UGC ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM PH 05 PHYSICAL SCIENCE TEST SERIES # 1. Quantum, Statistical & Thermal Physics

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1 UGC ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM BOOKLET CODE SUBJECT CODE PH 05 PHYSICAL SCIENCE TEST SERIES # Quantum, Statistical & Thermal Physics Timing: 3: H M.M: 00 Instructions. This test paper has a total of 50 questions. All Questions are compulsory.. Each questions are 4 marks 3. Read the Questions carefully and mark your appropriate response to the OMR sheet. 4. There is Negative marking of /4 for each wrong answer. 5. Mark the response by Black or Blue Ball Pen only. 6. Calculator and mobile phone is not allowed during exam

2 . The state of a quantum mechanical system is described by a wave function. Consider two physical observables that have discrete eigenvalues: observable A with eigenvalues {α}, and observable B with eigenvalues {β}. Under what circumstances can all wave functions be expanded in a set of basis states, each of which is a simultaneous eigenfunction of both A and B? (a) Only if the values {α} and {β} are nondegenerate (b) Only if A and B commute (c) Only if A commutes with the Hamitonian of the system (d) Only if B commutes with the Hamiltonian of the system. An ensemble of quantum harmonic oscillator is kept at a finite temperature T / kb Energy level of a Single oscillator is n, specific heat varies with temperature (T) at kt is (a) T (b) (c) (d) independent of T 3. Variational parameter of a particle of mass m in the potential V ( x) x 6m normalized trial wave function /4 / ( x) e x, is,value of n is (a) (b) / (c) /3 (d) 4, estimated using the 4. A measurement of energy E will always satisfy which of the following relationships? (a) E π²ħ²/8ma² (b) E π²ħ²/ma² (c) E = π²ħ²/8ma² (d) E = π²ħ²/ma² 5. A very simple square well potential is given x 0 V ( x) 0 0 x L x L Density of state of this quantum system depend on (a) L (b) (c) (d) 6. A system containing two identical particles is described by a wave function of the form ( x ) ( x) ( x ) ( x) Where x and x represent the spatial coordinates of the particles and α and β represent all the quantum numbers, including spin, of the states that they occupy. The particles might be (a) Electrons (b) Positrons (c) Protons (d) Deuterons

3 7. The figure above shows one of the possible energy eigenfunctions ψ(x) for a particle bouncing freely back and forth along the x-axis between impenetrable walls located at x = a and x = +a. The potential energy equals zero for x > a. If the energy of the particle is electron volts when it is in the quantum state associated with this eigenfunction, what is its energy when it is in the quantum state of lowest possible energy? (a) 0 ev (b) / ev (c). / ev (d) ev 8. The state in equation is linear combination of three orthonormal eigenstates 6 3 of the operator corresponding to eigenvalues -, and. What is the expectation value of for this state? (a) /3 (b) (7/6) (c) (d) 4/3 9. Which of the following functions could represent the radial wave function for an electron in an atom? (r is the distance of the electron from the nucleus; A and b are constants.) I. Ae br II. A sin (br) III. A/r (a) I only (b) II only (c) I and II only (d) I and III only 0. A particle of mass m is in a potential V m x, where is a constant. Let. = In the Heisenberg picture is given by (a) (b) i (c) - (d) - i. In one dimension, two metal A and B have Fermi temperature of free electron in ratio :4 ratio of number of electron per unit area is (a) :3 (b) : (c) :8 (d) :4. Hamiltonian of matrix 0 H 0 0 0

4 energy eigen value correct to I st order in perturbation is (a) 0,, (b) 0, (c) 0, 3, (d) 0, 3. If the Pressure of a free electron gas in - Dimension is increased by 8 times, its electron density change by (a) (b) 4 (c) 8 (d) 6 4. We have a planer sheet of Silicin gas (single layer of silicon, which treated as -D system).fill electron following dispersion relation, c is a constant, electron density of electron varies with T n, T is temp. of system, n is (a) 0 (b) (c) (d) ½ 5. Pressure of a non-relativistic free Fermi gas in one dimension depends at T 0, varies with particle density as (a) 53 n (b) (c) /3 n (d) 6. A system can have three energy levels: E = 0,. The level E=0 is doubly degenerate, while the others are non-degenerate. The specific heat at inverse temperature is (b). (d) None 7. A particle of mass m in three dimensions is in the potential n Ratio of ground states energy to first excited state energy is (a) (b) (c) (d) 8. Given that r the uncertainty p r in the ground state o(r) = of the hydrogen atom by uncertainty principle is( ) (b). (c) (d). 9. Consider black body radiation contained in a cavity whose walls are at temperature T. The radiation is in equilibrium with the walls of the cavity. If the temperature of the walls is increased to T and the radiation is allowed to come to equilibrium at the new temperature, the number of photon of the radiation increases by a factor of (a) (b) 4 (c) 8 (d) 6

5 0. The Coulomb potential V(r) = - of a hydrogen atom is perturbed by adding (where c is a constant) to the Hamiltonian. The first order correction to the ground state energy is in state (a) c (b) 4c (c) - 4c (d) none. The electron in newly discovered two-dimensional gas MoS with a linear energy-momentum relation. A is constant, If is the number of electrons per unit area, Energy density is proportional to (a) 3/ (b) (c) /3 (d). The Hamiltonian for a spin-/ particle at rest is given by, where and are Pauli spin matrices and A isconstants. The eigenvalues of this Hamiltonian are (a) (b) (c) A(doubly degenerate) (d) 3. At time t 0a particle in the potential ( x,0) A ( ) n ( x) n n V x m x ( ) / is described by the wave function Where n( x ) are eigenstates of the energy with eigenvalues En n. You are given that ( n, n' ) nn'. What is normalization constant A (a) (b) (c) m (d) none of these 4. Consider a linear harmonic oscillator and Let, 0 and be its real, normalized ground and first excited state energy eigenfunctions respectively. Let A0 B with A and B real numbers be the wave function of the oscillator at some instant of time. is maximum if (a) AB (b) AB (c) A B (d) A B 5. Consider electron in graphene obey dispersion relation of electron is proportional to wave vector (k) over the entire k-space, ground state energy per electron of system in term of Fermi energy is (a) E F (b) E F (c) E F (d) E F 6. The wave function for a particle confined to a one-dimensional box located between x = 0 and x = L is Ψ(x) =A sin (n π x/l) + B cos (n π x/l). The constants A and B are determined to be

6 (a),0 (b), (c) 0, (d), 7. Let represent the state of an electron with spin up and the state of an electron with spin down. Valid spin eigenfunctions for a triplet state I. II. 3 ( S ) of a two-electron atom include which of the following? (a) I only (b) II only (c) III only (d) I and III III. 8. A particle of mass m is acted on by a harmonic force with potential energy function V(x) = mω²x²/ (a one dimensional simple harmonic oscillator). If there is a wall at x = 0 so that V = for x < 0, then the energy levels are equals to (a) 0, ħω, ħω,... (b) 0, ¹ ₂ħω, ħω,... (c) ¹ ₂ħω, ³ ₂ħω, ⁵ ₂ħω,... (d) ³ ₂ħω, ⁷ ₂ħω, ¹¹ ₂ħω, The raising and lowering operators for the quantum harmonic oscillator satisfy a n ( n ) n a n n n for energy eigenstates n with energy E n. Which of the following gives the first-order shift in the n energy level due to the perturbation H V( a a ) where V is constant? (a) 0 (b) V (c) V (d) 5V 30. The operator a (defind bellow) when operating on a harmonic energy eigenstate ψ n with energy E n, produces another energy eigenstate whose energy is E n ħω 0. Which of the following is true? m 0 p a x i m 0 I. a commutes with the Hamiltonian. II. a is a Hermitian operator and therefore an observable. III. The adjoint operator a a (a) I only (b) II only (c) III only (d) I and II only

7 3. A Classical model of a diatomic molecule is a springy dumbbell, as shown above, where the dumbbell is free to rotate about axes perpendicular to the spring. In the limit of high temperature, what is the specific heat per mole at constant volume? (a) 3/ R (b) 5/ (c) 7/ R (d) 9/ R 3. A sample of N atoms of helium gas is confined in a.0 cubic meter volume. The probability that none of the helium atoms is in a 0 6 cubic meter volume of the container is (a) zero (b) 0 6 (c) ( 0 6 ) N (d) (0 6 ) N 33. For a system in which the number of particles is fixed, the reciprocal of the Kelvin temperature T is given by which of the following derivatives? (Let P= pressure, V = volume, S = entropy, and U = internal energy.) (a) ( P/ V) S (b) ( P/ S) V (c) ( V/ P) U (d) ( S/ U) V 34. A large isolated of N weakly interacting particles is in thermal equilibrium. Each particle has only 3 possible non degenerate states of energies 0, ε, and 3ε. When the system is at an absolute temperature T ε/k, where k is Boltzmann s constant, the average energy of each particle is (a) zero (b) ε (c) 4ε /3 (d) 3ε 35. In a gas of N diatomic molecules, two possible models for a classical description of a diatomic molecule are: Which of the following statements about this gas is true? (a) Model I has a specific heat c v = ³ ₂Nk (b) Model II has a smaller specific heat than Model I (c) Model II is always correct (d) The choice between Models I and II depends on the temperature 36. If the absolute temperature of a blackbody is increased by a factor of 3, the energy radiated per second per unit area does which of the following (a) Decreases by a factor of 8 (b) Decreases by a factor of 9 (c) Increases by a factor of 9 (d) Increases by a factor of The solution to the Schrödinger equation for a particle bound in a one-dimensional, infinitely deep potential well, indexed by quantum number n, indicates that in the middle of the well the probability density vanishes for (a) The ground state (n = ) only (b) States of even n (n =, 4,...) (c) States of odd n (n =, 3,...) (d) All states (n =,, 3,...)

38. At a given instant of time, a rigid rotator is in the state ψ(θ, ϕ) = (¾π) sin θ sin ϕ, where θ is the polar angle relative to the z-axis and ϕ is the azimuthal angle.

8 38. At a given instant of time, a rigid rotator is in the state ψ(θ, ϕ) = (¾π) sin θ sin ϕ, where θ is the polar angle relative to the z-axis and ϕ is the azimuthal angle. Measurement will find which of the following possible values of the z-component of the angular momentum L z? (a) 0 (b) ħ/, ħ/ (c) ħ, ħ (d) ħ, ħ 39. For an ideal diatomic gas in thermal equilibrium, the ratio of the molar heat capacity at constant volume at very high temperatures to that at very low temperatures is equal to (a) 3 (b) 5/3 (c) (d) 7/3 40. A diatomic molecules is initially in the state Ψ(Θ, Φ) = (5Y + 3Y 5 + Y 5 ) / (38) ½ where Y l m is a spherical harmonics. If measurements are made of the total angular momentum quantum number l and of azimuthal angular momentum quantum number m, what is the probability of obtaining the results l = 5? (a) 36/444 (b) 9/38 (c)3/38 (d) 5/(38) ½ 4. Suppose an e is in a state describe by the wave function Where g( r) r dr and polar, and azimuthal angles. 0 What is probability when m 0 (a) (b) 3 ( i e sin cos ) g ( r ) 4 (c) 3 4. A particle of mass is confined to a one dimensional region as (d) none At its normalized wave function is 8 x x ( x,0) cos sin 5a a a What is probability corresponding to (a) 0.8 (b) 0.6 (c) 0.64 (d) A quantum-mechanical state of a particle with cartesian coordinate x,y,z is described by the normalised / wave fucntion ( x, y, z) Az exp ( x y z ) Expectation value of L is (a) 0 (b) (c) (d) 6

9 3 44. We have a multiwall carbon nanotube, which follows the dispersion relation ( k) k over the enitre k- space fermi energy ( F ) depends on e density of electron (a) /3 /3 3 (b) (c) (d) 45. Suppose ( x,0) x A a For a and A are constant. What is x (a) a (b) a (c) a (d) a 46. Suppose you are given the following relation among the entropy S, volume V, internal energy E, number of particle N of a thermodynamics system 3 S A( NVU ) Relation between pressure (p) to temperature (T) is (a) p T (b) p /3 T (c) p 3/ T (d) p T 47. For a quantum mechanical harmonic oscillators with energy En n n 0,,,... Chemical potential of system is (a) sinh (c) KT n sinh (b) (d) NKT n sinh NK cot n sinh 48. If a particle is represented by unnormalised wave function C ( ) 5 a x x a ( x) a 0 x a Ground state energy of system is (a) (b) (c) (d)

10 49. An is confined to the interior of a hollow spherical cavity of radius with impenetrable walls expression for pressure exerted on walls of cavity by in its ground state is varies with radius as (a) (b) (c) (d) In case of spherical,pressure is independent of radius 50. The wave function ψ(x) = A exp ( b x /), where A and b are real constants, is a normalized eigenfunction of the Schrodinger equation for a particle of mass M and energy E in a one dimensional potential V(x) such that V(x) = 0 at x = 0. Which of the following is correct? (a) V = ħ b 4 /M (b) V = ħ b 4 x /M (c) V = ħ b 6 x 4 /M (d) E = ħ b ( b x ) Good luck ANSWERKEY. (b). (d) 3. (c) 4. (b) 5. (a) 6. (d) 7. (c) 8. (c) 9. (a) 0. (b). (d). (b) 3. (a) 4. (d) 5. (b) 6. (c) 7. (b) 8. (d) 9. (c) 0. (b). (a). (a) 3. (b) 4. (c) 5. (c) 6. () 7. (d) 8. (d) 9. (d) 30. (c) 3. (c) 3. (c) 33. (d) 34. (c) 35. (d) 36. (d) 37. (b) 38. (c) 39. (d) 40. (c) 4. (c) 4. (d) 43. (b) 44. (b) 45. (a) 46. (c) 47. (c) 48. (a) 49. (c) 50. (b)

SCIENCE VISION INSTITUTE For CSIR NET/JRF, GATE, JEST, TIFR & IIT-JAM Web:

SCIENCE VISION INSTITUTE For CSIR NET/JRF, GATE, JEST, TIFR & IIT-JAM Web: Test Series: CSIR NET/JRF Exam Physical Sciences Test Paper: Quantum Mechanics-I Instructions: 1. Attempt all Questions. Max Marks: 185 2. There is a negative marking of 1/4 for each wrong answer. 3. Each