Question 1:
The 2s-orbital of H-atom has a radial node at 2a 0 because $${psi _{2s}}$$ is proportional to
A.
$$left( {frac{1}{2} + frac{r}{{{a_0}}}}
ight)$$
B.
$$left( {2 + frac{r}{{{a_0}}}}
ight)$$
C.
$$left( {2 - frac{r}{{{a_0}}}}
ight)$$
D.
$$left( {2 - frac{r}{{2{a_0}}}}
ight)$$
Answer: _________
Question 2:
The region of electromagnetic spectrum employed in the electron spin resonance (ESR) spectroscopy is
A.
radiowave
B.
microwave
C.
infrared
D.
visible
Answer: _________
Question 3:
Which of the following pairs of operators commute?
A.
$$x{ ext{ and }}frac{d}{{dx}}$$
B.
$$frac{d}{{dx}}{ ext{ and }}frac{{{d^2}}}{{d{x^2}}} + frac{{2d}}{{dx}}$$
C.
$${x^2}frac{d}{{dx}}{ ext{ and }}frac{{{d^2}}}{{d{x^2}}}$$
D.
$${x^3}{ ext{ and }}frac{d}{{dx}}$$
Answer: _________
Question 4:
The set of eigen functions $$sqrt {frac{2}{a}} sin frac{{npi x}}{a}left( {0 leqslant x leqslant a,,n = 1,,2,,3,,...}
ight){ ext{is}}$$
A.
orthogonal
B.
normalized
C.
both orthogonal and normalized
D.
unnormalized
Answer: _________
Question 5:
The system for which energy (E) increases quadratically with the quantum number (n) is
A.
particle in a one-dimensional box
B.
hydrogen atom
C.
one-dimensional harmonic oscillator
D.
rigid rotor
Answer: _________
Question 6:
The uncertainty in the momentum (Δp x ) of the particle in its lowest energy state is
A.
$$Delta {p_x} = 0$$
B.
$$Delta {p_x} = frac{h}{a}$$
C.
$$Delta {p_x} = frac{h}{{2a}}$$
D.
$$Delta {p_x} = frac{h}{{2pi a}}$$
Answer: _________
Question 7:
A particle is confined to a one-dimensional box of length 1 mm. If the length is changed by 10 -9 m, the % change in the ground state energy is
A.
2 × 10 -4
B.
2 × 10 -7
C.
2 × 10 -2
D.
0
Answer: _________
Question 8:
The function e ax 2 (a > 0) is not an acceptable wave function for bound system because
A.
it is not continuous
B.
it is multi-valued
C.
it is not normalizable
D.
All of these
Answer: _________
Question 9:
The wave function for a harmonic oscillator described $$Nxexp left( { - frac{{a{x^2}}}{2}}
ight)$$ xa0 has
A.
one maximum only
B.
one maximum, one minimum only
C.
two maxima, one minimum only
D.
two maxima, two minima only
Answer: _________
Question 10:
Consider a particle of mass m moving in a one-dimensional box under the potential V = 0 for 0 ≤ x ≤ a and V = $$infty $$ outside the box. When the particle is in its lowest energy state, the average momentum (< p x >) of the particle is
A.
$$ < {p_x} > = 0$$
B.
$$ < {p_x} > = frac{h}{a}$$
C.
$$ < {p_x} > = frac{h}{{2a}}$$
D.
$$ < {p_x} > = frac{h}{{2pi a}}$$
Answer: _________
Question 11:
The de-Broglie wavelength for a He atom travelling at 1000 m/s (typical speed at room temperature) is
A.
99.7 × 10 -12 m
B.
199.4 × 10 -12 m
C.
199.4 × 10 -18 m
D.
99 × 10 -6 m
Answer: _________
Question 12:
The vibrational partition function for a molecule which can be described as a simple harmonic oscillator with fundamental frequency $$
u $$ is given by
A.
$$exp left( { - frac{{h
u }}{{{K_B}T}}}
ight)$$
B.
$${left[ {1 - exp left( { - frac{{h
u }}{{{K_B}T}}}
ight)}
ight]^{ - 1}}$$
C.
$$exp left( { - frac{{h
u }}{{{K_B}T}}}
ight){left[ {1 - exp left( { - frac{{h
u }}{{{K_B}T}}}
ight)}
ight]^{ - 1}}$$
D.
$$exp left( { - frac{{h
u }}{{2{K_B}T}}}
ight){left[ {1 - exp left( { - frac{{h
u }}{{{K_B}T}}}
ight)}
ight]^{ - 1}}$$
Answer: _________
Question 13:
The wave function for a quantum mechanical particle in a one-dimensional box of length a is given by $$psi = Asin frac{{pi x}}{a}.$$ xa0 The value of A for a box of length 200 nm is
A.
4 × 10 4 (nm) 2
B.
10$$sqrt 2 $$ (nm) 1/2
C.
$$sqrt 2 $$ /10 (nm) -1/2
D.
0.1 (nm) -1/2
Answer: _________
Question 14:
First order perturbation correction $$Delta varepsilon _n^{left( 1
ight)}$$ xa0to energy level $${varepsilon _n}$$ of a simple harmonic oscillator due to the anharmonicity perturbation $$gamma {x^3}$$xa0is given by
A.
$$Delta varepsilon _n^{left( 1
ight)} = gamma $$
B.
$$Delta varepsilon _n^{left( 1
ight)} = {gamma ^2}$$
C.
$$Delta varepsilon _n^{left( 1
ight)} = {gamma ^{ - 1}}$$
D.
$$Delta varepsilon _n^{left( 1
ight)} = 0$$
Answer: _________
Question 15:
Which one of the following is not a photodetector?
A.
Bolometer
B.
Charge-transfer device
C.
Photomultiplier tube
D.
Silicon diode
Answer: _________
Question 16:
An electron of mass m is confined to a one-dimensional box of length b. If it makes a radiative transition from second excited state to the ground state/the frequency of the photon emitted is
A.
$$frac{{9h}}{{8m{b^2}}}$$
B.
$$frac{{3h}}{{8m{b^2}}}$$
C.
$$frac{h}{{m{b^2}}}$$
D.
$$frac{{2h}}{{8m{b^2}}}$$
Answer: _________
Question 17:
As per the uncertainty principle, $$Delta x cdot Delta p$$ xa0 equals to
A.
$$frac{h}{{2pi }}$$
B.
$$frac{h}{2}$$
C.
$$lambda $$
D.
zero
Answer: _________
Question 18:
In units of $$frac{{{h^2}}}{{8m{l^2}}},$$ xa0the energy difference between levels corresponding to 3 and 2 node eigen functions for a particle of mass m in a one-dimensional box of length $$l$$ is
A.
1
B.
3
C.
5
D.
7
Answer: _________
Question 19:
The velocity of the electron in the hydrogen atom
A.
increases with increasing principal quantum number
B.
decreases with increasing principal quantum number
C.
is uniform for any value of the principal quantum number
D.
first increases and then decreases with principal quantum number
Answer: _________
Question 20:
The zero-point energy of the vibration of 35 CI 2 mimicking a harmonic oscillator with a force constant k = 2293.8 N/m is
A.
10.5 × 10 -21 J
B.
14.8 × 10 -21 J
C.
20.9 × 10 -21 J
D.
20.6 × 10 -21 J
Answer: _________
Answer Key
1:
C
2:
B
3:
B
4:
C
5:
A
6:
B
7:
A
8:
D
9:
B
10:
A
11:
A
Solution: This question is about the de Broglie wavelength , which tells us that even particles like atoms can behave like waves! The de Broglie wavelength (λ) is calculated using the formula: λ = h/mv where: h is Planck's constant (6.626 x 10 -34 Js) m is the mass of the particle (in kg) v is the velocity of the particle (in m/s) First, we need the mass of a Helium (He) atom. A He atom has a mass of approximately 4 amu (atomic mass units). To use the formula, we need to convert this to kilograms. Since 1 amu ≈ 1.66 x 10 -27 kg, the mass of a He atom is roughly 4 * 1.66 x 10 -27 kg = 6.64 x 10 -27 kg. Now, we can plug the values into the de Broglie wavelength formula: λ = (6.626 x 10 -34 Js) / (6.64 x 10 -27 kg * 1000 m/s) After calculating, you will get a wavelength around 99.7 x 10 -12 m.
12:
B
13:
D
14:
D
15:
A
16:
C
17:
B
18:
C
19:
B
20:
B