Variation Method for GATE 2026 – High-Yield Notes, PYQs & Step-by-Step Solutions

Understanding the Variational Principle in the Variation Method

The variational principle serves as the mathematical foundation for the Variation Method. It states that for any arbitrary, well-behaved normalized function used to describe a quantum system, the expectation value of the Hamiltonian (total energy) will never be lower than the actual ground state energy of that system. This provides a safe “ceiling” for energy estimates calculated via the Variation Method.

In the context of GATE Physics, understanding this inequality is crucial. If $E_0$ represents the true ground state energy and $\psi_{trial}$ is a guessed wavefunction, the principle guarantees:

$$E[\psi_{trial}] = \frac{\langle \psi_{trial} | \hat{H} | \psi_{trial} \rangle}{\langle \psi_{trial} | \psi_{trial} \rangle} \ge E_0$$

This inequality transforms a differential equation problem into a calculus minimization problem. By adjusting parameters within the function to minimize energy, you get closer to the true value. This approach classifies the Variation Method as one of the most reliable approximate methods for bound states.

Criteria for Selecting a Valid Trial Wave function

Choosing the right trial wave function is the most critical step in the Variation Method. A poorly chosen function will yield an energy value significantly higher than the true ground state, reducing the accuracy of your result. The trial function is not a random guess; it must adhere to specific physical constraints to be valid for quantum mechanical calculations using the Variation Method.

To ensure your trial functions are valid for GATE Physics problems, they must meet these conditions:

• Boundary Conditions: The function must vanish where the potential is infinite (e.g., at the walls of a box).
• Continuity: The function and its first derivative must be continuous and single-valued everywhere.
• Normalizability: The integral of the square of the function over all space must be finite (square-integrable).
• Symmetry: The parity of the trial function should match the symmetry of the potential (e.g., use an even function for a symmetric potential).

The Variation Theorem: Proof of Upper Bound Energy

The variation theorem provides the rigorous justification for why the Variation Method works. It confirms that the energy calculated is always an upper bound energy. This means you will never underestimate the ground state energy; you will only overestimate it or hit it exactly.

Consider the eigenvalues $E_n$ and eigen functions $\phi_n$ of the Hamiltonian $\hat{H}$ such that $\hat{H}\phi_n = E_n\phi_n$. Since the eigen functions form a complete set, any arbitrary trial wavefunction $\psi$ can be expanded as a linear combination:

$$\psi = \sum_n c_n \phi_n$$

The expectation value of the energy is:

$$\langle E \rangle = \frac{\sum |c_n|^2 E_n}{\sum |c_n|^2}$$

Subtracting the true ground state energy $E_0$ from both sides reveals that the difference is non-negative, proving that $\langle E \rangle \ge E_0$. This theorem ensures that minimizing the energy with respect to variational parameters systematically improves the approximation provided by the Variation Method.

Step-by-Step Procedure to Apply the Variation Method

Applying the Variation Method in an exam setting requires a structured algorithm. GATE Physics questions often ask for the optimal value of a variational parameter or the minimum energy estimate.

Follow this standard procedure to solve these problems efficiently using the Variation Method:

1. Construct the Hamiltonian ($\hat{H}$): Write down the kinetic and potential energy operators for the given system.
2. Select a Trial Wavefunction ($\psi$): Choose a function containing one or more adjustable parameters (e.g., $\alpha, \beta$). Ensure it satisfies boundary conditions.
3. Calculate Expectation Value ($\langle E \rangle$): Compute the integral $\langle \psi | \hat{H} | \psi \rangle$. If the function is not normalized, divide by $\langle \psi | \psi \rangle$.
4. Minimize Energy: Differentiate the energy expression with respect to the parameter (e.g., $\frac{\partial E}{\partial \alpha} = 0$) to find the optimal parameter value.
5. Determine Ground State Energy: Substitute the optimal parameter back into the energy expression to find the minimum energy estimate.

Application: Harmonic Oscillator Using Gaussian Trial Function

A classic textbook example often referenced in GATE Physics is the 1D Harmonic Oscillator. While this system has an exact solution, it perfectly demonstrates the accuracy of the Variation Method when using a Gaussian trial wavefunction.

Step 1: Hamiltonian
$$\hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + \frac{1}{2}m\omega^2 x^2$$

Step 2: Trial Wavefunction
We select a Gaussian function because the ground state is expected to be localized around the origin:
$$\psi(x) = A e^{-\alpha x^2}$$

Step 3: Expectation Value
Calculating the kinetic and potential energy integrals yields:
$$\langle E(\alpha) \rangle = \frac{\hbar^2 \alpha}{2m} + \frac{m\omega^2}{8\alpha}$$

Step 4: Minimization
Differentiating with respect to $\alpha$ and setting to zero gives the optimal $\alpha = \frac{m\omega}{2\hbar}$.

Substituting this back gives $E_{min} = \frac{1}{2}\hbar\omega$. In this specific case, the Variation Method yields the exact ground state energy because the trial function form mathematically matched the true eigenstate.

Solving the Helium Atom Ground State Example

The helium atom example is the standard test case for illustrating the necessity of approximate methods like the Variation Method in multi-electron systems. The electron-electron repulsion term makes the Schrödinger equation impossible to solve analytically.

Using the Variation Method, we treat the interaction as a perturbation or use a screened hydrogenic wavefunction as the trial wavefunction:

$$\psi(r_1, r_2) = \frac{Z_{eff}^3}{\pi a_0^3} e^{-Z_{eff}(r_1 + r_2)/a_0}$$

Here, $Z_{eff}$ (effective nuclear charge) is the variational parameter. Minimizing the energy with respect to $Z_{eff}$ yields a value of approximately $27/16 \approx 1.69$. This calculation gives a ground state energy of -77.5 eV, which is significantly closer to the experimental value (-79.0 eV) than a standard perturbation approach without screening. This highlights the utility of the Variation Method in realistic quantum chemistry problems.

Limitations: When the Variation Method Fails

While the Variation Method is robust, it is not without flaws. A critical analysis reveals specific limitations that every GATE Physics aspirant must understand to avoid pitfalls in conceptual questions.

The primary limitation is that the Variation Method provides an upper bound energy only. It does not tell you how close you are to the true answer; it only tells you that you are not below it. If you choose a physically incorrect trial wavefunction (e.g., wrong asymptotic behavior), your minimized energy can still be significantly higher than the true $E_0$, giving a false sense of accuracy.

Furthermore, the Variation Method is difficult to apply to excited states. To find the first excited state energy ($E_1$), the trial function must be orthogonal to the exact ground state function ($\psi_0$). Since $\psi_0$ is usually unknown (which is why we are using an approximation), ensuring strict orthogonality is computationally challenging. This makes the Variation Method primarily a tool for ground state energy estimation.

High-Yield Trial Functions for GATE Physics

To save time during the GATE Physics exam, memorizing the behavior of standard trial functions is advantageous for applying the Variation Method quickly. You do not always need to derive the integrals from scratch if you recognize the standard forms.

Common trial functions used in Variation Method problems include:

• Gaussian ($\psi = e^{-\alpha x^2}$): Best for smooth, symmetric potentials like the Harmonic Oscillator. Easy to integrate.
• Exponential ($\psi = e^{-\alpha |x|}$): Suitable for potentials with a cusp or sharp change at the origin, such as the Delta function potential.
• Sine Squared ($\psi = \sin^2(kx)$): Often used for infinite potential wells (1D box) to approximate the ground state, satisfying zero boundary conditions at walls.
• Polynomial ($\psi = x(L-x)$): A simple algebraic alternative for the particle in a box ($0 < x < L$). It is computationally cheaper than trigonometric functions but yields slightly higher energy.

Comparison with WKB Approximation

In GATE Physics, students often confuse the Variation Method with the WKB approximation. Both are approximate methods, but they serve different regimes.

Solved PYQ Application: Delta Potential using Variation Method

Let’s apply the Variation Method to a problem type often seen in competitive exams: A particle of mass $m$ in a 1D attractive delta potential $V(x) = -\lambda \delta(x)$.

Objective: Estimate the ground state energy using a Gaussian trial wavefunction $\psi(x) = A e^{-\alpha x^2}$.

1. Normalization:
   $$A^2 \int_{-\infty}^{\infty} e^{-2\alpha x^2} dx = 1 \implies A = \left(\frac{2\alpha}{\pi}\right)^{1/4}$$
2. Kinetic Energy Expectation:
   $$\langle T \rangle = \frac{\hbar^2 \alpha}{2m}$$
3. Potential Energy Expectation:
   $$\langle V \rangle = -\lambda \int_{-\infty}^{\infty} \delta(x) |\psi(x)|^2 dx = -\lambda |\psi(0)|^2 = -\lambda A^2 = -\lambda \sqrt{\frac{2\alpha}{\pi}}$$
4. Total Energy Minimization via Variation Method:
   $$E(\alpha) = \frac{\hbar^2 \alpha}{2m} – \lambda \sqrt{\frac{2}{\pi}} \alpha^{1/2}$$
   Differentiating regarding $\alpha$ and solving for $E_{min}$ yields:
   $$E_{min} = -\frac{m \lambda^2}{\pi \hbar^2}$$

The exact answer is $-m\lambda^2 / 2\hbar^2$. The Variation Method result is approximately $-0.318 (m\lambda^2/\hbar^2)$, while the exact is $-0.5 (m\lambda^2/\hbar^2)$. The variational result is higher (less negative), satisfying the upper bound energy condition inherent to the Variation Method.\

Learn More :

• Postulates of quantum mechanics GATE
• Operators and observables
• Particle in a box 1D 2D 3D
• Harmonic Oscillator
• Rigid Rotor
• Hydrogen Atom
• Angular Momentum