List of top Physics Questions on Quantum Mechanics

A force of 49 N acts tangentially at the highest point of a sphere (solid of mass 20 kg) kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is:

Match the LIST-I with LIST-II

LIST-I (Energy of a particle in a box of length L)LIST-II (Degeneracy of the states)
A.\( \frac{14h^2}{8mL^2} \)I.1
B.\( \frac{11h^2}{8mL^2} \)II.3
C.\( \frac{3h^2}{8mL^2} \)III.6


Choose the correct answer from the options given below:

Fermions have spin value equal to:

A particle of mass \(m\) is in an infinite square potential of length \(L\). The wave function is superimposed state of the first two energy eigenstates, given by:

\[ \Psi(x) = \sqrt{\frac{1}{3}} \Psi_{n=1}(x) + \sqrt{\frac{2}{3}} \Psi_{n=2}(x) \]
Identify the correct statements: 
A. \( \langle p \rangle = 0 \) 
B. \( \Delta p = \frac{\sqrt{3}h}{2L} \) 
C. \( \langle E \rangle = \frac{3h^2}{8mL^2} \) 
D. \( \Delta x = 0 \) 
Choose the correct answer from the options given below:

The de-Broglie wavelength of an electron moving with a velocity of \(10^7\) m/s is:
The Schrodinger wave equation is:


Consider two blocks A and B of masses \( m_1 = 10 \) kg and \( m_2 = 5 \) kg that are placed on a frictionless table. The block A moves with a constant speed \( v = 3 \) m/s towards the block B kept at rest. A spring with spring constant \( k = 3000 \) N/m is attached with the block B as shown in the figure. After the collision, suppose that the blocks A and B, along with the spring in constant compression state, move together, then the compression in the spring is, (Neglect the mass of the spring)

Let \( \lambda_e \), \( \lambda_p \), and \( \lambda_d \) be the wavelengths associated with an electron, a proton, and a deuteron, all moving with the same speed. Then the correct relation between them is:
A system of three non-identical spin-\(\frac{1}{2}\) particles has the Hamiltonian
\[ H = \frac{A \hbar^2}{2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3, \] where \( \vec{S}_1, \vec{S}_2, \vec{S}_3 \) are the spin operators of particles labelled 1, 2, and 3 respectively, and \( A \) is a constant with appropriate dimensions. The set of possible energy eigenvalues of the system is:
Which of the following option(s) is/are correct for photons?
The uncertainty in the velocities of two particles \( A \) and \( B \) are \( 0.03 \) and \( 0.01 \, \text{m/s} \), respectively. The mass of \( B \) is four times the mass of \( A \). The ratio of uncertainties in their positions is:
A satellite is launched into a circular orbit of radius \( R \) around the earth. A second satellite is launched into an orbit of radius \( 1.03R \). The time period of revolution of the second satellite is larger than the first one approximately by:
An electron with mass \( m \) and charge \( q \) is in the spin up state \[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \] at time \( t = 0 \). A constant magnetic field is applied along the \( y \)-axis, \( \mathbf{B} = B_0 \hat{j} \), where \( B_0 \) is a constant. The Hamiltonian of the system is \[ H = -\hbar \omega \sigma_y, \quad \text{where} \quad \omega = \frac{q B_0}{2m} < 0 \quad \text{and} \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}. \]
The minimum time after which the electron will be in the spin down state along the \( x \)-axis, i.e., \[ \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \] is:
A system of five identical, non-interacting particles with mass \( m \) and spin \( \frac{3}{2} \) is confined to a one-dimensional potential well of length \( L \). If the lowest energy of the system is \[ E_{{min}} = \frac{N \pi^2 \hbar^2}{2m L^2}, \] the value of \( N \) (in integer) is:
Let \( |m\rangle \) and \( |n\rangle \) denote the energy eigenstates of a one-dimensional simple harmonic oscillator. The position and momentum operators are \( \hat{X} \) and \( \hat{P} \), respectively. The matrix element \( \langle m | \hat{P}\hat{X} | n \rangle \) is non-zero when:

Two projectile protons \( P_1 \) and \( P_2 \), both with spin up (along the \( +z \)-direction), are scattered from another fixed target proton \( T \) with spin up at rest in the \( xy \)-plane, as shown in the figure. They scatter one at a time. The nuclear interaction potential between both the projectiles and the target proton is \( \hat{\lambda} \vec{L} \cdot \vec{S} \), where \( \vec{L} \) is the orbital angular momentum of the system with respect to the target, \( \vec{S} \) is the spin angular momentum of the system, and \( \lambda \) is a negative constant in appropriate units. Which one of the following is correct?


 

A particle of mass \( m \) is in a potential \( V(x) = \frac{1}{2} m \omega^2 x^2 \) for \( x > 0 \), and \( V(x) = \infty \) for \( x \leq 0 \), where \( \omega \) is the angular frequency. The ratio of the energies corresponding to the lowest energy level to the next higher level is:
A particle is scattered from a potential \( V_{\vec{r}} = g \delta^3(\vec{r}) \), where \( g \) is a positive constant. Using the first Born approximation, the angular \((\theta, \phi)\) dependence of the differential scattering cross section \( \frac{d\sigma}{d\Omega} \) is:
A particle is released from height \(s\) above the surface of the earth. At a certain height its K.E is 3 times of P.E. The height from the surface of the earth and the speed of the particle at the instant are respectively:
An electron projected perpendicular to a uniform magnetic field \( B \) moves in a circle. If Bohr’s quantization is applicable, then the radius of the electronic orbit in the first excited state is:
An electron projected perpendicular to a uniform magnetic field \( B \) moves in a circle. If Bohr’s quantization is applicable, then the radius of the electronic orbit in the first excited state is:
Monochromatic light of wavelength λ = 4770 ˚A is incident separately on the surfaces of four different metals A, B, C and D. The work functions of A, B, C, and D are 4.2 eV, 3.7 eV,3.2 eV and 2.3 eV, respectively. The metal/ metals from which electrons will be emitted is/are:
A transmitter of power 10 kW emits radio waves of wavelength 500 m. The number of photons emitted per second by the transmitter is of the order of:
An electron of mass \( m \) with initial velocity \( \vec{v} = v_0 \hat{i} \) (\( v_0>0 \)) enters in an electric field \( \vec{E} = -E_0 \hat{i} \) (\( E_0 \) is constant \(>0 \)) at \( t = 0 \). If \( \lambda \) is its de-Broglie wavelength initially, then the de-Broglie wavelength after time \( t \) is:
A \( \mu \)-meson of charge \( e \), mass \( 208 \, m_e \) moves in a circular orbit around a heavy nucleus having charge \( +3e \). The quantum state \( n \) for which the radius of the orbit is the same as that of the first Bohr orbit for the hydrogen atom is (approximately):