physical science test sereis # 3 · ugc point academy leading institue for csir-jrf/net, gate &...

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Instructions

1. This test paper has a total of 50 questions carrying 100 marks. All sections

are compulsory. Question in each section are different type.

2. Read the Questions carefully and mark your appropriate response to the

OMR sheet.

3. There is Negative marking of 1/4th for each wrong answer.

4. Mark the response by Black or Blue Ball Pen only.

5. Any other belongings like Book/ Notes / Electronic device etc are not

permitted in the examination hall.

6. Submit your answer sheet (OMR Sheet) to the invigilator before leaving the

examination hall.

UGC POINT ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM

BOOKLET CODE SUBJECT CODE

PHYSICAL SCIENCE

Quantum Mechanics + Electronics

Date: 29/11/2015 Timing: 2:00 H Maximum Marks: 100

TEST SEREIS # 3

05 A

mailto:[email protected]



1. Two identical Bosons are placed in an infinite square well. They interact weakly with one another

via the potential

1 2 0 1 2,V x x aV x x

(Where 0V is a constant with the dimensions of energy and a is the width of the well)

Use first order perturbation theory to estimate the effect of the particle-particle interaction on the

energy of the first excited state.

(1) 03V (2) 02V (3) 03V (4) 020V

2. Use the WKB approximation to find the allowed energies nE of an infinite square well with a

“shelf” of height V0 extending half-way across

0 0 / 2

0 / 2

V if x a

V x if a x a

if otherwise

Express your answer in terms of 0V and 20 2/ 2nE n ma

(1) 2

0 00 016

n

n

VE E V

E (2)

20 0

0 0/ 2

16n

n

VE E V

E

(3) 2

0 00 0

/ 216

n

n

VE E V

E (4)

20 0

0 016n

n

VE E V

E

3. Use a gaussion trail function 2bxAe to obtain the lowest upper bound you can on the ground state

energy of the potential: 4V x x

(1)

1/34

2

3 3

4 4m

(2)

1/34

2

3 3

2 4m

(3) 3

2

1 3

2 4m

(4)

1/33

2

3 3

2 4m

4. Consider a quantum system with just three linearly independent states. Suppose the Hamiltonian, in

matrix form is

0

1 0 0

0 1

0 2

H V

Where 0V is constant and is some small numbers 1 use to degenerate perturbation theory, find

the first order correction energy to the two initially degenerate Eigen values

(1) 0 0 0,V V V (2) 0 0 0, 2V V V

(3) 0 0 0, VV V (4) 0 0 0, 2V V V

5. Consider the isotopic three dimensional Harmonic oscillator for which the potential is

2 21

2V r m r

Find the first order corrected energy on the first excited state, if perturbed potential is 2H' x yz

(1) ,02m

(2) 2

,02m

(3) 2

,03m

(4) None of these




6. A free particle described by a plane wave and moving in the positive z-direction undergoes

scattering by a potential

0 ;

0 ;

V if r RV r

if r R

If 0V is changed to 02V , keeping R fixed then the differential cross-section in the Born approximation

(1) Increases to four times the original value (2) Increases to twice the original value

(3) Decreases to Half the original value (4) Decreases to one fourth the original value

7. An electron propagating along the x-axis passes through a slit of width 1y nm . The uncertainty in

the y-component of its velocity after passing through the slit is

(1) 57.322 10 / secm (2) 51.166 10 / secm

(3) 53.436 10 / secm (4) 42.326 10 / secm

8. What is the ground state energy of six non-interacting electrons in a 2-dimensional isotropic, simple

harmonic oscillator characterized by ?

(1) 3 (2) 5 (3) 10 (4) 12

9. A particle is scattered by a spherically symmetric potential. In the centre of mass (CM) frame the

wave function of the incoming particle is ikzAe , where K is the wave vector and A is a constant.

Then the differential scattering cross-section in centre of mass frame is

(1)

2

2

2

fA

r

(2)

22A f

(3) 2

f (4) A f

10. The time-dependent wave function of a particle of mass m moving in a one dimensional under the

influence of a potential V x is given to be

/ 0

,0 0

x i txe e for xx t

for x

Where , and are real numbers. For 0x , the potential V x is of the form ( 1K and 2K are

constants)

(1) 21

KK

x (2) 2

1 2

KK

x (3) 1 2K K x (4) 2

1 2K K x

11. The wave function of a quantum mechanical system described by a Hamiltonian H can be written

as a linear coordination of 1 and 2 which are the eigen functions of H with eigen value 1E and 2E

respectively. At 0t , the system is prepared in the state 0 1 2

4 3

5 5 and then allowed to evolve

with time. The wave function at time 1 22

ht

E E

will be (accurate to within a phase)

(1) 1 2

4 3

5 5 (2) 1 2

4 3

5 5 (3) 1 (4) 1 2

3 4

5 5

12. An atom is capable of existing in two states a ground state of mass M and an excited state of mass

M . If the transition from the ground state proceeds by the absorption of a photon, the photon

frequency in the laboratory frame (where the atom is initially at rest) is

(1) 2c

(2) 2

12

c

m

(3)

2Mc (4)

2c

h




13. A particle of mass m and energy E, moving in the positive x-direction encounter a one dimensional

potential barrier at 0x , the barrier is defined by

0 0 0

0 0

0

V for x

V V for x V is positiveand E V

If the function of particle in the region 0x is given as ikx ikxAe Be , if / 0.4B A , then the

transmission coefficient is

(1) 1.32 (2) 0.84 (3) 1.82 (4) 0.52

14. Let the Hamiltonian for two spin 1/2 particles of equal mass ,m momenta 1P and 2P position 1r and 2r

be 2 2

2 2 21 21 2 1 2

1.

2 2 2

P PH m r r K

m m , where 1 and 2 denote the corresponding Pauli matrices,

0.1eV and 0.2K eV . If the ground state has net spin zero, then the energy (in eV) is

(1) -0.3 eV (2) 0.82 eV (3) 1.52 eV (4) 2 eV

15. An operator for a spin 1/2 particle is given by .A B , where ˆ ˆ/ 2B B x y , denotes Pauli

matrices and is a constant. The eigen values of A are

(1) / 2B (2) B (3) 0, B (4) 0, B

16. Evaluate the commutator ˆ ˆˆ ˆ, ,A B C D

(1) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆCBDA BCDA ABCD ACBD (2) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆCBDA BCDA ABCD ACBD

(3) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆCBDA BCDA ABCD ACBD (4) None of these

17. Consider a state 1 2 3

1 1 1

2 5 10 , which is given in terms of orthonormal Eigen states

1 2, and 3 of an operator B such that 2ˆn nB n . Find the expectation value of B for the

state

(1) 13/4 (2) 5/4 (3) 7/4 (4) 11/4

18. Find the value of 3

0

cos 3 / 2 d

(1) 25 (2) 22 (3) 27 (4) 20

19. The momentum operators ˆHP t in the Heisenberg picture for one dimensional Harmonic oscillator is

(1) ˆ ˆcos sinp t m x t (2) ˆ ˆcos sinp t m x t

(3) ˆ cosp t (4) ˆ ˆsin cosp t m x t

20. A hydrogen atom is placed in a time dependent electric field pointing in the z-direction, with

magnitude

0

0 0

0t

tE t

E e t

What is the probability that at t the hydrogen atom, if initially in the ground state, makes a

transition to the 2P state?

(1)

2 2 2150 0

10 2 2 2

2

3

e E a

h (2)

2 2 2150 0

10 2 2 2

2

3

e E a

h

(3)

2 2 2150 0

10 2 3 3

2

3

e E a

h (4) None of these




21. Which of the following match is incorrect?

(1) Goudsmit and Unlenbeck → Electron spin

(2) Pauli spin principle → Anti symmetric state

(3) Stern-Gerlach experiment → Space quantization

(4) Davission –Germent Experiment → Particle as a wave nature

22. A step potential is defined as

0 0

16 0

V x

V eV x

A electron of total energy 25E eV coming from interact with potential. If A, B, C are amplitude

of incident wave reflected wave and transmitted wave then which one is true.

(1) A B C (2) A C B

(3) C A B (4) C B A

23. Parity operator is defined as r r where ˆˆ ˆr xi yj zk and I is identity operator defined as

I r r . The operators P and P are given as

11

2

11

2

P

P

Consider the following two statements:

0

&

I P p P P Null matrix

II P P P P

Of these

(1) Only I is correct (2) Only II is correct

(3) Both I and II are correct (4) None of them is correct

24. The kinetic energy operator given by 2

2

xPK

m and potential energy operator 2 21

2V m x where xp is

momentum operator and x is position operator in x direction respectively, then value of commutation

[K,V] is given by

(1) 2xi XP (2) 2, xK V i XP

(3)

,2

x xi XP P XK V

(4)

2

,2

x xi XP P XK V

25. If A is an operator and A is expectation value of A and if H(t) is Hamiltonian operator, then

,A H t is equal to

(1) d

i A Adt t

(2) d

i A Adt t

(3) d A

i Adt t

(4) None of these




26. Find the Eigen value and Eigen vector of operators x y z , where ,x y and z are ,x y and z

component of Pauli-spin matrix

(1) Eigen value 1, -1 and Eigen vector 1 0

0 1

(2) Eigen value 1, 1 and Eigen vector 1 0

0 1

(3) Eigen value i, -i and Eigen vector 1 0

0 1

(4) Eigen value i, i and Eigen vector 1 0

0 1

27. The spin part of the wave function of an electron is 1/2 1/2

13

2

where 1/2 is up spin

vector and 1/2 is down spin vector then what is the expectation value of xS in the direction at

(1) 0 (2) 3

2 (3)

3

2 (4)

3

4

28. The number of nodes in the plot of probability of finding the electron as a function of distance from

the nucleus of hydrogen atom in state 0/2

3300

1, cos

32

r arr e

aa

is

(1) 3 (2) 2 (3) 1 (4) 0

29. If an one dimensional harmonic oscillator of charge q is placed in an uniform electric field of

strength E, then

(1) All the energy levels are shifted by a constant amount proportional to 2E

(2) Only the ground state level is shifted

(3) All the energy levels are shifted differently as a function of E

(4) All the energy level are shifted by a constant amount proportional to E

30. For the attractive Delta function potential given as V x x . If E is total energy of the particle

of mass m then

(1) For E < 0 wave function is exponentially decaying and there is no bound state

(2) For E < 0 wave function is oscillatory and there is only one bound state

(3) For E < 0 wave function is exponentially decaying and there is only one bound state

(4) For E < 0 wave function is oscillatory and there is infinite number of bound state same as particle

in one dimensional rigid box

31. The simultaneous equation on the Boolean variables , ,x y z and w

1

0

1

0

x y z

xy

xz w

xy z w

(1) 0100 (2) 1101 (3) 1011 (4) 1000




32. Consider the following logic circuit whose inputs are functions 1 2 3, ,f f f and output is f , given that

1

2

3

, , 0,1,3,5

, , 6,5

, , 1,4,5

f x y z

f x y z

f x y z

f is

(1) 1,4,5 (2) 6,7 (3) 0,1,3,5 (4) None

33. ,f A B A B . Simplified expression for function , y ,f f x y z is

(1) x z (2) xyz (3) xy z (4) none

34. The literal count of a Boolean expression is the sum of the number of times each literal appears in

the expression for example, the literal count of xy xz is ‘4’ what are the minimum possible literal

counts of the product-of-sum and sum-of-product representations respectively of the function given

by the following Karnaugh map? Here, X denotes “don’t care”

(1) (11,9) (2) (9,13) (3) (9,10) (4) 11,11)

35. Which of the following sets of component (s) is/are sufficient to implement any arbitrary Boolean

function?

(1) XOR gates, NOT gates (2) 2 to 1 multiplexers

(3) AND gates, XOR gates (4) three-input gates that output (A.B)+C for the inputs

A,B and C

36. Which of the following functions implements the Karnaugh map show below?

(1) AB CD (2) D C A (3) AD AB (4) C D C D A B

37. What is the minimum number of gates required to implement the Boolean function AB C if we

have to use only 2-input NOR gates?

(1) 2 (2) 3 (3) 4 (4) 5




38. Consider the following circuit

Which of the following is TRUE?

(1) f is independent of x

(2) f is independent of y

(3) f is independent of z

(4) None of x,y,z is redundant

39. Let denote the exclusive OR(XOR) operation. Let ‘1’ and ‘0’ denote the binary constants consider

the following Boolean expression for F over two variables P and Q

, 1F P Q P P Q P Q Q O

The equivalent expression for F is

(1) P Q (2) P Q (3) P Q (4) P Q

40. If P, Q, R are Boolean variables, then . .P Q P Q P R PR Q simplifies to

(1) PQ (2) PR (3) PQ R (4) PR Q

41. Consider the following Boolean expression for F:

, , ,F P Q R S PQ PQR PQRS

The minimal sum-of-products form of F is

(1) PQ QR QS (2) P Q R S

(3) P Q R S (4) PR PRS P

42. Consider the following Boolean function of four variables

, , , 1,3,4,6,9,11,12,14f w x y z

The function is

(1) Independent of one variable (2) independent of two variables

(3) Independent of three variables (4) dependent on all the variables

43. Given the following Karnaugh map, which one of the following represents the minimal sum-of-

product of the map?

(1) xy yz (2) w x y xy xz

(3) wx yz xy (4) xz y

44. The minterm expansion of

, ,f P Q R PQ QR PR is

(1) 2 4 6 7m m m m (2) 0 1 3 5m m m m

(3) 0 1 6 1m m m m (4) 2 3 4 5m m m m




45. What is the Boolean expression for the output f of the combinational logic circuit of NOR gates

given below?

(1) Q R (2) P Q (3) P R (4) P Q R

46. What is the Boolean expression for the output f of the logic circuit given below

(1) ACD ABC (2) ACD ABC

(3) ACD ABC (4) ACD ABC

47. Which one of the following circuit is NOT equivalent to a 2-input XNOR (exclusive NOR) gate?

48. In an SR latch made by cross-coupling two NAND gates, if both S and R inputs are set to ‘0’ then it

will result in

(1) 0, 1Q Q (2) 1, 0Q Q

(3) 1, 1Q Q (4) Indeterminate states

49. The following arrangement of master-slave flip flops

Has the initial state of P, Q as 0, 1 (respectively). After the clock cycles the output state P, Q is

(respectively)

(1) 1, 0 (2) 1, 1 (3) 0, 0 (4) 0, 1




50. Two NAND gates having open collector outputs areticed together as shown in figure. The logic

function Y, implemented by the circuit is

(1) Y ABC DE (2) Y ABC DE

(3) .ABC DE (4) .Y ABC DE


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