# Quantum Physics I PHY471, Fall
1999

# Homework set
1

# Due Wednesday,
9/8/1999

#

*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)=kx*^{2}/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.

## 4. [3
pt] Ohanian, #1.18, p. 19