Quantum Mechanics JEST 2012-2019 PDF
Quantum Mechanics JEST 2012-2019 PDF
Quantum Mechanics JEST 2012-2019 PDF
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Quantum Mechanics
JEST-2012
Q1. The ground state (apart from normalization) of a particle of unit mass moving in a one-
dimensional potential V(x) is exp x 2 / 2 cosh 2 x . The potential V(x), in suitable
units so that h = 1, is (up to an addiative constant.)
(a) π2/2 (b) 2 / 2 2 x tanh 2 x
(c) 2 / 2 2 x tan 2 x (d) 2 / 2 2 x coth 2 x
Ans. : (b)
Q2. Consider the Bohr model of the hydrogen atom. If is the fine-structure constant, the
velocity of the electron in its lowest orbit is
or 1 c
c c
(a) (b) (c) 2 c (d) c
1 1 2
Ans. : (d)
Solution: mvr n
mv 2 1 ze 2 1 ze 2
r
r 4 0 r 2 4 0 mr 2
1 ze 2
mv n
4 0 mv 2
ze 2 e2
v and fine structure constant
4 0 n 4 0 c
ze 2 ze 2 c
For lowest orbit, v v
4 0 4 0 c
v c
Q3.
Define x f † f , and y i f † f , where the are Pauli spin matrices and
f , f † obey anti-commutation relations f , f 0, f , f † 1 . Then z is given by
i z x y
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
1
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
1 i †
f f f † f f † f † f ff † f 2
2
z x y
i i
f † f 1 f † . f 1 2 f † f 2 f † f 1
1
Q4. Consider a system of two spin- particles with total spin S S1 S2 , where S1 and S2
2
are in terms of Pauli matrices i . The spin triplet projection operator is
1 3 3 1
(a) S1 S2 (b) S1 S2 (c) S1 S2 (d) S1 S2
4 4 4 4
Ans. : (c)
Solution: S S1 S 2 S 2 S12 S 22 2S1 S 2
3 3
S 2 2.S1 S2 2 S 0, 1
4 4
3
S 2 2 S1 S2 2 for Triplet projection operator
4
3
s s 1 2 2 S1 S 2 2 S 1
4
3 3
11 1 2 S1 S 2 S1 S 2 I
4 4
1
Q5. Consider a spin- particle in the homogeneous magnetic field of magnitude B along z -
2
2
(a) t (b) t (c) t (d) Never
B B B B 2 B B
Ans.: (a)
1 1
Solution: E B B zˆ
2 1
1 1 iEtb
x, t e x, t
2 1
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
2
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
i B Bt
1 1 1 1
e
2 1 2 1
i B Bt
e
1
Bt
cos B cos
B B t
t
B B
1
Q6. The ground state energy of 5 identical spin- particles which are subject to a one-
2
dimensional simple harmonic oscillator potential of frequency ω is
15 13 1
(a) (b) (c) (d) 5
2 2 2
Ans. : (b)
1
Solution: Degeneracy 2 s 1 2 1 2
2
1 3 5 13
Eground 2 2 1
2 2 2 2
Q7. The spatial part of a two-electron state is symmetric under exchange. If and
represent the spin-up and spin-down states respectively of each particle, the spin-part of
the two-particle state is
(a) (b)
(c) / 2
(d) / 2
Ans. : (c)
Solution: Since, electrons are Fermions and Fermions have anti-symmetric wave function
spatial part is symmetric then its spin part is antisymmetric to maintain antisymmtric
wave function
x
1
2
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
3
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q8. The wave function of a free particle in one dimension is given by
x A sin x B sin 3 x . Then x is an eigenstate of
(a) the position operator (b) the Hamiltonian
(c) the momentum operator (d) the parity operator
Ans. : (d) x x
x {parity (even and odd)
x A sin x B sin 3 x A sin x B sin 3 x
x x negative parity i.e. parity operator
Q9. The quantum state sin x expi cos x , where 0 and x, are, real, is
orthogonal to:
(a) sin x (b) cos x expi sin x
Ans.: (d)
Solution: 0 , sin x expi cos x
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
4
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
JEST-2013
Q10. A particle of mass m is contained in a one-dimensional infinite well extending from
2 L/2 2
2 1 L / 2 3 x x 2 1 2L 3 x 2 L x
cos cos dx sin sin
L 2 L / 2 2L 2L L 2 3 2L 2 L L / 2
2 2 2
2 1 2 L 3 3 2L 2 2 8
sin sin sin sin
L 2 3 4 4 4 4 3 3
Q11. A quantum mechanical particle in a harmonic oscillator potential has the initial wave
function 0 x 1 x , where 0 and 1 are the real wavefunctions in the ground and
first excited state of the harmonic oscillator Hamiltonian. For convenience we take
m 1 for the oscillator. What is the probability density of finding the particle at
x at time t ?
(a) 1 x 0 x (b) 1 x 0 x
2 2 2
(c) 1 x 0 x (d) 1 x 0 x
2 2 2
Ans.: (a)
Solution: x 0 x 1 x
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
5
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Et Et
x, t 0 x e i 0 1 x e i 1
Now probability density at time t
t
x, t * x, t x, t 0 x 1 x 2 Re 0* x 1 x cos E1 E0
2 2 2
putting t
x, t 0 x 1 x 2 Re 0* x 1 x cos E1 E0 1
2 2 2
x, t 0 x 1 x 2 Re 0* x 1 x 1 x 0 x
2 2 2 2
Q12. If J x , J y and J z are angular momentum operators, the eigenvalues of the operator
J x Jy
are:
(a) real and discrete with rational spacing
(b) real and discrete with irrational spacing
(c) real and continuous
(d) not all real
Ans.: (b)
1 i 0 1 0 0
Solution: J x J J , J y J J J , J 1 0
2 2 0 0
0 1 i 0 1 J J y 1 0 1 i
Jx , Jy x
2 1 0 2 1 0 2 1 i 0
1 1 i
eigen value 2 2 0 2
2 1 i
Q13. A simple model of a helium-like atom with electron-electron interaction is replaced by
Hooke’s law force is described by Hamiltonian
2 2
2m
1
1 22 m 2 r12 r22
2 4
2
m 2 r1 r2 .
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
6
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
1 / 2
Q14. Consider the state 1 / 2 corresponding to the angular momentum l 1 in the L z basis
1 / 2
of states with m 1, 0, 1 . If L2z is measured in this state yielding a result 1, what is the
state after the measurement?
1 1/ 3 0 1 / 2
(a) 0 (b) 0 (c) 0 (d) 0
0 1
2/3 1 / 2
Ans.: (d)
1 0 0 1 0 0 1 0 0
Solution: L z 0 0 0 , L z 0 0 0 , eigenvector
2
0 , 1, 0
0 0 1 0 0 1 0 0 1
Corresponding eigenvalue 1, 0, 1
1 1
1
Now state after measurement yielding 1 1 3 0 0
1 2
1
Q15. What are the eigenvalues of the operator H a , where are the three Pauli matrices
and a is a vector?
(c) a x a y a z
(a) a x a y and a z (b) a x a z ia y (d) a
Ans.: (d)
Solution: H a x .a x y .a y z .a z
0 1 0 i 1 0
az a x ia y
ax ay az
0 1 ax ia y
1 0 i 0 az
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
7
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q16. The hermitian conjugate of the operator is
x
(a) (b) (c) i (d) i
x x x x
Ans.: (a)
x
† *
Solution: * x x x
x x
x *
x x dx * x x
*
x dx
x x
* x
x dx
x
Q17. If the expectation value of the momentum is p for the wavefunction x , then the
Ans.: (c)
Solution: * x i x dx p
x
Now
ikx
ikx
ikx
ikx ik ikx
e
x i e x dx e x i e
* *
x e x
x
x
ikx
ikx
ik ikx
e
x i x e i. e * x x dx
*
x
* x i x k * x x p K
x
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
8
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q18. Two electrons are confined in a one dimensional box of length L . The one-electron states
2 n x
are given by n x sin . What would be the ground state wave function
L L
x1 , x2 if both electrons are arranged to have the same spin state?
1 2 x1 2 x2 2 2 x1 x2
(a) x1 , x2 L sin L sin L L sin L sin L
2
1 2 x1 2 x2 2 2 x1 x2
(b) x1 , x2 L sin L sin L L sin L sin L
2
2 x 2 x2
(c) x1 , x2 sin 1 sin
L L L
2 2 x1 x2
(d) x1 , x2 sin sin
L L L
Ans.: (b)
1
Solution: Electrons are Fermions of spin and its wave functions are anti-symmetric
2
Since, spin part is symmetric, therefore, space part will be anti-symmetric (since as total
wave function is anti-symmetric)
Then,
1 2 x1 2 x2 2 2 x1 x2
x1 , x2 L sin L .sin L L sin L .sin L
2
d d
Q19. The operator x x is equivalent to
dx dx
d2 d2
(a) 2
x2 (b) 2 x 2 1
dx dx
d2 d d2 d
(c) 2
x x2 1 (d) 2
2x x 2
dx dx dx dx
Ans.: (b)
d d d d
Solution: x x f x x f x xf x
dx dx dx dx
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
9
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
d d
f x xf x x f x x 2 f x
d
dx dx dx
d2 df x d
2
f x f x x x f x x2 f x
dx dx dx
d2 d2
f x x 2
f x f x
dx 2 x 2
1 f x
dx 2
JEST-2014
Q20. Suppose a spin 1 / 2 particle is in the state
1 1 i
6 2
If S x ( x component of the spin angular momentum operator) is measured what is the
probability of getting / 2 ?
(a) 1 / 3 (b) 2 / 3 (c) 5 / 6 (d) 1 / 6
Ans.: (c)
0 1 1 1
Solution: S x with eigenvalues and eigenvector corresponding to is
2 1 0 2 2 2 1
Now probability getting
2
2
1 1 1 i
1 1 2 1
1 i 2
2
2 6 5
p 12
2 1 1 i 1 6
1 i 2 2 6
6
6
Q21. The Hamiltonian operator for a two-state system is given by
H 1 1 2 2 1 2 2 1 ,
where is a positive number with the dimension of energy. The energy eigenstates
corresponding to the larger and smaller eigenvalues respectively are:
(a) 1
2 1 2 , 1 2 1 2 (b) 1
2 1 2 , 1 2 1 2
(c) 1 2 1 2 , 2 1 1 2 (d) 1 2 1 2 , 2 1 1 2
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
10
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Ans.: (b)
Solution: H 1 1 2 2 1 2 2 1 H 1 1 2 , H 2 1 2
Lets check for option (b): 1
2 1 2 , 1 2 1 2
Now H H 1 2 1 2 H 1 H 2 1 2
H 1
2 1 2 H 1
2 1 H 2 1 2
2 1 1 2
1 2 1 1 1
2 1 2 2 1 2 2 2
2 1
2 1 2
Now H 1 2 1 2 H 1
2 1 2 H 1 H
2 1 2
1 2
2 1 1 2 1 2 1 1 1 2 1 2
2 1 2 2 2 2 1 1 2 2
2
Q22. Consider an eigenstate of L and Lz operator denoted by l, m . Let A nˆ L denote an
l l 1 m 2 l l 1 m 2
(a) cos (b) sin
2 2
2
Now A A2 A
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
11
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
A Lx sin cos Ly sin sin Lz cos
A m cos Lx 0, Ly 0
L 2
x
L2y sin 2 L2z cos 2
L 2
L2z sin
2
L2z cos 2
A A2 A
2
l l 1 m
2 2
sin 2 m 2 2 cos 2 m 2 2 cos 2
A l l 1 m 2 sin
Q23. Consider a three-state system with energies E , E and E 3g (where g is a constant) and
1
If the system is initially (at t 0 ), in state i 0
0
0
what is the probability that at a later time t system will be in state f 0
1
4 2 3 gt
(a) 0 (b) sin
9 2
4 3 gt 4 2 E 3gt
(c) cos 2 (d) sin
9 2 9 2
Ans.: (b)
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
12
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q24. The lowest quantum mechanical energy of a particle confined in a one-dimensional box
of size L is 2 eV . The energy of the quantum mechanical ground state for a system of
1
three non-interacting spin particles is
2
(a) 6 eV (b) 10 eV (c) 12 eV (d) 16 eV
Ans.: (c)
22
Solution: E1 2eV , E2 4 E1 8 eV
2ml 2
1 1
Spin, spin is , therefore, degeneracy gi 2S 1 2 1 2
2 2
ground state energy = 2 2 eV 1 8 eV 12 eV
Q25. A ball bounces off earth. You are asked to solve this quantum mechanically assuming the
earth is an infinitely hard sphere. Consider surface of earth as the origin implying
V 0 and a linear potential elsewhere (i.e. V x mgx for x 0 ). Which of the
following wave functions is physically admissible for this problem (with k 0 ):
2 2
(a) e kx / x (b) xe kx (c) Axekx (d) Ae kx
Ans.: (b)
2
Solution: xe kx
For given potential, at x 0 and x wave function must vanish.
Q26. The operator A and B share all the eigenstates. Then the least possible value of the
product of uncertainties AB is
(a) (b) 0 (c) / 2 (d) Determinant (AB)
Ans.: (b)
Solution: A B
AB
2
A B 0 [ A and B have share their eigen values, so AB 0 ]
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
13
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q27. Consider a square well of depth V0 and width a with V0 as fixed. Let V0 and
JEST-2015
2
Q28. Consider a harmonic oscillator in the state e 2
e a 0 , where 0 is the ground
state, a is the raising operator and is a complex number. What is the probability that
the harmonic oscillator is in the n th eigenstate n ?
2
n
2n
2
2
(a) e (b) e n!
n!
2 2n
n
2
(c) e (d) e 2
n! n!
Ans.: (a)
a a
n n
2
2
a n
Solution: e 2
e 0 e 2
0 and n 0 a 0 n n
n n n
n
2 *
n n
n n
2 2 2 2
e 2
n e n n e e e 1
n
2
n n n n
2
n 2
Probability that is in n state is, n
2 2
n
n 1
e 2
n n
n e 2
n
n
n
n
2
2 2n
1 e 2
2 2
n e 2 n
nn n n e
n n n n
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
14
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q29. A particle of mass m moves in 1 dimensional potential V x , which vanishes at infinity.
d 2
A 2 2sec3 h x sec h x
dx 2
d 2 2 d 2
Now put the value in equation V x x E x
dx 2 2m dx 2
2 2
A 2sec3 h x sec h x V x A sec h x EA sec h x
2m
V x 0 as x
2 2 2 2
A sec h x 2 A sec3 h x EA sec h x
2m 2m
Now we have to do approximation i.e. sec3 h x dacays very fastly as x so second
term
2 2 2 2 2 2
2 A sec h x 0 . Thus
3
A sec h x EA sec h x E
2m 2m 2m
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
15
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
1
Q30. Consider a spin particle characterized by the Hamiltonian H S z . Under a
2
perturbation H gS x , the second order correction to the ground state energy is given by,
g2 g2 g2 g2
(a) (b) (c) (d)
4 4 2 2
Ans.: (a)
1 0
Solution: H sz sz
2 0 1
and
1 0 g 1 0
H and H gsx
2 0 1 2 0 1
0
Ground state energy is with eigenvector 1
2 1
1
and first excited state energy is with eigenvector 2
2 0
2 2
m H 1 m H 1
Second order correction in ground state E12
E10 Em0
m 1
2 2
2
0 10
1 0
g 22 1 01 g 22 g2
E1
2
4 2 4 4
2
Q31. Given that 1 and 2 are eigenstates of a Hamiltonian with eigenvalues E1 and E2
1
(a) E1 E 2 (b) E1 E 2
2
1 1
(c) E1 E 2 (d) E 2 E1
2 2
Ans.: (b)
Solution: E 2 1 2 1 2
E1 E2
E12 E22 and E
1 1
E1 E2
2 2 2 2 2
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
16
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
E E2 E
2
E 1
2
E22 1 E E 2
2 E12 2 E22 E12 E22 2 E1 E2
1 2
2 4 4
E12 E22 2 E1 E2 1
E E1 E2
4 2
kr 2
Q32. A particle moving under the influence of a potential V r has a wavefunction
2
r , t . If the wavefunction changes to r , t , the ratio of the average final kinetic
energy to the initial kinetic energy will be,
1 1
(a) (b) (c) (d) 2
2
Ans.: (c)
2 2
r dr , is
Solution: For r , t the average kinetic energy T * r , t 2 2
0
2m
2 2
r , t r dr
T * r , t 2
0
2m
r dr
Put r r or r dr and 2r 2 r2
2
2
2
1 2 2
T 3 0 r , t 2m r , t r dr 0 r , t 2m r , t r dr
* 2 * 2
T T 1
T
T
Q33. If a Hamiltonian H is given as H 0 0 1 1 i 0 1 1 0 , where 0 and 1 are
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
17
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
H 0 0 i 1 and H 1 1 i 0
0 H 0 0 H 1 1 i
The matrix representation of H is
1H 0 1H 1 i 1
1 i
0 1 1 0 2
2
Eigenvalue of H
i 1
L L
Q34. A particle of mass m is confined in a potential well given by V x 0 for x
2 2
2 x
For ground state 0 cos
L L
0 H 0 2 L/2 x
E01 x cos 2 0
0 0 L L/2 L
2
m H 0
E0
2
E0 0
2
E00 Em0
m0 E E
0
0
0
m
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
18
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
JEST-2016
Q35. The wavefunction of a hydrogen atom is given by the following superposition of energy
eigen functions nlm r ( n, l , m are the usual quantum numbers):
2 3 1
r 100 r 210 r 322 r
7 14 14
The ratio of expectation value of the energy to the ground state energy and the
expectation value of L2 are, respectively:
229 12 2 101 12 2
(a) and (b) and
504 7 504 7
101 229
(c) and 2 (d) and 2
504 504
Ans.: (a)
2 E 9 E 1 E 229
Solution: E 0 0 0 E0
7 1 14 4 14 9 504
2 9 1 24 12
L2 0 2 2 2 6 2 2 2
7 14 14 14 7
1
Q36. A spin- particle in a uniform external magnetic field has energy eigenstates 1 and 2 .
2
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
19
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
E E1 t
exp i 2 1
E2 E1 T E E h
2 1
T 2T
Q37. The energy of a particle is given by E p q where p and q are the generalized
momentum and coordinate, respectively. All the states with E E0 are equally probable
and states with E E0 are inaccessible. The probability density of finding the particle at
(a)
E0 q (b)
q
(c)
E0 q (d)
1
2
E0 E02 E 2
0 E0
Ans.: (c)
Solution: For condition, E p q total number of accessible state upto energy E0 for q 0
1
is area under the curve 2 E02 E02
2
The probability density of finding the particle at coordinate q , with q 0
dpdq pdq E q dq
2 0 2
2
E0 E0 E0
Q38. Consider a quantum particle of mass m in one dimension in an infinite potential well, i.e.,
a a a
V x 0 for x and V x for x . A small perturbation,
2 2 2
2 x
V x is added. The change in the ground state energy to O is:
a
(a)
2 2
2 4 (b)
2 2
2 4
2 2 2 2
(c)
2
4 (d)
2
4
Ans.: (a)
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
20
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
a a
2 2 2 2 x
2
Solution: E11 1*V ' x 1dx x cos dx
a a
a a a
2 2
a a a
2 2
2 x 2 4 22 x 2 x
4 2
.2. x cos 2 dx 2 x cos 1dx 2 x cos 1dx
a 0 a a a 0 2 a a 0 a
a
4 2 2 x
2 x cos
a 0 a
1dx
2 2
2 4
1
Q39. If Yxy
2
Y2,2 Y2,2 where Yl ,m are spherical harmonics then which of the following
is true?
(a) Yxy is an eigenfunction of both L2 and Lz
Ans.: (b)
Solution: The L2Yxy l l 1 2Yxy , where l 2 and LzYxy mYxy
2
1
Q40. A spin-1 particle is in a state described by the column matrix 2 in the S z
10
2i
basis. What is the probability that a measurement of operator S z will yield the result
1 1 1 1
(a) (b) (c) (d)
2 3 4 6
Ans.: (c)
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
21
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
0 1 0 2
1
Solution: S x 1 0 1 , 2
2 10
0 1 0 2i
1
2
Sx 1 i
10
1
1 0 0
Sz 0 0 0
0 0 1
1
The eigen state corresponding to eigen value of S z is 0
0
2
1
2
1 0 0 1 i
10
1 1
P
1 4
2 2
1 1 i 1 1 i
10 1
Q41. The Hamiltonian of a quantum particle of mass m confined to a ring of unit radius is:
2
2
H i
2m
where is the angular coordinate, is a constant. The energy eigenvalues and
eigenfunctions of the particle are ( n is an integer):
ein 2 sin n 2
(a) n n (b) n n
2 2
and En and En
2 2m 2 2m
cos n 2 ein 2
(c) n n (d) n n
2 2
and En and En
2 2m 2 2m
Ans.: (a)
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
22
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
2
2
2 2
Solution: H i 2 2i 2 E
2m 2m
ein
By inspection, n , which will also satisfy boundary condition
2
2 n
2
E
2m
d
Q42. The adjoint of a differential operator acting on a wavefunction x for a quantum
dx
mechanical system is:
d d d d
(a) (b) i (c) (d) i
dx dx dx dx
Ans.: (c)
Q43. In the ground state of hydrogen atom, the most probable distance of the electron from the
nucleus, in units of Bohr radius a0 is:
1 3
(a) (b) 1 (c) 2 (d)
2 2
Ans.: (d)
r
1
Solution: 100 e a0
a03
r
1 dP
P 3 e a0 rp
*
0 rp a0
a0 dr
Ans.: (b)
Solution: P, Q 1 PQ 1 Q 1 P
Q 1 P, Q Q 1 Q 1 PQ QP Q 1 Q 1 PQQ 1 QPQ 1 Q 1 P PQ 1 P, Q 1
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
23
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q45. A spin
1
particle is in a state
where and are the eigenstates of S z
2 2
operator. The expectation value of the spin angular momentum measured along x
direction is:
(a) (b) (c) 0 (d)
2
Ans.: (d)
1
Solution:
2
,
0 1
Sx
2 1 2 1 0
2
1
1 1 0 1 2
Sx
2 2 2 1 0 1 2
2
JEST 2017
Q46. What is the dimension of , where is a wavefunction in two dimensions?
ix
(a) kg m 1s 2 (b) kg s 2 (c) kg m 2 s 2 (d) kg s 1
Ans. : (d)
dim of kg m sec 2 sec
Solution: Dimension of kg sec 1
ix dim of x m
Q47. Suppose the spin degrees of freedom of a 2 - particle system can be described by a 21 -
dimensional Hilbert subspace. Which among the following could be the spin of one of the
particles?
1 3
(a) (b) 3 (c) (d) 2
2 2
Ans. : (b)
Solution: Dimension of Hilbert space 2 s1 1 2 s2 1 7 3 21
So, s1 3, s2 1
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
24
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q48. If the ground state wavefunction of a particle moving in a one dimensional potential is
1 , is proportional to
Ans. : (b)
Solution: From figure, we can conclude that option (b) is the correct answer.
V
x x
Q49. A particle is described by the following Hamiltonian
pˆ 2 1
Hˆ m 2 xˆ 2 xˆ 4
2m 2
where the quartic term can be treated perturbatively. If E0 and E1 denote the energy
correction of O to the ground state and the first excited state respectively, what is the
Ans. : 5
ˆ Pˆ 2 1
Solution: H m 2 xˆ 2 xˆ 4
2m 2
Now, energy correction of O to ground state is
2 2
E0 0 xˆ 0
4
0 6n 6n 3 0
2
3
2m 2m
And energy correction of O to first excited state is
2
E1 1 xˆ 4 1 1 6n 6 n 3 1
2
2m
2 2
E1 15
6 6 3
15 . Hence, 5
2m 2m E0 3
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
25
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q50. If x̂ t be the position operator at a time t in the Heisenberg picture for a particle
pˆ 2 1
ˆ
described by the Hamiltonian, H m 2 xˆ 2 what is ei t 0 xˆ t xˆ 0 0 in units of
2m 2
where 0 is the ground state?
2m
Solution: Operator X̂ t in Hisenburg picture is written as
Xˆ t eiHt / Xˆ 0 eiHt /
Here, Xˆ 0 0 1
2m
So, above equation reduces as,
0 Xˆ t Xˆ 0 0 0 eiHt / Xˆ 0 e iHt / 1
2m
In integral form,
0 Xˆ t Xˆ 0 0 0* t Xˆ 0 1 t dx
2m
i t i 3 t
it *
0* e 2 Xˆ 0 1e 2
dx e 0 x1 dx
2m 2m
2
Therefore, e 0 Xˆ t Xˆ 0 0
i t
0 a a 1
†
2 m
eit 0 Xˆ t Xˆ 0 0
2m
Q51. Consider a particle confined by a potential V x k x , where k is a positive constant.
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
26
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
E/k E/k
1
b
k
2m E V x dx n 2 2m E kxdx 2 2m E 1 x dx
0 2 0 0
E
1 1
E 2E 2m 2
2 2mE 1 t dt 2mE 1 t dt 2 E 3 / 2
0
k k 0
k 3
1 3 k 1
n En3 / 2 n
2 4 2m 2
2/3
3 k 1
En n
4 2m 2
Q52. Consider the Hamiltonian
1 0 0 0 0 1
H t 0 2 0 t 0 0 0
0 0 3 1 0 2
The time dependent function t for t 0 and zero for t 0 . Find
2
t 0 t 0 , where t 0 is the normalised ground state of the system at
a time t 0 and t 0 is the state of the system at t 0 .
1 1
(a)
2
1 cos 2 t (b)
2
1 cos t
1 1
(c)
2
1 sin 2 t (d)
2
1 sin t
Ans. : (a)
1 0 0 0 0 1
Solution: H t 0 2 0 t 0 0 0
0 0 3 1 0 2
, t 0
Time dependent function t
0 , t 0
When t 0
1 0 1
H t 0 2 0
1 0 1
Eigen value are 0 , 2 , 2 .
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
27
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
1
1
For Eigen value zero, the ground state wave function is t 0 0 .
2
1
1 0
1 i t 1 i 3 t
And t 0 0 e 0 e
2 2
0 1
2
2 1 i t i 3 t
Now, t 0 t 0 e e
4
1
2 2
t 3 t t 3 t
cos cos sin sin
4
1 t 3 t t 3 1 2 t 1 2 t
1 1 2 cos .cos sin cos 2 2.cos 1 cos
4 4 2
JEST-2018
Q53. If x is an infinitely differentiable function, then D̂ x , where the operator
d
Dˆ exp ax , is
dx
Ans. : (c)
Q54. A one dimensional harmonic oscillator (mass m and frequency ) is in a state such
the energy of the n -th excited state. If H is the Hamiltonian of the oscillator,
3 11 2 2
H and H 2
, then the probability that the energy
2 4
measurement yields E0 is
1 1 1
(a) (b) (c) (d) 0
2 4 8
Ans. : (b)
Solution: a 0 b 1 c 2 let us assume a, b, c is real
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
28
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
3 5
a2 b2 c2 2 2 2
H 2 2 2 3 a 3b 5c 3 .…(i)
a 2 b2 c 2 4 2 2 2 4
2 2 2
2 2 3 2 5
a b c
2 11
2 2
H 2 2
a 2 b2 c2 4
a 2 9b 2 25c 2 11 2 2
.....(ii)
4 4 4 4
a 2 b2 c 2 1 .…(iii)
1 1 1
Solving a 2 , b 2 , c 2
4 2 4
a2 1
P 2 2 2
a2
2 a b c 4
Q55. A quantum particle of mass m is moving on a horizontal circular path of radius a . The
particle is prepared in a quantum state described by the wavefunction
4
cos 2 ,
3
being the azimuthal angle. If a measurement of the z -component of orbital angular
momentum of die particle is carried out, the possible outcomes and the corresponding
probabilities are
1 1 1
(a) Lz 0, , 2 with 0 P 0 , P and P 2
5 5 5
(b) Lz 0 with P 0 1
1 1
(c) Lz 0, with P 0 and P
3 3
2 1
(d) Lz 0, 2 with P 0 and P 2
3 6
Ans. : (d)
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
29
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
2 1 1
0 2 2
3 6 6
2 1
Lz 0, 2 with P 0 and P 2
3 6
Q56. Consider two canonically conjugate operators X̂ and Ŷ such that Xˆ , Yˆ iI , where I
is identity operator. If Xˆ 11Qˆ1 12Qˆ 2 , Yˆ 21Qˆ1 22Qˆ 2 , where ij are complex
numbers and Qˆ1 , Qˆ 2 zI , the value of 11 22 12 21 is
i
(a) iz (b) (c) i (d) z
z
Ans. : (b)
Q58. The normalized eigenfunctions and eigenvalues of the Hamiltonian of a Particle confined
to move between 0 x a in one dimension are
2 n x n 2 2 2
n x sin and En
a a 2ma 2
respectively. Here 1, 2,3... . Suppose the state of the particle is
x x
x A sin 1 cos
a a
where A is the normalization constant. If the energy of the particle is measured, the
2 2 x
probability to get the result as 2
is . What is the value of x ?
2ma 100
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
30
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Ans. : 80
a 2 x 2 2 x x
x A sin sin cos
2 a a 2 a a a
a 1
A 1 2
2 2
a 2 1 a 5 8
1 A 1 A2 1 A
2 4 2 4 5a
a 8 1 4 1
. 1 2 1 2
2 5a 2 5 5
22 4 x 4
P 2
x 100 80
2ma 5 100 5
0 xˆ 2 0 C
0 xˆ 2 2 2C
1 xˆ 2 1 3C
1 xˆ 2 3 6C
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
31
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Ans. : 5
2
Solution: For nth state En n X 4 n
4m
2 2
6n 2 6n 3
3 2 2 15 2
E0 0 X 4 0
4m 2 2
1 X 4
1
4m 2 2
6.12
6.1
3
4m 2 2
E1
5
E0
Q60. Consider a wavepacket defined by
x dkf k exp i kx
K K
Further, f k 0 for k and f k a for k . Then, the form of normalized
2 2
x is
Kx
sin
8 K Kx 2 2
(a) sin (b)
x 2 K x
Kx
sin
8 K Kx 2 2
(c) cos (d)
x 2 K x
Ans. : (b)
Solution: Given x dkf k eikx
K /2 K
x dK a eiKx K
K / 2 2
K K
q q i x i x K
f K 0
K /2
eikx e 2 e 2 K
ix K / 2 ix 2
a
2 kx
x sin
x 2
K K
22 Kx
A2 dx 1 2 2
x 2 2
h 2 Kx / 2
4 A2 1
x2
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
32
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
x
4 A2 1
2
1 1
A2 A
2 K 2 K
2 1 Kx
x sin
x 2 K 2
Kx
sin
2 2
x
K x
JEST-2019
Q61. What is the binding energy of an electron in the ground state of a He ion?
(a) 6.8eV (b) 13.6 eV (c) 27.2 eV (d) 54.4 eV
Ans. : (d)
13.6 2
Solution: E z eV
n2
He : z 2
13.6 4
E eV
n2
The binding energy of an electron in ground state is
13.6 4
E eV 54.4 eV
1
2
b2 x 2
Q62. The wave function x A exp (for real constants A and b ) is a normalized
2
correct?
2b 4 x 2 2b 4 x 2 2b 2 2b 2
(a) V (b) V (c) E (d) E
m 2m 4m m
Ans. : (b)
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
33
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
1/ 4
m m x 2
Solution: Comparing with harmonic oscillator x exp the potential is
2
1
V x m 2 x 2 and energy is E
2 2
b x
2 2 2
b b4 2 x 2 b2 2
x A exp so V x and energy E
2 m 2m 2 2m
where is a constant. Which one of the following represents the possible ground state
wave function of the particle?
1 1
(a) (b)
0 0
0 x 0 x
1 1
1 1
(c) 0 (d) 0
0 x 0 x
1 1
Ans. : (b)
1
Q64. For a spin particle placed in a magnetic field B , the Hamiltonian is
2
H BS y S y , where S y is the y -component of the spin operator. The state of the
system at time t 0 is t 0 , where S z . At a later time t , if S z
2
measured then what is the probability to get a value ?
2
t
(a) cos 2 t (b) sin 2 t (c) 0 (d) sin 2
2
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
34
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Ans. : (d)
1
Solution: H BS y S y Eigen value is , with eigen vector 1
2 2 2
1
and 2 respectively.
2
1 1
t 0 I 1 1 2 2 1 2
2 2
1 it 1 it
t t 1 exp 2 exp
2 2 2 2
If S z is measured on t then probability to find is
2
2
t it it
2
1 2 t
P exp exp sin
2 t t 4 2 2 2
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
35
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q66. Consider a quantum particle of mass m and a charge e moving in a two dimensional
potential given as:
k
V x, y x y k x y
2 2
2
The particle is also subject to an external electric field E iˆ ˆj , where is a
constant iˆ and ĵ corresponds to unit vectors along x and y directions, respectively. Let
E1 and E0 be the energies of the first excited state and ground state, respectively. What
is the value of E1 E0 ?
2k 2k 2k 2k
(a) (b) e 2 (c) 3 (d) 3 e 2
m m m m
Ans. : (a)
Solution: For constant electric field we know there is not any change in frequency and energy of
each level is changed by constant value.
The total potential is
k 3 3
V x, y x y k x y x y V x, y kx 2 ky 2 kxy x y
2 2
2 2 2
m 0 3k k
T and V
0 m k 3k
Secular equation is given by
4k 2k
2
V 2 m 0 3k 2 m k 2 0 x ,y
m m
1 1
The equivalent quantum mechanical energy is Enx ,n y nx x n y y V0
2 2
Where nx 0,1, 2,3... and n y 0,1, 2,3...
4k 2k
The ground state energy E0 E0.0
2 m 2 m
4k 3 2k
The first excited state energy E1 E0.1
2 m 2 m
2k
E1 E0
m
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
36
fiziks
Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Q67. A one-dimensional harmonic oscillator is in the state
1
n
n 0 n!
1
where n is the normalized energy eigenstate with eigenvalue n . Let the
2
1
expectation value of the Hamiltonian in the state be expressed as . What is
2
the value of ?
Ans. : 3
1
n 1
2 n 1
Solution: H e 3.2
n 0 n 2 n 1 n 2
Q68. Consider a system of 15 non-interacting spin-polarized electrons. They are trapped in a
1
two dimensional isotropic harmonic oscillator potential V x, y m 2 x 2 y 2 . The
2
angular frequency is such that 1 in some chosen unit. What is the ground state
energy of the system in the same units?
Ans. : 55
Solution: Non-interacting spin-polarized electrons means direction of spin is fixed
1 2 2 3 3 4 4 5 5 55
H.No. 40-D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi-110016
Phone: 011-26865455/+91-9871145498
Website: www.physicsbyfiziks.com | Email: fiziks.physics@gmail.com
37