Problem set 97

1. In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?
1. Each normal coordinate has an eigen-frequency associated with it
2. The normal coordinates are orthogonal to one another
3. The normal coordinates are all independent
4. The potential energy of the system is a sum of squares of the normal coordinates with constant coefficients

The normal coordinates are orthogonal to one another

Hence, answer is (B)

2. The eigenvalues of a Hermitian matrix are all
1. real
2. imaginary
3. of modulus one
4. real and positive

real

Hence, answer is (A)

3. Given the following table,
   Group-I
   P. Stem-Gerlach experiment
   Q. Zeeman effect
   R. Frank-Hertz experiment
   S. Davisson-Germer experiment
   Group-II
   1. Wave nature of particles
   2. Quantization of energy of electrons in the atoms
   3. Existence of electron spin
   4. Space quantization of angular momentum
   Which one of the following correctly matches the experiments from Group-I to their inferences in Group-II ?
1. P-2, Q-3, R-4, S-1
2. P-1, Q-3, R-2, S-4
3. P-3, Q-4, R-2, S-1
4. P-2, Q-1, R-4, S-3

P. Stem-Gerlach experiment : Existence of electron spin
Q. Zeeman effect:Space quantization of angular momentum
R. Frank-Hertz experiment:Quantization of energy of electrons in the atoms
S. Davisson-Germer experiment:Wave nature of particles


Hence, answer is (C)

4. Which of the following atoms cannot exhibit Bose-Einstein condensation, even in principle?
1. $^{1}H_1$
2. $^{4}H_2$
3. $^{23}Na_{11}$
4. $^{30}K_{19}$
For Bose-Einstein condensation:
Number of electron + number of proton + number of neutron = Even
For $^{30}K_{19}$
Number of proton = 19, Number of electron = 19, Number of neutron = 11.
19 + 19 + 11 = 49 this is odd.
So it will not exhibit Bose-Einstein condensation.

Hence, answer is (D)

5. Two gases separated by an impermeable but movable partition are allowed to freely exchange energy. At equilibrium, the two sides will have the same
1. pressure and temperature.
2. volume and temperature.
3. pressure and volume.
4. volume and energy.

pressure and temperature

Hence, answer is (A)