Asymptotic Series: Often, the perturbation series is asymptotic, meaning it converges for a few terms and then diverges as order increases. Calculating up to the second order correction is usually safe, but higher orders may yield physical nonsense in specific field theories.

Solved PYQs on Perturbation Theory for GATE

Solving previous year questions (PYQs) is the most effective way to internalize perturbation theory concepts. The following examples demonstrate how to apply first-order correction formulas and symmetry arguments to solve typical GATE-level physics problems efficiently.

PYQ Example 1: The Delta Potential (1D Box)

Problem: A particle is in a 1D infinite square well of width $a$ ($0 \le x \le a$). A perturbation $H’ = V_0 a \delta(x – a/2)$ is applied. What is the first order correction to the ground state energy?

Solution:

1. Identify the Unperturbed State:
   Ground state ($n=1$): $\psi_1(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{\pi x}{a}\right)$.
2. Apply Formula:
   $$E_1^{(1)} = \langle \psi_1 | H’ | \psi_1 \rangle = \int_0^a \psi_1^*(x) [V_0 a \delta(x – a/2)] \psi_1(x) dx$$
3. Evaluate Integral:
   $$E_1^{(1)} = V_0 a \left( \sqrt{\frac{2}{a}} \right)^2 \int_0^a \sin^2\left(\frac{\pi x}{a}\right) \delta(x – a/2) dx$$

   Using the property $\int f(x)\delta(x-x_0)dx = f(x_0)$:

   $$E_1^{(1)} = 2 V_0 \sin^2\left(\frac{\pi (a/2)}{a}\right) = 2 V_0 \sin^2(\pi/2) = 2 V_0 (1)^2$$
   $$E_1^{(1)} = 2 V_0$$

Answer: The energy shifts upward by $2 V_0$.

PYQ Example 2: Perturbed Harmonic Oscillator

Problem: A harmonic oscillator with frequency $\omega$ is perturbed by $H’ = \beta x$. Find the energy shift to the second order.

Solution:

1. First Order Correction:
   $$E_n^{(1)} = \langle n | \beta x | n \rangle$$

   Since $x$ is an odd operator and $|n\rangle$ has definite parity, the integral is zero. $E_n^{(1)} = 0$.

2. Second Order Correction:
   $$E_n^{(2)} = \sum_{m \neq n} \frac{|\langle m | \beta x | n \rangle|^2}{E_n^{(0)} – E_m^{(0)}}$$

   Selection rules for $x$ allow transitions only to $m = n+1$ and $m = n-1$.

   • Term 1 ($m=n+1$): $\Delta E = -\hbar \omega$. Matrix element squared $\propto (n+1)$.
   • Term 2 ($m=n-1$): $\Delta E = +\hbar \omega$. Matrix element squared $\propto n$.

   Using exact matrix elements $\langle n+1 | x | n \rangle = \sqrt{\frac{\hbar}{2m\omega}}\sqrt{n+1}$:

   $$E_n^{(2)} = \beta^2 \left[ \frac{\frac{\hbar}{2m\omega}(n+1)}{-\hbar\omega} + \frac{\frac{\hbar}{2m\omega}(n)}{\hbar\omega} \right] = \frac{\beta^2}{2m\omega^2} [ -(n+1) + n ]$$
   $$E_n^{(2)} = -\frac{\beta^2}{2m\omega^2}$$

Answer: The second order correction is constant for all levels: $-\frac{\beta^2}{2m\omega^2}$. This matches the exact solution obtained by completing the square in the Hamiltonian.

Quick Tricks Summary for GATE

• Odd Potentials: If $H’$ is an odd function of $x$ (like $x, x^3$) and the domain is symmetric (like HO or box $-a$ to $a$), $E^{(1)} = 0$.
• Constant Perturbation: If $H’ = C$, then $E^{(1)} = C$.
• Degenerate Lifting: If a perturbation does not break the symmetry causing degeneracy, the degeneracy remains. You need a symmetry-breaking perturbation (like an electric field for the Hydrogen atom Stark effect) to lift it.
• Infinite Wall Shifts: Changing the width of an infinite square well from $a$ to $a+\epsilon$ can be treated as a perturbation where the wall moves. Use the virial theorem or adiabatic invariants for quick checks.

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