1. Two identical bosons of mass m are placed in a one-dimensional potential . The bosons interact via a weak potential, where x₁ and x₂ denote coordinates of the particles. Given that the ground state wavefunction of the harmonic oscillator is . The ground state energy of the two-boson system, to the first order in V₀, is:

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The Hamiltonian of the system of two identical Bosons

= H₀ + H_p

The unperturbed energy of system

The unperturbed ground state energy

The first order correction in energy of ground state

The value of the integral

On substituting the value of the integral from equation (3) into equation (2), we get

Consider:

Then ground state energy corrected to first order

2. Ten grams of ice at 0ºC is added to a beaker containing 30 grams of water at 25ºC. What is the final temperature of the system when it comes to thermal equilibrium? (The specific heat of water is 1 cal/gm/ºC and latent heat of melting of ice is 80 cal/gm)

(a) 0ºC

(b) 7.5ºC

(c) 12.5ºC

(d) –1.25ºC

Ans. (a)

Sol. The heat flows spontaneously from hot body to cold body. It means heat will be flowed from water to ice.

If water temperature changes from 25ºC to 0ºC, then heat given by the water = 30 × 1 × 25 = 750 cal.

To convert ice into water at 0ºC completely the heat required = mL = 10 × 80 = 800 Cal Heat given < heat required

So ice is not converted into water completely so the final temperature is 0ºC.

3. A vessel has two compartments of volume V₁ and V₂, containing an ideal gas at pressures p₁ and p₂, and temperature T₁ and T₂ respectively. If the wall separating the compartments is removed, the resulting equilibrium temperature will be

(a)

(b)

(c)

(d) (T₁T₂)^1/2

Ans. (c)

Sol.

If T₂ > T₁, on removing the wall the heat will flow from ideal gas at T₂ towards ideal gas at T₁

Heat given = Heat taken (Calorimetry principle).

Let equilibrium temperature be T_f then

Cp₂ (T₂ – T_f) = Cp₁ (T_f – T₁), n₂R (T₂ – T_f) = n₁R (T_f – T₁)

where C_p = heat capacity at constant pressure

4. For temperature T₁ > T₂, the qualitative temperature dependence of the probability distribution F(v) of the speed v of a molecule in three dimensions is correctly represented by the following figure:

(a)

(b)

(c)

(d)

Ans. (a)

Sol. For N-particles system in thermal equilibrium, the number of molecules having speed between v to v + dv is given by

If we increase temperature, then and v_rms all will increase and particles have more values of velocities. More particles will distribute in large velocity region but total number of particles in the system will be constant (N). In F(v)v_sv plot area under the curve remain constant. So an increasing temperature peak will decrease and shifted towards higher velocity side.

5. A system of non-interacting spin-1/2 charged particles are placed in an external magnetic field. At low temperature T, the leading behavior of the excess energy above the ground state energy, depends on T as: (c is a constant)

(a) cT

(b) cT³

(c) e^–c/T

(d) c(is independent of T)

Ans. (c)

Sol.

6. The acceleration due to gravity g is determined by measuring the time period T and the length L of a simple pendulum. If the uncertainties in the measurements of T and L are T and L respectively, the fractional error g/g in measuring g is best approximated by

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

relative error in g is given by

7. A silicon transistor with built-in voltage 0.7V is used in the circuit shown, with V_BB = 9.7V, R_B = 300kΩ, V_CC = 12 V and R_C = 2k . Which of the following figures correctly represents the load line and the quiescent Q point?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For DC load line

Calculation of I_C

8. If the analog input to an 8-bit successive approximation ADC is increased from 1.0 V to 2.0 V, then the conversion time will

(a) remain unchanged

(b) double

(c) decrease to half its original value

(d) increase four times

Ans. (a)

Sol. Successive approximation converter are also called as uniform converter.

Hence remain unchanged.

9. The insulation resistance R of an insulated cable is measured by connecting it in parallel with a capacitor C, a voltmeter, and battery B as shown. the voltage across the cable dropped from 150V to 15 V, 1000 seconds after the switch S is closed. If the capacitance of the cable is 5 µF then its insulation resistance is approximately

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

10. The approximation is valid up to 3 decimal places as long as is less than: (take 180º/ 57.29°).

(a) 1.28º

(b) 1.81º

(c) 3.28º

(d) 4.01º

Ans. (b)

Sol.

11. The area of a disc in its rest frame S is equal to 1 (in some units). The disc will appear distorted to an observer O moving with a speed u with respect to S along the plane of the disc. The area of the disc measured in the rest frame of the observer O is (c is the speed of light in vacuum)

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

12. A particle of charge e and mass m is located at the midpoint of the line joining two mixed collinear dipoles with unit charges as shown in the figure. (The particle is constrained to move only along the line joining the dipoles). Assuming that the length of the dipoles is much shorter than their separation, the natural frequency of oscillation of the particle is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The dipole moment of each dipole is p = e2d.

The charge is displaced by a small distance x towards right. Then force on charge e is given by

But x << R hence

By equation (1) and (2),

Frequency of oscillation

13. A current I is created by a narrow beam of protons moving in vaccum with constant velocity . The direction and magnitude, respectively, of the Poynting vectors outside the beam at a radial distance r (much larger than the width of the beam) from the axis, are

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Electric field at distance r is

Equ. (1) put in equ. (3)

We know that

Equ. (2), (4) put in equ. (5)

14. If the electric and magnetic fields are unchanged when the vector potential changes (in suitable units) according to , where , then the scalar potential must simultaneously change to

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

If electric and magnetic field needs to be unchanged.

where is gauge function

From eq (1) and (2)

Equ (4) put in eq (3)

15. Consider an axially symmetric static charge distribution of the form, . The radial component of the dipole moment due to this charge distribution is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

For dipole moment

16. In a basis in which the z-component S_z of the spin is diagonal, an electron is in a spin state .

The probabilities that a measurement of S_z will yield the values and are, respectively.

(a) 1/2 and ½

(b) 2/3 and 1/3

(c) 1/4 and ¾

(d) 1/3 and 2/3

Ans. (d)

Sol. The eigenfunctions of S_Z in a basis in which S_Z is diagonal are

The given state can be written as linear superposition of and as

The probability of yielding the value

and the probability of yielding the value

17. Consider the normalized state of a particle in a one-dimensional harmonic oscillator:

where and denote the ground and first excited states respectively, and b₁ and b₂ are real constants. The expectation value of the displacement x in the state will be a minimum when

(a) b₂ = 0, (b₁ = 1)

(b)

(c)

(d) b₂ = b₁

Ans. (a)

Sol. Given : Normalized state of a particle in a one-dimensional harmonic oscillator is

Normalization condition :

Expectation value of the displacement x in the state

Since, x is an odd function, therefore first two terms of the above expression will be zero.

Now P will be minimum if b₁b₂ is minimum.

So, b₁b₂ will be minimum if b₂ = 0 or b₁ = 0

So, correct option should be (a)

In the answer key correct option is given to be (d) i.e. b₁ = b₂

If b₁ = b₂ b₁b₂ will be maximum and its maximum value will be then, will be maximum and its maximum value will be

18. A muon (µ^–) from cosmic rays is trapped by a proton to form a hydrogen-like atom. Given that a muon is approximately 200 times heavier than an electron, the longest wavelength of the spectral line (in the analogue of the Lyman series) of such an atom will be

(a) 5.62 Å

(b) 6.67 Å

(c) 3.75 Å

(d) 13.3 Å

Ans. (b)

Sol. Reduced mass of muonic-hydrogen is given by

Longest wavelength of Lyman series corresponds to transition

19. The un-normalized wavefunction of a particle in a spherically symmetric potential is given by where f(r) is a function of the radial variable r. The eigenvalue of the operator (namely the square of the orbital angular momentum) is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The un-normalized wavefunction

In spherical polar co-ordinate system

has only power of so the orbital quantum number

The eigenvalue of

20. Given that the value of H₄(0) is

(a) 12

(b) 6

(c) 24

(d) –6

Ans. (a)

Sol. On putting x = 0, we get

On putting n = 4 and comparing the coefficients of t⁴, we get

H₄(0) = 12

21. A unit vector on the xy-plane is at an angle of 120º with respect to . The angle between the vectors and will be 60º if

(a)

(b)

(c) b = a/2

(d) b = a

Ans. (c)

Sol. Since, the unit vector makes an angle 120º with x-axis, then it makes an angle 30º with y-axis.

Therefore,

So, and can be written as, and

Since, the angle between and is 60º, therefore

22. With z = x + iy, which of the following functions f(x, y) is NOT a (complex) analytic function of z?

(a) f(x, y) = (x + iy – 8)³(4 + x² – y² + 2ixy)⁷

(b) f(x, y) = (x + iy)⁷ (1 – x – iy)³

(c) f(x, y) = (x² – y² + 2ixy – 3)⁵

(d) f(x, y) = (1 – x + iy)⁴ (2 + x + iy)⁶

Ans. (d)

Sol. Any complex function f(x,y) will not be analytic in nature if it contains term.

Given options can be written as

(a) f(x, y) = (z – 8)³ (4 + z²)⁷ (b) f(x, y) = z⁷ (1 – z)³

(c) f(x, y) = (z² – 3)⁵ (d)

Only option (d) contains term. Therefore, option (d) will not be analytic in nature.

23. A planet of mass m and an angular momentum L moves in a circular orbit in a potential, V(r) = –k/r, where k is a constant. If it is slightly perturbed radially, the angular frequency of radial oscillations is

(a)

(b) mk²/L³

(c)

(d)

Ans. (b)

Sol. Total energy of planet is written as

E = T + V(r)

Effective potential is

gives radius of stationary orbit.

Force constant of oscillation

Therefore, frequency of oscillation,

24. The Lagrangian of a particle of mass m moving in one dimensions is given by where b is a positive constant. The coordinate of the particle x(t) at time t is given by: (in the following c₁ and c₂ are constants)

(a)

(b) c₁t + c₂

(c)

(d)

Ans. (a)

Sol.

This is same as uniform gravitational potential. Therefore, solution must be of the form.

25. A uniform cylinder of radius r and length , and a uniform sphere of radius R are released on an inclined plane when their centres of mass are at the same height. If they roll down without slipping, and if the sphere reaches the bottom of the plane with a speed V, then the speed of the cylinder when it reaches the bottom is:

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Acceleration of a rolling object ... (1)

For solid cylinder

I = mr²/2 ... (2)

Eq. (2) put in Eq. (1)

For solid sphere

Eq. (3) put in Eq. (1)

According to Eq. of motion

given V_s = V;

26. The components of a vector potential A_µ (A₀, A₁, A₂, A₃) are given by A_µ = k (–xyz, yzt, zxt, xyt) where k is a constant. The three components of the electric field are

(a) k(yz, zx, xy)

(b) k(x, y, z)

(c) (0, 0, 0)

(d) k(xt, yt, zt)

Ans. (c)

Sol. Vector potential in four vector representation is

A_µ = k (–xyz, yzt, zxt, xyt)

It is equivalent to

Electric field is given by

27. In the Born approximation, the scattering amplitude for the Yukawa potential is given by: (in the following b = 2k)

(a)

(b)

(c)

(d)

Ans. (b)

Sol. In the Born approximation, the scattering amplitude

28. If denotes the eigenfunction of the Hamiltonian with a potential V = V(r) then the expectation value of the operator in the state is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

The expectation value of the operator

On substituting the values of and in equation (i), we get

29. An oscillating current I(t) = I₀ exp(–it) flows in the direction of the y-axis through a thin metal sheet of area 1.0 cm² kept in the xy-plane. The rate of total energy radiated per unit area from the surfaces of the metal sheet at a distance of 100m is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Current per unit length on sheet is

Current per unit area of the sheet

Dipole moment per unit length per unit time

Power radiated per unit area of the sheet is given by

30. Consider a two-dimensional infinite square will . Its normalized eigenfunctions are where n_x, n_y = 1, 2, 3, ....... If a perturbation is applied, then the correction to the energy of the first excited state to order V₀ is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The energy of n^th state for 2-dimensional infinite square well

The energy of first excited state

and state is 2 fold degenerate with (n_x, n_y) = (1, 2) & (2, 1)

To find first order corection in every of first excited state we use degenerate perturbation theory.

We construct secular determinant

on substituting these values of matrix elements in secular determinant, we get

31. Consider a system of two Ising spins S₁ and S₂ taking value +1 with interaction energy given by = –JS₁S₂, when it is in thermal equilibrium at temperature T. For large T, the average energy of the system varies as C/k_BT, with C given by

(a) –2J²

(b) –J²

(c) J²

(d) 4J

Ans. (b)

Sol. The interaction energy given by

where S₁ and S₂ taking values +1.

The possible value of the energy of the system are

E₁ = –J (+1) (+1) = –J E₂ = –J (–1) = –J

E₃ = –J (+1) (–1) = +J E₄ = –J (–1) (+1) = +J

When the system is in thermal equilibrium then the average energy of the system is given by

32. Consider three particles A, B and C, each with an attribute S that can take two value +1. Let S_A = 1, S_B = 1 and S_C = –1 at a given instant. In the next instant, each S value can change to –S with probability 1/3. The probability that S_A + S_B + S_C remains unchanged is

(a) 2/3

(b) 1/3

(c) 2/9

(d) 4/9

Ans. (d)

Sol. At a given instant, S_A = 1, S_B = 1, S_C = –1 S_A + S_B + S_C = 1

At the next instant, S_A + S_B + S_C will remain unchanged for the following cases:

Required probability,

33. The bound on the ground state energy of the Hamiltonian with an attractive delta-function potential, namely

using the variational principle with the trial wavefunction is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The Hamiltonian,

The trial wave function with b variational parameter.

34. Consider two different systems each with three identical non-interacting particles. Both have single particle states with energies and . One system is populated by spin-½ fermions and the other by bosons. What is the value of E_F – E_B where E_F and E_B are the ground state energies of the fermionic and bosonic systems respectively ?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

According to Pauli's exclusion principle

35. The input to a lock-in amplifier has the form where V_i, are the amplitude, frequency and phase of the input signal respectively. This signal is multiplied by a reference signal of the same frequency , amplitude V_r and phase . If the multiplied signal is fed to a low pass filter of cut-off frequency , then the final output signal is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. After multiplication

For low pass filter cut-off frequency , 2^nd term can be neglected.

36. The solution of the partial differential equation

satisfying the boundary conditions u(0, t) = 0 = u(L, t) and initial conditions u(x, 0) = sin() and is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The last initial condition is satisfied by only options (c) and (d).

Out of option (c) and (d), only option (d) satisfies the given partial differential equation.

37. Consider the hydrogen deuterium molecule HD. If the mean distance between the two atoms is 0.08 nm and the mass of the hydrogen atom is 938 MeV/c², then the energy difference between the two lowest rotational states is approximately

(a) 10^–1eV

(b) 10^–2eV

(c) 2 × 10^–2 eV

(d) 10^–3 eV

Ans. (b)

Sol. r_e = 0.08 nm

The energy separation between two rotational states is given by

38. Four digital outputs V, P, T and H monitor the speed v, tyre pressure p, temperature t and relative humidity h of a car. These outputs switch from 0 to 1 when the values of the parameters exceed 85 km/hr, 2 bar, 40ºC and 50%, respectively. A logic circuit that is used to switch ON a lamp at the output E is shown below.

Which of the following conditions will switch the lamp ON?

(a) v < 85km / hr, p < 2bar, t > 40ºC, h > 50%

(b) v < 85km / hr, p < 2bar, t > 40ºC, h < 50%

(c) v > 85km / hr, p < 2bar, t > 40ºC, h < 50%

(d) v > 85km / hr, p < 2br, t < 40ºC, h > 50%

Ans. (a)

Sol.

For the lamp to become on either V-p = 1 or T.H = 1 or both = 1

hence, only option 1 have t > 40º and H > 50%

hence, 0 lamp is on only for option (a).

39. The solution of the differential equation with the initial condition x(0) = 1 will blow up as t tends to

(a) 1

(b) 2

(c) 1/2

(d)

Ans. (a)

Sol.

Therefore, the solution of the given differential equation will blow up as t tends to 1.

40. Let u be a random variable uniformly distributed in the interval [0, 1] and V = –c ln u, where c is a real constant. If V is to be exponentially distributed in the interval [0, ) with unit standard deviation, then the value of c should be

(a) ln 2

(b) ½

(c) 1

(d) –1

Ans. (c)

Sol. Let u be a random variable uniformly distributed in the interval [0, 1] i.e. the distribution function will be given as

f(u) = u for [0, 1]

= 0 otherwise

V is exponentially distributed in the interval (0, ) with unit standard deviation i.e. the distribution function will be given as

Here, standard deviation of an exponential distribution is

To convert uniform distribution into exponential distribution, we have to assume

Therefore, the value of c will be 1.

41. The inverse Laplace transform of is

(a)

(b)

(c) t – 1 + e^–t

(d)

Ans. (c)

Sol.

Alternative way, Find the laplace transform of the all options and it will be seen that laplace transform of option (c) will be the one which is given in the question.

42. The number of degrees of freedom of a rigid body in d space-dimensions is

(a) 2d

(b) 6

(c) d(d + 1)/2

(d) d!

Ans. (c)

Sol. , d = dimension of space

In 1 – d, d = 1, DOF = 1

In 2 – d, d = 2, DOF = 3

In 3 – d, d = 3, DOF = 6

43. A particle of mass m is at the stable equilibrium position of its potential energy V(x) = ax – bx³ where a, b are positive constants. The minimum velocity that has to be imparted to the particle to render its motion unstable is

(a) (64a³/9m²b)^1/4

(b) (64a³/27m²b)^1/4

(c) (16a³/27m²b)^1/4

(d) (3a³/64m²b)^1/4

Ans. (b)

Sol. V(x) = ax – bx³

for stable position

potential energy at stable position

To make motion unstable particle should at least reach unstable point.

Therefore, loss in K.E. = gain in

This gives,

44. If the operators A and B satisfy the commutation relation [A, B] = I, where I is the identity operator, then

(a) [e^A, B] = e^A

(b) [e^A, B] = [e^B, A]

(c) [e^A, B] = [e^–B, A]

(d) [e^A, B] = 1

Ans. (a)

Sol.

[1, B] = B – B = 0

[A², B] = 2[A, B]A = 2A

[A³, B] = 3A²[A, B] = 3A²

45. A system is governed by the Hamiltonian where a and b are constants and p_x, p_y are momenta conjugate to x and y respectively. For what values of a and b will the quantities (p_x – 3y) and (p_y + 2x) be conserved?

(a) a = –3, b = 2

(b) a = 3, b = –2

(c) a = 2, b = –3

(d) a = –2, b = 3

Ans. (d)

Sol. P_x – 3y and P_y + 2x are constants of motion.

Similarly using {p_y + 2x, H} = 0

We get value of a.

46. Using the frequency-dependent Drude formula, what is the effective kinetic inductance of a metallic wire that is to be used as a transmission line? [In the following, the electron mass is m, density of electrons is n, and the length and cross-sectional area of the wire are and A respectively]

(a)

(b) zero

(c)

(d)

Ans. (c)

Sol. According to energy conservation

Where N total number of electrons, v_d drift velocity of electron, L inductance

47. The phonon dispersion for the following one-dimensional diatomic lattice with masses M₁ and M₂ (as shown in the figure)

is given by

where a is the lattice parameter and K is spring constant. The velocity of sound is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Take –sign for acoustical branch

48. The binding energy of a light nucleus (Z, A) in MeV is given by the approximate formula

where N = A – Z is the neutron number. The value of Z of the most stable isobar for a given A is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

For most stable isobar

(Note: In nuclear physics, the sign of the last term should be negative in that case the answer will be )

49. Muons are produced through the annihilation of particle a and its antiparticle, namely the process . A muon has a rest mass of 105 MeV/c² and its proper life time is 2µs. If the center of mass energy of the collision is 2.1 GeV in the laboratory frame that coincides with the center-of-mass frame, then the fraction of muons that will decay before they reach a detector placed 6km away from the interaction point is

(a) e^–1

(b) 1 – e^–1

(c) 1 – e^–2

(d) e^–10

Ans. (b)

Sol.

(Note: If the correct proper life time = 2.2 µs is given, then the answer will be exact)

50. The conductors in a 0.75 km long two-wire transmission line are separated by a centre-to-centre distance of 0.2m. If each conductor has a diameter of 4cm, then the capacitance of the line is

(a) 8.85 µF

(b) 88.5 nF

(c) 8.85 pF

(d) 8.85 nF

Ans. (b)

Sol. Capacitance per unit length of two wire line is

Total capacitance C = C' = 0.75 × 10³ Farad

C = 80.11 × 10^–9 Farad = C = 80.11 nF

51. The electron dispersion relation for a one-dimensional metal is given by where k is the momentum, a is the lattice constant, is a constant having dimensions of energy and . If the average number of electrons per atom in the conduction band is 1/3, then the Fermi energy is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Correct option is (a)

52. The electronic energy levels in a hydrogen atom are given by E_n = –13.6/n² eV. If a selective excitation to the n = 100 level is to be made using a laser, the maximum allowed frequency line width of the laser is

(a) 6.5 MHz

(b) 6.5 GHz

(c) 6.5 Hz

(d) 6.5 kHz

Ans. (b)

Sol.

Frequency line-width

53. If the energy dispersion of a two-dimensional electron system is where u is the velocity and k is the momentum, then the density of states D(E) depends on the energy as

(a)

(b)

(c) E

(d) constant

Ans. (c)

Sol. If , where u is the velocity and k is momentum, k is the momentum,

The volume of phase space in 2-dimension

= position space volume × momentum space volume

[for sharp value of momentum]

The number of microstates

(for sharp value of k)

(for sharp value of energy)

If energy lies between E to E + dE, then number of microstates,

The density of states = Number of microstates or energy states per unit energy interval.

54. Consider the laser resonator cavity shown in the figure.

If I₁ is the intensity of the radiation at mirror M₁ and is the gain coefficient of the medium between the mirrors, then the energy density of photons in the plane P at a distance x from M₁ is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Energy density of the standing wave between the mirrors is given by

55. A spin-1/2 particle A undergoes the decay A B + C + D where it is known that B and C are also spin-1/2 particles. The complete set of allowed values of the spin of the particle D is

(a)

(b) 0, 1

(c)

(d)

Ans. (c)

Sol. A B + C + D

It is like the decay of a neutron

, where spin of each particle = 1/2.

Spin of the left side and the combined spin of the products must be same to conserve the spin angular momentum conservation law.

So from b, C, D, we can form either two doublets or a quarlet spin state