PYQ Bank

Derive an expression for the Compton shift (\Delta\lambda) and show that it is independent of the wavelength of incident radiation. Establish parallelism between Compton and Raman scattering

An infinite well potential is V(x) = \begin{cases} 0, & -a/2 < x < a/2 \\ \infty, & x < -a/2, \quad x > a/2 \end{cases} Solve the time independent Schrodinger equation and find the closed form expressions for the eigenvalues and eigenfunction of potential. Give a schematic representation of the first three energy levels and corresponding wave functions and probability distribution functions. Show that the zero-point energy is in accord with the uncertainty principle.

Write down the time independent Schrodinger equation for the x < 0 and x > 0 in case of a step potential V(x) = \begin{cases} 0, & x < 0 \\ V_o, & x > 0 \end{cases} Discuss the solutions thus obtained for the case E > V_o. Obtain the following relations. |R|^2 = \left(\frac{1-\mu}{1+\mu}\right)^2 \quad \text{and} \quad \frac{k}{k_o}|T|^2 = \mu \left(\frac{2}{1+\mu}\right)^2 There \mu = \sqrt{1 - \frac{V_o}{E}}, R and T are reflection and transmission coefficients respectively. Other symbols have their usual meaning. Plot the behaviour of |R|^2 and k/k_o |T|^2 with \mu.

Using Heisenberg Uncertainty principle, find the ground state energy and Bohar radius of hydrogen atom.

Calculate the velocity and direction of recoil electron for back scattered X-ray photon of \text{Mo } K_\alpha of 0.707\ \text{\AA} in Compton effect.

An electron moving with energy E encounters one dimensional potential step as given below: V(x) = 0 \quad x < 0 \qquad V(x) = V_0 \quad x > 0 (a) Suppose the electron has the energy E > V_0 and is incident from (-x) direction, find the normalized wave function so corresponds to unit incident flux. (b) Solve the above problem for the case E < V_0 and discuss the significance of the result with the help of a suitable example.

An electron is confined in a one dimensional box 1 \text{\AA} width. Draw the energy level diagram upto three energy stats of also draw the corresponding normalized eigen function. Derive the expressions for eigen functions and eigen values used in the calculations. Show that eigen functions are othogonal.

Using Heisenberg Uncertainty principle, calculate the mass of meson being exchanged between nucleons inside the nucleus, taking the range of nuclear force as 1.7 F. Distinguish between pion and muon.

Describe how Davisson-Germer electron diffraction experiment confirms the de Broglie hypothesis of matter wave.

Find the energy Ligenvalues of a one dimensional harmonic oscillator whose Hamiltonian is given by H = \frac{P^2}{2m} + \frac{1}{2} k x^2

Using de Broglie's relation, deduce an expression for energy values of a particle enclosed in a one-dimensional box. 'Free electrons' in a long-chain molecule may be treated as particles in a one-dimensional box. From a long-chain molecule of length 60\text{ \AA} deduce the lowest three energy state values in eV units.

What values of J quantum number may be observed in normal atoms of alkalis and alkaline earths ? Which of the spectral lines in alkaline earth would show normal Zeeman effect and why?

With incident X-rays of wavelength 1.618\text{ \AA} Compton scattering is observed at 90^\circ from the incident beam. Calculate. (i) the wavelength of Compton radiation, and (ii) the angle at which the recoil electron may be observed. In what state should the electrons be in Compton scattering?

Using Schrodinger's equation show that a free particle in a box can have only discrete energy values.

Derive an expression for the magnetic moment of an electron in an atom taking into account electron-spin. Explain anomalous Zeeman effect by considering the above magnetic moment.

State the important characteristics of photoelectric effect and indicate how they are in conflict with classical ideas but are explained on the basis of quantum theory.

Draw energy diagram for transverse Zeeman effect for D line of sodium 3^2P_{3/2} \rightarrow 3^2S_{1/2}

Set up Schrodinger's equation for a free particle confined in a cubical box and find the energy eigenvalues. What form will Pauli principle take for electrons in such a box? Deduce the zero point energy if the length of the box be 10^{-10}\text{ metre} and there be 10 electrons in it. (The coulomb interaction maybe disregarded.)

A photon of energy 10\text{ eV} is scattered by an electron having an initial velocity of 1000\text{ km s}^{-1}. After the scattering the electron velocity is reduced to 200\text{ km s}^{-1}. Deduce the energy of the scattered photon.

What are de Broglie waves? State the uncertainty principle and explain its relation with de Broglie waves by considering a free particle.