A particle of mass m moves in a circular orbit in a
central potential field U(r) = U₀r⁴. If Bohr's quantization conditions
are applied, radii of possible
orbitals r_n vary with $${n^{{1 \over \alpha }}}$$, where $$\alpha$$ is ____________.

The first three spectral lines of H-atom in the Balmer series are
given $$\lambda$$₁, $$\lambda$$₂, $$\lambda$$₃ considering the Bohr atomic model, the wave lengths of first and third spectral lines $$\left( \frac{\lambda_{1} }{\lambda_{3} } \right) $$ are related by a factor of approximately 'x' $$\times$$ 10^$$-$$1.

The value of x, to the nearest integer, is _________.

A particle of mass 200 MeV/c² collides with a hydrogen atom at rest. Soon after the collision the particle comes to rest, and the atom recoils and goes to its first excited state. The initial kinetic energy of the particle (in eV) is
$${N \over 4}$$. The value of N is :
(Given the mass of the hydrogen atom to be 1 GeV/c²) ______ .

The first member of the Balmer series of hydrogen atom has a wavelength of 6561 Å. The wavelength of the second member of the Balmer series (in nm) is: