If $$\sigma $$ is the total cross-section and f(θ), θ being the angle of scattering, is the scattering amplitude for a quantum mechanical elastic scattering by a spherically symmetric potential, then which of the following is true? Note that k is the magnitude of the wave vector along the $${{\bf{\hat z}}}$$ direction.

Rock mass parameters for a shale rock, m_b, s, and a of the Hoek and Brown failure criterion as given in the equation are 2.4, 0.01 and 0.52 respectively. The uni-axial compressive strength (σ_Ci) of intact sh!le is 50 MPa.
$${\sigma _1} = {\sigma _3} + {\sigma _{ci}}{\left( {{m_b}\frac{{{\sigma _3}}}{{{\sigma _{ci}}}} + S} \right)^a}$$
The global rock mass strength of shale in MPa is

A

1.2

B

10.29

C

20.83

D

50,00

Which of the following statements concerning the quantum numbers is correct? (i) The Angular quantum number determines the three-dimensional shape of the orbital. (ii) The Principal quantum number determines the orientation and energy of the orbital. (iii) The Magnetic quantum number determines the size of the orbital. (iv) Spin quantum number of an electron determines the orientation of the spin of the electron relative to the chosen axis.

A

ii and iv

B

i and iv

C

i and iii

D

ii and iii

How many quantum numbers describe the spherically symmetric potential field of an electron?

A

One

B

Two

C

Three

D

Four

A force vector is along 4i – 4k direction and has a magnitude 100N and another force vector is along 4i +2j -4k and has a magnitude of 120N. What is the resultant of both forces?

A

80i + 40j – 80k N

B

80i – 40j – 80k N

C

151i + 40j – 80k N

D

151i+ 40j – 151k N

A particle of mass m is confined in an infinite potential well \[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\text{if }}0 It is subjected to a perturbing potential $${V_P}\left( x \right) = {V_0}\sin \left( {\frac{{2\pi x}}{L}} \right)$$ within the well. Let E⁽¹⁾ and E⁽²⁾ be the corrections to the ground state energy in the first and second order in V₀.
Quantum Mechanics mcq question image

Which of the following is correct?

A

[

A. E⁽¹⁾ = 0; E⁽²⁾

B. E⁽¹⁾ > 0; E⁽²⁾ = 0

B

E⁽¹⁾ = 0; E⁽²⁾ depends on the sign of V₀

C

E⁽¹⁾ (2)

A particle of mass m is confined in the potential
Quantum Mechanics mcq question image

\[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{2}m{\omega ^2}{x^2},}&{{\text{for }}x Let the wave function of the particle be given by $$\psi \left( x \right) = - \frac{1}{{\sqrt 5 }}{\psi _0} + \frac{2}{{\sqrt 5 }}{\psi _1}$$
where $${\psi _0}$$ and $${\psi _1}$$ are the eigen functions of the ground state and the first excited slate respectively. The expectation value of the energy is

A

$$\frac{{31}}{{10}}\hbar \omega $$

B

$$\frac{{25}}{{10}}\hbar \omega $$

C

$$\frac{{13}}{{10}}\hbar \omega $$

D

$$\frac{{11}}{{10}}\hbar \omega $$

There are only three bound states for a particle of mass m in a one-dimensional potential well of the form shown in the figure. The depth V₀ of the potential satisfies
Quantum Mechanics mcq question image

A

[

A. $$\frac{{2{\pi ^2}{\hbar ^2}}}{{m{a^2}}}

B. $$\frac{{{\pi ^2}{\hbar ^2}}}{{m{a^2}}}

B

$$\frac{{2{\pi ^2}{\hbar ^2}}}{{m{a^2}}}

C

$$\frac{{2{\pi ^2}{\hbar ^2}}}{{m{a^2}}}

Consider a system described by ẋ = Ax + Bu y = Cx + Du The system is completely output controllable if and only if Where: x = State vector (n-vector) u = Control vector (r-vector) y = Output vector (m-vector) A = n × n matrix B = n × r matrix C = m × n matrix D = m × r matrix

A

The matrix [ CB | CBA | CB2A| … | CBn-1 A | D ] is of rank n

B

The matrix [ CB | CAB | CA2B | … | CAn-1 B | D ] is of rank m

C

The matrix [ BC | BAC | BA2C | … | BAn-1 C | D ] is of rank m

D

The matrix [ BC | ABC | CA2 B | … | CAn-1 B | D ] is of rank n

A particle is incident with a constant energy E on a one-dimensional potential barrier as shown in the figure.
Quantum Mechanics mcq question image

The wave functions in regions I and II are respectively

A

decaying, oscillatory

B

oscillatory, oscillatory

C

oscillatory, decaying

D

decaying, decaying

A vector quantity has both magnitude and direction, whereas a scalar quantity has only magnitude and no direction. Which of the following is a vector quantity?

A

Work

B

Speed

C

Displacement

D

Energy

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