UPSC Civil Services (Main) Examination 2026 — Physics Optional Paper II Analysis, Trends & Comparison | OurEducation

Last Updated: Sep 8, 2026

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An analysis of Physics Optional Paper II of UPSC Civil Services (Main) Examination 2026 (held 2026-08-30) — difficulty, topic spread, and how it compares with previous years.

Official paper: official source.

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UPSC Civil Services (Main) Examination 2026 — Physics Optional Paper II: Analysis

1. Overall Difficulty & Balance

The 2026 Physics Optional Paper II maintained a high standard of rigor while ensuring a balanced distribution across core topics. The paper tested both conceptual depth and problem-solving agility, with a slight tilt toward quantum mechanics and atomic/molecular physics. The questions were well-structured, requiring multi-step reasoning rather than rote application. The difficulty level was comparable to previous years, with no single question standing out as excessively trivial or overly complex. The balance between theory, derivations, and numerical problems was well-maintained, though some sections (e.g., nuclear physics) had more weightage than others.

2. Section/Topic Distribution

The paper exhibited the following distribution of topics:

• Quantum Mechanics (30-35%): Dominated by questions on wavefunctions, commutators, spin systems, and harmonic oscillators. Notably, the half-harmonic oscillator (Q2(b)(ii)) and spin-½ particle wavefunction (Q1(a)(ii)) were key focal points.
• Atomic & Molecular Physics (20-25%): Heavy emphasis on hydrogen atom quantum numbers (Q3(c)(i)), Stern-Gerlach separation (Q1(c)), and vibrational spectroscopy (Q4(a)). The Lamb shift question (Q2(c)(ii)) was a conceptual highlight.
• Nuclear & Particle Physics (15-20%): Binding energy, gamma decay, and effective mass (Q5) were tested, along with the two-fluid model in superconductivity (Q5(e)). The magic number failure of the 3D harmonic oscillator (Q6(a)) was a recurring theme.
• Thermal & Statistical Physics (10-15%): Debye model (Q8(a)) and diamagnetism (Q7(i)-(iii)) were the primary contributors. The T³ law application was straightforward but required precise calculations.
• Solid-State & Electronics (10-15%): Photovoltaic effect (Q8(b)) and op-amp comparator (Q7(b)) tested applied physics. The SU(3) symmetry (Q8(c)) was a niche but important topic.

3. Comparison with Last Year & Multi-Year Trend

Shifts:

• Increased Quantum Mechanics Weightage: Compared to 2025, quantum mechanics questions were more frequent, particularly on spin systems and commutators. The half-harmonic oscillator (Q2(b)(ii)) was a new addition, testing advanced understanding.
• Nuclear Physics Simplification: While nuclear physics remains a staple, the 2026 paper avoided overly complex decay chains or reactor kinetics, focusing instead on binding energy and gamma transitions (Q5).
• More Applied Problems: The inclusion of the op-amp comparator (Q7(b)) and photovoltaic effect (Q8(b)) suggests a trend toward applied physics, aligning with UPSC’s emphasis on real-world relevance.
• Consistent Themes: Topics like the Debye model (Q8(a)), Stern-Gerlach (Q1(c)), and magic numbers (Q6(a)) have appeared repeatedly over the years, reinforcing their importance.

Repeats:

• Harmonic Oscillator Variations: The full and half-harmonic oscillator problems (Q2(a), Q2(b)(ii)) have been tested in different forms in past papers, emphasizing their foundational role.
• Magnetic Resonance: EPR vs. NMR (Q2(c)(i)) and relaxation processes (Q3(b)(i)) are perennial favorites.
• Nuclear Models: The 3D harmonic oscillator’s failure to explain magic numbers (Q6(a)) is a classic topic revisited periodically.

4. Notable/Unexpected Questions & Why They Matter

• Half-Harmonic Oscillator (Q2(b)(ii)): This question tested the ability to adapt known solutions (infinite square well + harmonic oscillator) to a modified potential. It was unexpected because such problems are rare in standard textbooks but align with UPSC’s push for creative problem-solving.
• Commutator [x², cos(p)] (Q1(d)): While commutators are common, the inclusion of a cosine function made this question non-trivial. It required series expansion or Taylor series approximation, testing advanced mathematical physics skills.
• SU(3) Symmetry (Q8(c)): This was a high-difficulty, niche topic. Its inclusion suggests UPSC’s willingness to test advanced particle physics concepts, which are often overlooked by aspirants focusing solely on classical topics.
• Lamb Shift (Q2(c)(ii)): A conceptual question probing the limitations of the Dirac theory. It was unexpected because UPSC typically avoids highly specialized topics like the Lamb shift, but its inclusion underscores the need for a deep understanding of quantum electrodynamics (QED) fundamentals.

5. Key Takeaways for Aspirants

• Master Quantum Mechanics Fundamentals: Given the weightage, aspirants must prioritize wavefunctions, spin systems, commutators, and harmonic oscillators. The half-harmonic oscillator and spin-½ problems are must-practice.
• Revisit Classical Topics with a Twist: Questions like the commutator [x², cos(p)] and Stern-Gerlach separation require more than rote learning. Aspirants should focus on problem-solving techniques and mathematical tools (e.g., Taylor series, perturbation theory).
• Stay Updated on Applied Physics: The inclusion of op-amp circuits (Q7(b)) and photovoltaic effects (Q8(b)) signals UPSC’s interest in applied topics. Aspirants should supplement theory with practical problem-solving.
• Balance Depth and Breadth: While advanced topics like SU(3) symmetry (Q8(c)) are important, aspirants must also ensure they cover foundational areas like nuclear binding energy (Q5) and Debye model (Q8(a)).
• Practice Numerical Precision: Questions like the vibrational spectroscopy (Q4(a)) and Stern-Gerlach separation (Q1(c)) require meticulous calculations. Aspirants should practice dimensional analysis and unit conversions.
• Understand Conceptual Limits: The Lamb shift (Q2(c)(ii)) and limitations of the Dirac theory highlight the importance of knowing where standard theories break down. This is crucial for answering “why” questions in the exam.

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Questions asked

1. (i) The de Broglie wavelength of a particle in thermal equilibrium at a temperature of $27^{\circ}\text{C}$ is $\lambda$ . At what temperature will it be $\frac{\lambda}{2}$ ? (ii) The wave function of a spin $\frac{1}{2}$ particle is given by $\left(\frac{-i}{2}\right)$ . What is the probability of finding the particle in the spin-up state? 5+5 (b) The OH-radical has a moment of inertia of $1.48 \times 10^{-46} \text{ kg.m}^2$ . For $J = 6$ , calculate its angular velocity and angular momentum. Find the energy absorbed in the $J = 6 \rightarrow J = 7$ transition. (c) In a Stern-Gerlach experiment, the magnetic field varies with distance in the $z$ -direction according to $\frac{\text{dB}_z}{\text{d}z} = 1.5 \text{ T/mm}$ . The speed of silver atoms coming from the oven is $725 \text{ m/s}$ . Silver atoms travel a distance of $3 \text{ cm}$ through the magnet. Calculate the separation of the two beams of atoms when they leave the magnet. [Given : The mass of a silver atom = $1.8 \times 10^{-25} \text{ kg}$ and its magnetic moment is 1 Bohr magneton.] (d) If the position and momentum operators in 1-D are represented by $\hat{x}$ and $\hat{p}$ respectively, calculate the commutator $[\hat{x}^2, \cos(\hat{p})]$. (e) Find the eigenvalues and eigenfunctions of a system described by Hamiltonian $H = ae^{b\sigma_x}$, where a, b are constants and $\sigma_x$ is the x-component of Pauli matrix $\vec{\sigma}$.
2. (a) A particle is in the ground state of a 1-D simple harmonic oscillator potential, $V(x) = \frac{1}{2} m\omega^2 x^2$. Calculate the position uncertainty of the particle. (b) $$H = \frac{p^2}{2m} + V(x), \text{ जहाँ}$$ $$V(x) = \frac{1}{2} m\omega^2 x^2 \quad x \ge 0 \text{ के लिए}$$ $$= \infty \quad x < 0 \text{ के लिए}$$ (i) A particle is in the normalized state $\psi$ which is a superposition of the energy eigenstates $\psi_1$ and $\psi_2$ with energies 10 eV and 30 eV, respectively. The average value of the energy of the particle in the state $\psi$ is 24 eV. Find the state $\psi$ in terms of $\psi_1$ and $\psi_2$. (ii) Obtain the allowed eigenenergies of the half 1-D simple harmonic oscillator defined by the Hamiltonian $$\begin{array}{l} H = \frac{p^2}{2m} + V(x), \text{ where} \\ V(x) = \frac{1}{2} m\omega^2 x^2 \quad \text{for } x \ge 0 \\ = \infty \quad \text{for } x < 0 \end{array}$$ Express its eigenstates in terms of eigenstates of the full 1-D oscillator. 10+10 (c) (i) Why does the electron paramagnetic resonance (EPR) spectroscopy utilize the microwave region of the electromagnetic spectrum, while nuclear magnetic resonance (NMR) uses radio waves ? (ii) Why can the experimental observation of the Lamb shift not be explained by the Dirac theory ? 7+8
3. (i) Calculate the number of permitted electrons in a sub-shell and a shell in an atom. (ii) Explain how Pauli's exclusion principle helps in determining the electronic configuration in a many-electron system. 8+12 (b) (i) Why is the relaxation process so important in nuclear magnetic resonance (NMR) ? (ii) Why are $^{12}\text{C}$ and $^{16}\text{C}$ nuclei not suitable for the study of NMR ? 10+5 (c) (i) The wave function of a hydrogen atom is written as $\psi_{n, l, m} (r, \theta, \phi)$. Explain the quantum numbers $n, l, m$ and mention their ranges. Show that $\psi_{n, l, m} (r, \theta, \phi)$ is $n^2$ degenerate. (ii) The electron in the hydrogen atom is found in the state $\psi (r, \theta, \phi) = A R(r) \sin \theta \cos \theta e^{-i\phi}$. Find the z-component of angular momentum of the electron. 10+5
4. (i) What are hot bands in vibrational spectroscopy ? Why are they called so ? (ii) The fundamental and 2$^{nd}$ harmonic transitions of $^{14}\text{N}^{16}\text{O}$ are recorded at $1876.06 \text{ cm}^{-1}$ and $3724.2 \text{ cm}^{-1}$, respectively. Calculate the equilibrium vibrational frequency, the anharmonicity constant and the zero-point energy of the molecule. [Given : Mass of $^{14}\text{N}$ = $23.25 \times 10^{-27} \text{ kg}$, Mass of $^{16}\text{O}$ = $26.56 \times 10^{-27} \text{ kg}$] 8+7 (b) A particle of mass m is in the ground state of a 1-D box of infinite potential of length a. If the length of the box is changed to 2a symmetrically without disturbing the wave function of the particle, find the probability that the particle will be in the ground state of the new box. (c) (i) Explain why the rotational transition J = 0 → J = 1 is often not the most intense. (ii) Find the positions of the first four rotational Raman lines in the spectrum of H₂, if its bond length is 0·742 Å. What will be the effect of nuclear spin on the spectrum ? [Given : Mass of ¹H = 1·673 × 10⁻²⁷ kg] 8+7
5. Are the nucleons inside the nucleus governed by the laws of quantum physics ? Justify your answer. (b) Binding energy and rest mass energy of a two-nucleon bound state are denoted by B and $Mc^2$, respectively, where c is the speed of light. Calculate the minimum energy of a photon required to dissociate this bound state in terms of B and $Mc^2$. (c) A nucleus of rest mass M is initially in an excited state whose energy is $\Delta E$ above its ground state. The nucleus emits a $\gamma$-ray of energy $h\nu$ and makes a transition to its ground state. Calculate the fractional change in energy for the nucleus. (d) (i) What is the effective mass of an electron ? (ii) What are the physical reasons that the effective mass of an electron can be infinite and negative ? (iii) The energy near the valence band edge of a crystal is given by $E = -10^{-39} \text{ k}^2 \text{ Jm}^2$. An electron with wave vector $10^{10} \hat{k}_x \text{ m}^{-1}$ is removed from an orbital in the completely filled valence band. Determine its effective mass and momentum. 2+3+5 (e) Explain briefly the two-fluid model proposed by the London brothers and obtain the expressions for the penetration depth and the number of superelectrons in a superconducting specimen.
6. By considering the three-dimensional harmonic oscillator potential, explain the energy levels of nucleons inside the nucleus. Also, derive the zero-point energy of nucleons. Why did this assumption fail to explain the existence of nuclei with higher magic number ? (b) What do you understand by the packing fraction 'f' and the binding energy ' $E_b$ ' of a nucleus ? Draw the graphs for packing fraction f versus mass number (A) and binding energy fraction $f_b \left( = \frac{E_b}{A} \right)$ versus mass number (A). Further, explain how the graphs of the variation of f and $f_b$ with A have complementary approaches. (c) (i) What are the different methods used to increase the probability of exposure for the atomic planes with right orientation to X-rays in the X-rays diffraction studies ? (ii) Explain the principle and working of the Laue's diffraction method. Comment on the origin of Laue spots and the utility of Laue's diffraction pattern. 5+15
7. (i) What is diamagnetism ? Why do diamagnetic materials have negative magnetic susceptibility ? (ii) Discuss the limitations of the quantum theory of diamagnetism. (iii) Draw a diagram of magnetic susceptibility as a function of temperature. Pay attention to the relative scales in your diagram so that the relative strengths of the magnetic phenomena are presented. 5+5+10 (b)   (I) (II) The input signal as shown in Figure (I) is applied to the comparator circuit given in Figure (II). Make a sketch of the output signal showing its proper relationship to the input signal. Assume that the maximum output levels of the op-amp are ± 12 V.   (I) (II) (c) What is the criticality of a self-sustained nuclear reactor ? Write the basic processes affecting the nuclear chain reactions of a finite-size nuclear reactor. Also, write the critical size of reactors of different shapes in terms of geometrical buckling constant.
8. (i) Discuss the Debye model of lattice specific heat. What are the limitations of the Debye model ? (ii) At very low temperatures, the specific heat of rock salt varies with temperature according to the Debye T$^{3}$ law. The Debye temperature for rock salt is 281 K. How much heat will be required to raise the temperature of 2 kilo-mol of rock salt from 10 K to 50 K ? 10+5 (b) What is the photovoltaic effect ? What are the processes that are required to obtain a useful power output from photon interactions in a semiconductor ? List the factors that affect the efficiency of a solar cell. (c) What do you understand by the SU(3) symmetry for the classification of baryons and mesons ? Draw the octets for baryons and mesons along with their quarks and antiquarks.

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