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Electronic Spectrum of Hydrogen atom II - Particle on a Ring (Dry Lab)

Electronic Spectrum of Hydrogen atom II - Particle on a Ring (Dry Lab)

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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 were a wave associated with every moving particle. The formula relating the associated wave length λ to the velocity of the particle is:

\[λ = \dfrac{h}{mv} \label{2a}\]

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: indeed, 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

\[A = k \cos \left(π(\dfrac{x}{λ} - vt) \right) \label{3a}\]

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 \(\lambda\) 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!

The only stable orbits are ones where there is no discontinuity.

Q1

Your first task is to find the radii of the orbits which are allowed. Open the simulation, 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. * Coarse adjustment of the radius can be made with the slider, then use the left or right arrow keys for fine adjustment.

Number of wavelengths (j)	Radius (r)	Wavelength (2πr/j)
1	52.90	332.4
2	211.6	664.7
3	472.9	990.5
4	844.3	1326

Q2

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

Q3

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 \(\ref{2a}\)).