PYQ Bank

Using uncertainty principle, calculate the size and energy of the ground state hydrogen atom.

A typical atomic radius is about 5 \times 10^{-15}\,m and the energy of \beta-particle emitted from a nucleus is at most of the order of 1\,\mathrm{MeV}. Prove on the basis of uncertainty principle that the electrons are not present in nuclei.

In a nuclear experiment, beams of electrons and protons are moving with velocities c / 10 and c / 20 respectively. Calculate the de Broglie wavelengths of both the particles in metre (c is the speed of light = 3 \times 10^8\text{ m / s}).

An electron is confined to move between two rigid walls separated by 10^{-9}\,\mathrm{m}. Compute the de Broglie wavelengths representing the first three allowed energy states of the electron and the corresponding energies.

Solve the Schrodinger equation for a step potential and calculate the transmission and reflection coefficient for the case when the kinetic energy of the particle E_0 is greater than the potential energy V (i.e., E_0>V).

Calculate the density of states for an electron moving freely inside a metal with the help of quantum mechanical Schrodinger's equation for free particle in a box.

Obtain the solution of one-dimensional free particle Schrödinger equation. Show that it corresponds to plane monochromatic (constant angular frequency) wave.

Consider a one-dimensional harmonic oscillator potential V(x) = \frac{1}{2} m \omega_0^2 x^2 where m represents the mass of a particle, \omega_0 is the classical frequency of the oscillator and x is the displacement from equilibrium position. Obtain the solution of the corresponding Schrödinger equation. Using the normalization condition on the solution, show that the ground-state wave function has the form \psi_0(x) = \left( \frac{\alpha}{\pi} \right)^{1/2} e^{-\frac{1}{2} \alpha^2 x^2} \alpha = \left( \frac{m \omega_0}{\hbar} \right)^{1/2}

Find the energy, momentum and wavelength of photon emitted by a hydrogen atom making a direct transition from an excited state with n = 10 to the ground state. Also find the recoil speed of the hydrogen atom in this process.

Recall the Pauli matrix representation of two-state system of electron, \sigma_x, \sigma_y, \sigma_z, and show that they do not commute.

Write down the matrix representation of the three Pauli matrices \sigma_x, \sigma_y and \sigma_z. Prove that these matrices satisfy the following identities:

(i) [\sigma_x,\sigma_y]=2i\sigma_z

(ii) [\sigma^2\cdot\sigma_x]=0

(iii) (\vec{\sigma}\cdot\vec{A})(\vec{\sigma}\cdot\vec{B})=\vec{A}\cdot\vec{B}+i\vec{\sigma}\cdot(\vec{A}\times\vec{B}) if \vec{A} and \vec{B} commute with Pauli matrices.

Establish that: hc = 1240\ \mathrm{eV\,nm} = 1240\ \mathrm{MeV\,fm}

Determine the values of the total angular momentum for a 3d electron.

Deduce the commutation relations between the components of angular momentum operator \mathbf{L} [L_x, L_y] = i \hbar L_z [L_y, L_z] = i \hbar L_x [L_z, L_x] = i \hbar L_y using the commutation relations [x, p_x] = [y, p_y] = [z, p_z] = i \hbar

Find the de Broglie wavelength of a neutron moving with a kinetic energy of 500\text{ eV}. (1\text{ eV} = 1\cdot 602 \times 10^{-19}\text{ J})

Write the time independent Schrödinger equation for a bouncing ball.

Solve the Schrödinger equation for a particle in a three-dimensional rectangular potential barrier. Explain the terms degenerate and non-degenerate states in this context.

A particle trapped in an infinitely deep square well of width a has a wave function \psi=\left(\frac{2}{a}\right)^{1/2}\sin\left(\frac{\pi x}{a}\right). The walls are suddenly separated by infinite distance. Find the probability of the particle having momentum between p and p+dp.

Give an account of Heisenberg's Uncertainty principle. Outline an idealised experiment to bring out its significance.

What is Zeeman effect? How can it be understood on the basis of quantum mechanics?