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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 φ_{1} and φ_{2} 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 φ_{1} = 1.5eV. Find φ_{2}. (in eV)

Solution:

Solution=

eV_{s1}=hv- Φ_{1}

eV_{s2}=hv- Φ_{2}

eV_{s1}= eV_{s2}

eV_{s1}/eV_{s2 }= hv- Φ_{2}/ hv- Φ_{1}=1/2

2hv-2Φ_{2 }= 1hv- Φ_{1}

2Φ_{2 }= hv+ Φ_{1}

2Φ_{2 }= 5+1.5

2Φ_{2 }= 6.5eV

Φ_{2 }= 3.25eV

The correct answer would be 3.25eV

*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 A^{2} = 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 E_{2}. The lowest would be E_{1}.

E_{n} = (const)n^{2}

Since E_{2} = 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 V_{0} = 16eV

With * a* = 0.5 × 10

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 1**Å**.

Solution:

The eigenvalue of energy

when the particle is in the least energy state (* n* = 1), the energy

with

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 a_{0} is the Bohr radius and r is the distance from the origin. The most probable value of r in units of a_{0}.

Solution:

For a hydrogen atom,

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

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

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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