THE "PARTICLE ON A RING"

**INTRODUCTION**

It was Louis de Broglie who suggested in 1924, that the arbitrariness of the
way in which Bohr introduced quantization in his theory of atomic structure
could be removed, and quantum numbers derived in a more "natural" way, if it was
assumed that there was a *wave* associated with every moving particle. His
formula relating the associated wave length l to the
velocity of the particle is:

where h is Planck's constant, m is the mass of the particle, and v is its velocity.

This part of the computer experiment has to do with the general properties of a particle moving in a circle: the so called "particle on a ring".

**EXPERIMENT**

The applet below shows the relation between the Bohr electron in an orbit and the associated "electron - wave". You can adjust the radius of the orbit with the "scroll bar" and choose between displaying the electron as a particle or as a wave with the button. In keeping with the theory, you cannot see both the particle and the wave at the same time: there are no laboratory experiments which can simultaneously show both properties.

The amplitude of the wave at some point around the ring (orbit) is given by

where k is a constant, x is the distance travelled around the ring, v is the
velocity of the electron (when considered as a particle), t is the time after an
arbitrary t_{o} and l is the wavelength
associated with the electron. The position x_{o}, which marks the
beginning and end of the calculation of the wave form, is labelled to make it
easier to find. The only stable orbits are ones where there is no discontinuity
at this point, so that if the wave were continued around the circle it would
superimpose perfectly on earlier passes. Otherwise, there would be destructive
interference, and the wave would be annihilated. This is not allowed since the
electron cannot just disappear!

**Question 10.**

Your first task is to find the radii of the orbits whichareallowed. Go down to the applet, display the electron wave and carefully adjust the orbit radius until the junction between the beginning of the wave and its end is undetectable. There must be no hint of a discontinuity or cusp (kink) visible. There are several such situations in the range allowed by the scroll bar. Report the radius, which is displayed next to the scroll bar, the number of complete waves around the orbit, and a calculated wave length for each case.

Write down a simple formula for the wavelengths l which are allowed in terms of the radius (r) and a quantum number (j).

**Question 12.
**

Do the wave lengths which you observed vary with the radii of the stable orbits? If so, how and why? Your answer should be given in terms of the Bohr theory (described and illustrated in the first part of this experiment) and the de Broglie relation (equation (2) above).