Show that for free electron gas, the density of states in three dimensions (3D) varies as E^{1/2}, and this dependence changes to E^0 for 2D (quantum well), E^{-1/2} for 1D (quantum wire) and \delta function for 0D (quantum dot).

Which of the following functions is/are acceptable solution(s) of the Schrodinger equation?

(i) \psi(x)=Ae^{-ikx}+Be^{ikx}

(ii) \psi(x)=Ae^{-kx}+Be^{kx}

(iii) \psi(x)=A\sin 3kx+B\cos 5kx

(iv) \psi(x)=A\sin 3kx+B\sin 5kx

(v) \psi(x)=A\tan kx Explain your answer.

The wave function of a particle is given as \psi(x)=\frac{1}{\sqrt{a}}e^{-\lvert x\rvert/a}. Find the probability of locating the particle in the range -a\leq x\leq a.

Explain why the square of the angular momentum (L^2) and only one of the components (L_x,L_y,L_z) of L are regarded as constants of motion.

Draw a schematic diagram of the single particle energy levels in a shell model including the effect of spin-orbit coupling. Show how it explains magic numbers in nuclei. Give two examples to show how this scheme predicts the spins and parities of odd A nuclei.

Using Schrodinger equation, obtain the eigenfunctions and eigenvalues of energy for a 1- dimensional harmonic oscillator. Sketch the profiles of eigenfunctions for first three energy states.

Evaluate the most probable distance of the electron from nucleus of a hydrogen atom in its 2p state. What is the probability of finding the electron at this distance?

Estimate the de Broglie wavelength of the electron orbiting in the first excited state of the hydrogen atom.

A beam 4.0 keV electrons from a source is incident on a target 50.0 cm away. Find the radius of the electron beam spot due to Heisenberg’s uncertainty principle.

Calculate the lowest energy of an electron confined to move in a 1-dimensional potential well of width 10\,\mathrm{nm}.

Calculate the probability of transmission of an electron of 1.0\,\mathrm{eV} energy through a potential barrier of 4.0\,\mathrm{eV} and 0.1\,\mathrm{nm} width.

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.

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.

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.

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

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.

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.

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.