Obtain an expression for the magnetic dipole moment of a revolving electron in a Bohr model.

(a) State Bohr’s postulate to define stable orbits in hydrogen atom. How does de Broglie’s hypothesis explain the stability of these orbits ?

(b) A hydrogen atom initially in the ground state absorbs a photon which excites it to the n = 4 level. Estimate the frequency of the photon.

The Bohr model for the H-atom relies on the Coulomb’s law of electrostatics. Coulomb’s law has not directly been verified for very short distances of the order of angstroms. Supposing Coulomb’s law between two opposite charge +q₁, –q₂ is modified to

Calculate in such a case, the ground state energy of a H-atom, if ε = 0.1, R0 = 1Å.

Consider an excited hydrogen atom in state n moving with a velocity v (v << c). It emits a photon in the direction of its motion and changes its state to a lower state m. Apply momentum and energy conservation principles to calculate the frequency v of the emitted radiation. Compare this with the frequency v₀ emitted if the atom were at rest.

Suppose in an imaginary world the angular momentum is quantized to be even integral multiples of h/2π. What is the longest possible wavelength emitted by hydrogen atoms in visible range in such a world according to Bohr’s model?

In the Auger process an atom makes a transition to a lower state without emitting a photon. The excess energy is transferred to an outer electron which may be ejected by the atom. (This is called an Auger electron). Assuming the nucleus to be massive, calculate the kinetic energy of an n = 4 Auger electron emitted by Chromium by absorbing the energy from a n = 2 to n = 1 transition.

A uniform magnetic field B exists in a region. An electron projected perpendicular to the field goes in a circle. Assuming Bohr’s quantization rule for angular momentum, calculate

(a) the smallest possible radius of the electron

(b) the radius of the nth orbit and

(c) the minimum possible speed of the electron.

Consider a neutron and an electron bound to each other due to gravitational force. Assuming Bohr’s quantization rule for angular momentum to be valid in this case, derive an expression for the energy of the neutron-electron system.

Deuterium was discovered in 1932 by Harold Urey by measuring the small change in wavelength for a particular transition in ¹H and 2H. This is because, the wavelength of transition depend to a certain extent on the nuclear mass. If nuclear motion is taken into account then the electrons and nucleus revolve around their common center of mass. Such a system is equivalent to a single particle with a reduced mass μ, revolving around the nucleus at a distance equal to the electron-nucleus separation. Here μ = m_eM/ (m_e + M) where M is the nuclear mass and m_e is the electronic mass. Estimate the percentage difference in wavelength for the 1st line of the Lyman series in ¹H and ²H. (Mass of 1H nucleus is 1.6725 × 10^–27 kg, Mass of ²H nucleus is 3.3374 × 10^–27 kg, Mass of electron = 9.109 × 10^-31 kg.)

Light from Balmer series of hydrogen is able to eject photoelectrons from a metal. What can be the maximum work function of the metal?

A neutron having kinetic energy 12.5 eV collides with a hydrogen atom at rest. Neglect the difference in mass between the neutron and the hydrogen atom and assume that the neutron does not leave its line of motion. Find the possible kinetic energies of the neutron after the even.