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Physical chemistry 228

particle is confined to a finite space its momentum, and hence kinetic energy, cannot be
zero. There is no zero point energy for a particle in a circular orbit.

Particle in a circular orbit

A particle of mass m moving around a circular orbit of radius r with a velocity υ has
linear momentum p=mυ and angular momentum=mυr. If the potential energy is zero,
then the total energy is entirely kinetic and given by:

2
where I=mr is called the moment of inertia of the particle about the center of its path.
The moment of inertia is the rotational equivalent of mass in linear motion. The de
Broglie equation is used to express the angular momentum of the particle in terms of the
wavelength of its associated wavefunction:

hence

The boundary condition of the system is that the shape of the wavefunction must repeat
after each circuit of 360° around the circumference of the trajectory along which the
particle travels (Fig. 4). If this condition is not satisfied then the wavefunction cancels
out, or destructively interferes, on each circuit. Therefore physically acceptable
wavefunctions must have wavelengths:

Fig. 4. An allowed wavefunction for a
particle in a circular orbit must repeat
after 360° (solid line).