(i) Show that the radial probability density of the ground state of the hydrogen atom has a maximum at r = a. The ground state wave function of the hydrogen atom is given by \psi (r) = \frac{1}{\sqrt{\pi a^{3/2}}} e^{-r/a} where a is the Bohr radius. (ii) Calculate the Larmor frequency of a spin 1/2 particle in a magnetic field B.
Number at non-interacting electrons are confined in a cube of volume L^3. Obtain an expression for the Fermi energy.
Distinguish between a classical and a quantum mechanical harmonic oscillator. Explain the existence of zero point energy.
Derive an expression for the electrical conductivity of metals on the basis of free electron theory of metals.
(i) The wave function of a particle is . \psi(x) = A \exp \left( -\frac{x^2}{a^2} + i k_0 x \right) Find the expectation values of position (x) and momentum (p) for the particle. (ii) A 200 eV increase in the energy of an electron changes its De Broglie wavelength by a factor of two. Calculate the initial De Broglie wavelength of the electron.
Solve the one dimensional Schrodinger wave equation with potential: V(x) = \begin{cases} 0 & \text{for } x < -a \\ V_0 & \text{for } -a < x < a \\ 0 & \text{for } x > 0 \end{cases}
A particle is in a quantum system with a parabolic potential well. (i) Using appropriate method find the ground state energy. (ii) Obtain an expression for the ground state.
(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) 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.
(i) Write down the matrix representation of the energy operator of a linear oscillator. (ii) A linear oscillator is prepared in the state given by \psi(x) = 1/\sqrt{5} \{\hat{\psi}_0(x) + \sqrt{2} \hat{\psi}_1(x) + \sqrt{2} \hat{\psi}_2(x)\} Evaluate the energy of the oscillator in this state.
(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?
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.
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.
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.
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.
What are Compton effect and Compton wavelength ? Determine Compton shift. Show that maximum Compton shift is twice the Compton wavelength.
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.
Find the de Broglie wavelength associated with an electron of energy (i) 10 eV and (ii) 10 MeV
