# Please clearly state your assumptions, number the equations and indicate logical connections between different lines.

## 1.                  [3+3+1 pt] Bohr hydrogen-like atom

Consider a classical hydrogen-like atom, i.e. a nucleus of N neutrons and Z protons with only one atomic electron.

a)      From the equality of the Coulomb and centripetal forces between the nucleus and the atomic electron derive the classical relationship for the orbital energy.

b)      Now assume that the orbital angular momentum L is quantized and derive the quantized orbital energies of possible stationary states for the hydrogen-like atom.

c)      What is the qualitative difference in the energy spectra of the two solutions a) and b)?

## 2.                  [1+1+1+1+1 pt] De Broglie wavelength

a)      Calculate the de Broglie wavelength for a billiard ball of 0.05 kg moving at 0.1 m/s.

b)      Calculate the de Broglie wavelength for the Earth orbiting the Sun.

c)      Calculate the de Broglie wavelength for thermal neutrons (T=500 K) from a reactor.

d)      Calculate the de Broglie wavelength for electrons at room temperature (T=293K).

e)      For which of these systems is a quantum-mechanical description appropriate?  Make some convincing arguments that your answer is correct.

## 3.                  [3+2 pt] Uncertainty relation

A particle of mass m is moving in a one-dimensional harmonic-oscillator potential V(x)=kx2/2.  Consider the particle’s total energy, i.e. the sum of its kinetic and potential energy.

a)      Estimate the lowest total energy of the lowest state compatible with the uncertainty principle.

b)      Estimate this lowest energy in eV for the case of an electron.  If you make any assumptions, give some convincing arguments that your assumptions are reasonable.