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# Electrons as Waves, De Broglie Wavelength

Electrons as Waves

Einstein (and others) showed that electromagnetic radiation  has properties of matter as well as waves. This is known as the wave-particle duality for light.  Einstein’s evidence came by way of explaining the photoelectric effect for which he won the Nobel Prize in 1921.   In 1924, the French scientist Lois de Broglie wondered that since light, normally thought to be a wave, could have particle properties, could matter, specifically the electron, normally thought to be a particle, have wave properties as well?  He took Einstein’s famous equation  E=mc2, Planck’s equation  E=hn, and the relationship between wave speed, frequency and wavelength  c=fl  and combined them algebraically to derive the equation:

 h=6.626068 × 10-34 J s;    m=mass in kg; v=velocity in m/s

The calculated wavelength, l, for an 11g ping pong ball traveling at 2.5 meters per second is 2.4 x 10-32m.

This answer makes sense : With large objects traveling at slow speeds, the wavelengths are not able to be seen and are so small they are unimportant. The wavelength calculated is much smaller than the shortest known wavelength of gamma rays (10-11 m). If we use the mass of the electron traveling at 1 x 105 meters per second, we get a wavelength of about 7.3 x 10-9m, which is about the same size as the radius of an atom. At this speed, the electron can “orbit” the hydrogen nucleus over 3 million times in one second! It would appear that the electron is everywhere at once! Treating the electron as a wave just might be the right way to handle this problem. But the question remained how this could be applied to the atom.

If an electron really could exist as a wave inside the atom, where exactly was it?  The German scientist Heisenberg determined that it was impossible to experimentally determine both the position and the speed of the electron at the same time.  This became known as the Heisenberg Uncertainty Principle.  It simply means that the electron is so small and moving so fast, that the simple act of trying to measure its speed or position would change either quantity.  Trying to detect the electron by shining some type of wave at the electron would be energetic enough to move it and thus change its position or speed.  We can see that this principle would only apply to extremely small particles.  If we shine a flashlight at a truck in the dark, we can surely tell its position, or if we want to determine its speed by radar (radio waves) we can do so.  In each case, our measuring tool will not affect the speed or position of the truck; it is too massive.  So we were out of luck finding exactly where the electron is in the atom.  And if we assumed it acts like a wave, well, how does one tell the position of a wave?

The Austrian scientist, Erwin Schrödinger, pursued de Broglie’s idea of the electron having wave properties and it seemed to him that the electron might be like a standing wave around the nucleus.  A standing wave is like a string stretched between 2 points and plucked, like a guitar string.   The wave does not travel between the 2 points but vibrates as a standing wave with fixed wavelength and frequency.  There is a limitation on the number of waves that will fit in between the two points.  There must be  a whole number of waves to be a standing wave; there cannot be, for instance, a 2.3 waves.  So, only certain, or allowed wavelengths (or frequencies) can be possible for a given distance between the 2 points.  The same could be said about the atom.  At any given distance from the nucleus, only a certain number of whole waves would “fit” around the nucleus and not overlap in between waves.  For a given circumference, only a fixed number  of whole waves of specific wavelength would work.  Most wavelengths would not work and thus would not be observed.

This idea agreed very well with Bohr's idea of quantized energy levels: only certain energies and therefore, wavelengths would be allowed in the atom.  This explained why only certain colors (wavelengths) were seen in the spectrum of the hydrogen atom.  We are on to something!  Schrodinger set out to make a mathematical model that assumed the electron was a standing wave around the nucleus.  His solutions to that model agreed not only with the experimental evidence for hydrogen (as Bohr’s did too), but gave excellent results for all atoms when compared to their actual spectrum.

Energy, wavelength and electron transitions

/ Heisenberg / Planck / Schrödinger / Chadwick