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QUESTION: 1

Consider a two state system with normalized energy eigen state ψ_{1} & ψ_{2} and energy E_{1} < E_{2} what is the possible range for the expectation value of on an orbitrary linear combination of two state

Solution:

(H) __>__ E_{1} ......(1)

similarly

......(2)

form equation (1) & (2)

QUESTION: 2

Suppose a wave function and an operator is given by is given by

Solution:

We have operator,

Now, my operator will be

∴

∴

∴

QUESTION: 3

Consider the following Stem-Gerlach apparatus incoming beam consist of electron 2/3 of them having spin and other 1/3 have spin in the z-direction

what fraction of the incident beam are detected in the up and down outputs of the apparatus

Solution:

*Answer can only contain numeric values

QUESTION: 4

Consider a system whose initial state at r = 0 is given in term of a complete and orthonormal set of three vectors as follows the probability of find tlie system at tune t in state is _______ (upto two decunalplaces)

Solution:

We have wave function at t = 0

Therefore, the wave function at t = t is given by

(where, E_{1}, E_{2} and E_{3}, are energy of difference states)

The probability of finding the system at time t in state

*Answer can only contain numeric values

QUESTION: 5

A proton is confined in an infinite square well width 10 fin. calculate the wavelength of photon emitted when photon undergoes a transition from fixed state (n = 2) to ground state (n = 1) _____ (fm)

(h = 6 627 x 10^{-34} Js, m = 1.672 x 10^{-27} kg) (answer should be an integer).

Solution:

The energy of the particle in the box of width L is given by

λ = 201 fm

*Answer can only contain numeric values

QUESTION: 6

Consider a tliree dimensional harmonic oscillator with Hamiltonian

The number of distinct eigenstates with energy eigenvalue 5/2 ℏω is_________

(Answer should be an integer).

Solution:

Energy expression for 3-D harmonic oscillator,

The degneracyofnth state of an isolated 3D harmonic oscillator is given by

QUESTION: 7

Consider an operator for a system of total angular momentum j = 1 then winch of the following state having non zero uncertainty

Solution:

Operator J_{x} in matrix form for J = 1

The eigenvalue equation is given by

Consider is the eigen function of J_{x} corresponding the eigen value λ = 0

And according normalization condition,

⇒

∴

∴

Similarly the eigen function corresponding to others two eigen values are

As are the eigen state of operator J_{x}.

Therefore, uncertainty of J_{x} will be zero for these three states and is not an eigen states of J_{x}.

So, uncertainty of J_{x} will be non-zero for this state.

QUESTION: 8

In the simple harmonic oscillator

what are the condition on m and m' for to be non-zero

Solution:

For non zero of

⇒ m = n ± 1

And

QUESTION: 9

Find the differential cross-section for die scattering of slow (low velocity) particle from a spherical delta poten- tial V (r) = V_{0} δ (r - a)

Solution:

In case, the incident particle have low velocities only S-waves, L = 0 contributes in the scattering

Therefore, scattering amplitude is given by

Differential cross-section

*Answer can only contain numeric values

QUESTION: 10

For case of n = 2, ℓ = 1, m = 0 the value of r at which the radial probability density of the hydrogen atom reaches its maximum is ____ a_{0}. (answer should be an integer)

Solution:

Tlie probability density is given by

For the maximum of tlie probability density

*Answer can only contain numeric values

QUESTION: 11

Consider a system winch is mtially in the state was measured with value -ℏ, the probability is_______(upto one decimal place)

Solution:

for eigenvalue m_{L} = -1, eigenfunction probability of measuring L_{z} in state is

*Answer can only contain numeric values

QUESTION: 12

An electron is confined in the ground state of a one dimensional harmonic oscillator such that energy required to excite to its first excited state is _____ (MeV)

Solution:

The virial theorem states that

for ground state

for harmonic oscillator

Energy required to excite the electron to its first excited state

QUESTION: 13

A particle of mass m coming in from the left with energy E > 0, encounters barrier potential

The wave function is given by

The constant A and B satisfy which one of the following relation?

Solution:

The barrier potential is given as

The wave function is given as

Hie quantity B in equation (i) represent the amplitude ot reflected ray in medium 1 and A in equation (ii) represent the amplitude of transmitted ray in medium III

Hence, Reflectivity R = |A|^{2} and transmitivity T = | B |^{2}

smce R + T = 1

⇒

QUESTION: 14

Calculate the width of the probability density distribution for r(i.e Br) for hydrogen atom for the state

Solution:

Tlie width of the probability distribution is given by

width ot probability distribution

QUESTION: 15

At t = o, a state is given by

Where are ortlionomial stationary states of energy E_{l} & E_{2} respectively with E_{2} > E_{l} what is the shortest tune T > 0 for w hich is orthogonal to

Solution:

At t = 0, the wave function is given by

Therefore, the wave function at later time t = T is given by

These two states ortliogonal at time T

For shortest time m = 0

QUESTION: 16

If the state of a particle moving in one dimensional harmonic oscillator is given by

Where represent the normalized n^{th} energy eigenstate find the expectation value of number operator

Solution:

QUESTION: 17

A particle of mass m moves in a one dimensional potential box

Consider the V_{0} part as perturbation, using first order perturbation method calculate the energy of ground state.

Solution:

Tlie energies and wavefunction of a particle in flat box of length 6a

energy correction to first order perturbation

*Answer can only contain numeric values

QUESTION: 18

A spin state precesses in a magnetic field same way as the classical magnetic dipole precesses in magnetic field with lasmor frequency given by consider the Hamiltonian Larrnor frequency is (in tem is of ω_{0})

(answer should b e an mteger)

Solution:

From equation (i) and (ii)

*Answer can only contain numeric values

QUESTION: 19

Consider a system of four non-interacting identical spin 1/2 particles that are in same state and confined to move in a one-diniension infinite potential well of length a: V(x) = 0 for 0 < x < a and V(x) = ∞ forotlier values of x. The ground state energy of the system in units of is (answer should be an integers).

Solution:

The energy of the particle in one dimensional infinite potential box is given by

4 particles having same state

ground state energy'of system is

QUESTION: 20

Consider a system whose intial state and hamiltonian are given by

find the total energy of a system

Solution:

eigenvalues of energy

λ = -5,3, 5; E_{1 }= - 5, E_{2} = 3, E_{3} = 5

and their correspondmg eigenvectors

expanding in terms of eigenvectors.

Total energy of the system

Second method:

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