(i) Quantum theoretical methods have established themselves to explain natural phenomena in contemporary Physics. Comment. (ii) Formulate the axioms of quantum theory. (iii) State Born's condition on a wave function. What is its physical meaning?
(i) Give briefly an account on the important historical developments to establish the concept of electron spin. (ii) Write down the eigenvalue and spin state of an electron for the spin operator \hat{S}. (i) Show that the spin states of electron are orthogonal to each other. (ii) State the spin angular momentum commutation relations. (iii) Give the explicit forms of all the Pauli spin matrices.
(i) Find the eigenstates of the angular momentum vector component L_z of a spherically symmetric system. (ii) Write down any two properties of the Pauli matrices after defining through an expression.
Determine the eigenvalues and eigenfunctions of the operators D + 137, D^2 - 2xD + 4. where D = d/dx. Assume the eigenfunction of the second operator to be a quadratic.
The momentum of an electron is 600\text{ keV/c}. Determine its de Brogue wavelength and the phase and group velocities of its de Brogue waves.
X-ray photons of wavelength 2.0\text{ pm} are incident on free electrons. They are scattered at an angle of 60^\circ from the incident direction. Determine the following: (i) Momentum of the incident photon in keV/c (ii) Compton shift (iii) Kinetic energy and recoil angle of the electron
A 500\ \mu\text{A} beam of electrons of kinetic energy 1.5\text{ eV} enter a region with a sharply defined boundary in which their energy is reduced to 10\text{eV} by a difference of potential. Determine the reflected and transmitted currents. Derive the formulae used.
Find the de Broglie wavelength associated with an electron of energy (i) 10 eV and (ii) 10 MeV
What is spin-orbit coupling? Considering the sodium doublet (5890\text{ \AA} and 5896\text{ \AA}), calculate the effective magnetic field experienced by the electron in the 3p state.
For a particle confined in a one dimensional potential well of length L the wave-function is \psi(x) = c \sin (\pi x/L), 0 < x < L and \psi(x) = 0. \text{ outside} Calculate the expectation values of x and p.
What are Compton effect and Compton wavelength ? Determine Compton shift. Show that maximum Compton shift is twice the Compton wavelength.
Using the uncertainty principle \Delta x . \Delta p \sim h/2, estimate the minimum energy of a particle in a simple harmonic potential U = 1/2 k x^2.
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.
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.
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.
Describe how Davisson-Germer electron diffraction experiment confirms the de Broglie hypothesis of matter wave.
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.