Write down Schrodinger's equation in one-dimension and explain the significance of the eigenvalues and eigenfunction of this equation. Solve the Schrodinger's equation if the potential function V is given as V(x) = V_e(\text{a constant}) for |x| < a/2 = 0 for |x| > a/2 State the boundary conditions you use along with their justification (In the solution consider particles incident from one side only.) If he total energy F < V_0 show that there is an important difference with classical mechanics. Explain with the help of the uncertainty principle that E < V_0 does not mean that the kinetic energy is negative.

An electron moves with a speed of 10^{3}\text{ m/sec}, accurate to 0.008\%. Find the accuracy with which the position of the electron can be located.

Derive Schrodinger equation for a linear harmonic oscillator, and determine its eigenvalue and eigenfunctions. Discuss the significance zero point energy.

State and explain Einstein's photoelectric emission equation, Distinguish between extrinsic and intrinsic photo electric effects. What is "quantum field"? On what factors does the quantum yield depend?

State and explain Heisenberg uncertainty principle using situations where one of the parameters is (a) liner distance x (b) angle \phi (c) energy E. Prove that according to the principal the presence of an electron within the atomic nucleus is not possible.

Define Compton wavelength for an electron. Calculate in electron-volts the photon energy corresponding to radiations of the Compton wavelength. What light does the Compton effect throw on the nature of X-rays?

Compare the energy of a photon with that of a neutron when both are associated with wavelength of 1\text{\AA}.

Find the Fermi energy in eV of electrons in sodium(_{11}^{23}\text{C Na}), given its density to be 0.97\text{ g cm}^{-3}. Treat the electrons as a free gas.

The stopping potential for electrons emitted from a metal, due to photoelectric effect, is found to be 1\text{ volt} for light of 2,500\text{ Angstrom units}. Calculate the work function of the metal in electron volts.

Construct solutions of Schrodinger equation for a particle moving the potential \begin{aligned} V(x) &= 0 & \text{for } x < 0 \\\\ &= V_0 & \text{for } x > 0 \end{aligned} Deduce the reflection coefficient for particles incident from left to right (i.e. from negative to positive x regions) with energy E > V_0. Do you expect any reflection for particles of same energy incident from right to left? Explain qualitatively.

What is the importance of study of deuteron? Obtain the solution of Schrodinger equation for ground state of deuteron and show that deuteron is a loosely bound system.

$. Interpret your result.