DPP 4 Quantum Mechanics C4U Sahendra Kumar
Q 1 A system is described by the following real wavefunction
The probability (P) of finding the particle in a region dx around point I ,II and III in the figure
obeys the trend
IIT GATE 2016
a) P(I) > P(II) > P(III)
b) P(II) > P(III) > P(I)
c) P(II) > P(I) > P(III)
d) P(III) > P(I) > P(II)
Q 2 Consider a particle of mass m moving n a one dimensional box under the potential V=0 for
0<x<a and V=infinite outside the box . When the particle is in lowest energy state , the average
momentum <px> of the particle is
Gate 2007
a) <px>=0 b) <px>=(h/a) c) <px>=(h/2a) d) <px>=(h/2πa)
Q 3 The uncertainty in the momentum Δpx of the particle in its lowest energy state is
a) Δpx =0
b) Δpx =(h/a)
c) Δpx =(h/2a)
d) Δpx =(h/2πa) Gate 2007
Q 4 For a particle of mass m confined in a box of length L , assume ∆x=L , .Assume further that
∆pmin=<p2>1/2 . Use the uncertainty principle to obtain an estimate of energy of the particle the
value will be
CSIR Dec 2014
a) h2 / 8mL2 b) ħ2/8mL2 c) h2 / 32mL2 d) h2 / 2mL2
Q 5 Suppose, the ground stationary state of a harmonic oscillator with force constant ‘k’ is given
Ψ0=exp[-Ax2] Then, A should depend on k as
CSIR JRF Chemical science 2013
-1/2
1/2
(a) A ∝ k
(b) A ∝ k
(c) A ∝ k
(a) A ∝ k1/3
𝑖𝜀𝑡
Q 6 If Ψ(x, t) can be expressed as product of two function f(t) g(x) has the form of 𝑒 − ħ then it
can be shown that probability of finding the particle between x and x + dx is
(a) equal to 1
(b) is independent of time TIFR-12
(c) can be obtained by solving time dependent Schrödinger equation
(d) can be obtained by solving time independent Schrödinger equation
Q 7 Given three systems, A, B, and C, what could be they if the spacing between the
neighbouring energy levels in A decreases with increasing energy, while that for B is
constant, and that for C increases with increasing energy?
TIFR 2013
A) A= particle in 1 D box, B= harmonic oscillator, C=electron in hydrogen atom
B) A= electron in hydrogen atom, B= harmonic oscillator, C= particle in a 1D box
C) A= particle in a one-dimensional box, B= electron in hydrogen atom, C= harmonic
oscillator
D) A= electron in hydrogen atom, B= particle in a one-dimensional box, C= harmonic
Oscillator
Q 8 The mathematical expression e i(kx−wt) represents
TIFR
A) a wave stationary in space but oscillating in time B) a travelling wave moving in the positive
x direction C) a wave stationary in time but periodically varying in space
D) a travelling wave moving in the negative x direction
Q 9 The unit of wave function corresponding to particle confined to move in 1 D box is
a) L1/2
b) L-1/2
c) L d) dimensionless
Q 10 The lowest allowed energy is equal to zero for
IIT GATE 2009
a) the hydrogen atom
b) rigid rotor
c) a harmonic oscillator
d) a particle in 3 D box
Q 11 The enegy of second excited state of the hydrogen atom is
a)-27.2 eV b) -6.8 eV c) -1.5 eV d) -4.5 eV
Q 12 Determine the probability of finding the particle between L/4 to 3L/4 for a particle in 1D
box of length (0<x<L) in the first excited state .
Q 13 For hydrogen-like atom with a nuclear charge Z, the energy of orbital with principal
quantum number ‘n’ follows the relation.
June CSIR
𝑍2
𝑍
𝑍2
(a) En ∝ n2Z2
(b) En ∝ − 𝑛
(c) En ∝ − 𝑛
(c) En ∝ − 𝑛2
Q 14 For the particle in a box problem in (0,L) , the value of <x3> in the n→∞ would be
a)L3/6 (b) L3/3 (c) L3/4 (d) L4/4
CSIR JRF NET JUNE 2013
Q 15 For a hydrogen atom, which of the following orbital(s) have the lowest energy:
a) 4s
b) 4 p
c) 4d
d) all of these have same energy TIFR 2017
Q 16 Consider the statement
i)If g is an eigenfunction of the linear operator B , then cg is an eigenfunction of B , where c is an
arbitrary constant.
ii) For a stationary state, the probability density is independent of time.
iii) If f is an eigenfunction of the linear operator A with eigenvalue a, then f is an eigenfunction
of A 2 with eigenvalue a2
.The correct statement above is / are a) I b) I,ii c) I,ii,iii d) I,iii
Q 17 Which of the following functions are eigenfunctions of d2/dx2?
a) ex; (b) x2 ; (c) sin x; (d) 3 cos x; (e) sin x + cos x. Give the eigenvalue for each eigenfunction.
Q 18 The value of commutator [Sin x , d/dx] is equal to
a) Cos x b) –Cos x c) tan x d) 0
Q 19 When we operate the operator –x-2(δ/δx) on the function e-x3 the corresponding eignvalue
a) -3
b) +3
c) -1
d) none
Q 20 The energy of 2S orbital and 2p orbital is same for CSIR DEC 2013
a) Li
b) Li2+
c) Be2+
d) HQ 21 Which of the following statement is /are true TIFR 2015
i)The state function ψ(x,t) is always equal to a function of time multiplied by a function of
coordinate.
ii) If f1 and f2 are eign functions of operator B then C1 f1 +C2 f2 must always be an eignfunction of B .
where C1 and C2 are constants.
iii)The operator L2 commute with Lx + Ly .
a)i,iii (b) i,ii (c) ii,iii (d) only iii
Q 22 The wavefunction for a quantum mechanical particle in a 1 dimmensional box of length ‘a’ is given
by ψ=A sin (πx/a) . The value of A for a box of length 200 nm is IIT GATE 2011 C
(1)4×104(nm)2 (2) 10 √2 (nm)1/2 (3) √2 /10 nm-1/2 (4) 0.1 nm-1/2
Q 23 T/F i) any linear combination of the functions sin(3x) and cos(3x) is an eigenfunction of the operator
d2/dx2
ii) All three position coordinates can be specified simultaneously.
iii) The value of commutator [x,(x,H)] is zero.
Q 24 A very general form of the uncertainty principle was developed by H.P. Robertson in 1929 for two
observables A and B is
a) ΔA ΔB > {0.5 <[A,B]> }
b) ΔA ΔB > {0.5 [A,B] }
c) ΔA ΔB ≥ {0.5 <[A,B]> }
d) ΔA ΔB ≥ {0.5 [A,B] }
Q 25 The value of [H,px] in terms of V(x) is ………