Postulates Of Quantum Mechanics NAT


10 Questions MCQ Test Modern Physics for IIT JAM | Postulates Of Quantum Mechanics NAT


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*Answer can only contain numeric values
QUESTION: 1

The wave function  where A and B are real constant, is a normalised eigenfunction of the Schrödinger equation for a particle of mass M and energy E in a one dimensional potential V(x) such that V(x) = 0 at x = 0. Find the value for V in units of 


Solution:

Applying time independent Schrödinger equation

Substituting (2) in (1)

Putting V(x = 0) = 0  in (3)

The correct answer is: 0.5

*Answer can only contain numeric values
QUESTION: 2

In the spectrum of hydrogen, what is the ratio of the longest wavelength in the Lymann series to the longest is Balmer series ?


Solution:

Energy emitted due to transition of electron -

Wherin

Longest wavelength in balmer series is corresponding to transition between 

Lets say it is 

Longest wavelength in lymen series is corresponding to transition between  2 ---> 1

Lets say it is 

=0.1852

*Answer can only contain numeric values
QUESTION: 3

In a photoelectric effect set up two metals with work function  φ1 and φ2 are used. For the same value of incident frequency, the stopping potential for metal 1 is twice that for metal 2. If the incident energy is 5eV and the φ1 = 1.5eV.  Find  φ2. (in eV)


Solution:

Solution= 

eVs1=hv- Φ1

eVs2=hv- Φ2

eVs1= eVs2

eVs1/eVs2= hv- Φ2/ hv- Φ1=1/2

2hv-2Φ2=1hv- Φ1

Φ2=hv+ Φ1

=5+1.5

Φ2=6.5eV

The correct answer would be 6.5eV

*Answer can only contain numeric values
QUESTION: 4

A particle is in an infinite square well potential with walls at x = 0 and x = L. If the particle is in the state    where A is a constant, what is the probability that the particle is between x = L/3 and  x = 2L/3 upto 2 decimal places.


Solution:



Now A2 = 2/L = from normalization of ψ
∴    Probability = 1/3 = 0.33
The correct answer is: 0.33

*Answer can only contain numeric values
QUESTION: 5

The figure shows one of the possible energy eigenfunction ψ(x) for a particle bouncing freely back and forth along the x-axis between impenetrable walls located at x ± a The potential energy equals zero for |x| < a. If the energy of the particle is 2eV when it is in the quantum state associated with its eigen function, what is its energy when it is in the quantum state of lowest energy (in eV)?


Solution:

The number of nodes in a wavefunction determine the energy level. In this case there is just one node. Thus, this is E2. The lowest would be E1.
En = (const)n2
Since E2 = 2 = (constant) 4

The correct answer is: 0.5

*Answer can only contain numeric values
QUESTION: 6

What is the probability of finding the particle in classically forbidden region in ground state of simple harmonic oscillator


Solution:

The correct answer is: 0.18

*Answer can only contain numeric values
QUESTION: 7

An electron in a metal encounters a barrier layer of height 6eV and thickness 0.5 nm. If the electron energy of 5eV, what is the probability of tunneling through the barrier?


Solution:

We know,

With  E = 5eV  and  V0 = 16eV

With  a = 0.5 × 10–9 m,  m = 9.1 × 10–31 kg,  V0 = 6eV,  E = 5V   and 



Hence the percentage of probability of tunneling is 1.33.
The correct answer is: 1.33

*Answer can only contain numeric values
QUESTION: 8

The energy of an electron moving in one dimension in an infinitely high potential box of width 1Å.


Solution:

The eigenvalue of energy


when the particle is in the least energy state (n = 1), the energy
with h = 6.62 × 10–34 Jsm = 9.1 × 10–31 kg and a = 1 × 10–10 m

After calculating,

The correct answer is: 37.62 eV

*Answer can only contain numeric values
QUESTION: 9

The solution of the schrödinger equation for the ground state of hydrogen is  where a0 is the Bohr radius and r is the distance from the origin. The most probable value of r  in units of a0.


Solution:

For a hydrogen atom,

Now, for most probable r. We need to minimize the probability w.r.t. r.

Putting (2) in (1)

The correct answer is: 1

*Answer can only contain numeric values
QUESTION: 10

The wave function of a particle is given by ψ(x) = Ce-ax,  -∞ < x < + ∞ where C and a are the some constants. The probability of finding the particle in the region  0 < x < ∞


Solution:

The probability of finding the particle in the region  is 

Applying normalising condition i.e.,

The correct answer is: 0.5

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