Theories of the Universe
Just a Little Math Won't Hurt
The traditional establishment in any area of study, whether it's medicine, physics, or even art and music, often takes a dim view of new ideas or forms of expression. More than one scientist's theories have been scoffed at and ridiculed because they went beyond the accepted understanding of how things are supposed to be. Einstein met with disbelief at first, as did Louis de Broglie. Many considered his explanation of matter waves as crazy and absurd. Of course, he was later proved to be correct when experiments were done that showed that matter did indeed have a wave property. It's a good thing that the innovators and visionaries of society don't pay attention to conventional ideas, otherwise we could still be making tools from stone.
De Broglie had discovered a new formula, one as important and startling as Planck's formula. It stated that the momentum of a particle p was equal to Planck's constant h, divided by the wavelength λ. That is, p = h ÷ λ. I would like to walk you through the math for this formula because it was a brilliant insight. It's just a matter of substituting some letters and rewriting Einstein's famous, E = mc2.
We begin with
Because mc is mass times speed, or in other words momentum (p), we can substitute p for mc. So,
Because the term c stands for speed of light, we can substitute it with f (frequency) times λ (the symbol lambda, which represents wavelength) because that's the formula for finding the speed at which waves travel. Then,
This is almost the same formula just different terms.
Erwin Schrödinger had a disdain for convention, especially regarding sex, but it extended beyond that. Many times when he attended a conference, he would walk from the train station to the hotel where the delegates stayed, carrying only a rucksack and looking like a tramp. It always required a great deal of argument at the reception desk before he could claim his room. But this same disdain for convention made him very popular with his students and he was highly regarded as a teacher.
Now if we use the Planck/Einstein equation that relates a particle's energy to its frequency, or E = h × f (you were introduced to it as h = E ÷ f to show you how Planck's constant was derived, but we just rewrote it in another form) we get the following:
We do a little simple algebra (divide both sides by f λ , which cancels out the two f λ 's on the right and the two f's on the left):
And we get h ÷ λ = p or turned around p = h ÷ λ.
That means that a particle (p) is equal to Planck's constant (h) divided by the wavelength (λ). Cool, huh? Well maybe not to some of you. But don't worry if you didn't follow it, you'll still be able to understand why it was important. With this new mathematical discovery, Bohr's orbits could be explained. Each orbit was a standing wave pattern. The atom could be understood as a finely tuned instrument (maybe Pythagoras's and Plato's music of the spheres wasn't that far off). These mathematical relations balanced the tiny electron into a tuned standing wave pattern. Orbits had determined and fixed sizes in order that these distinct, quantized wave patterns could exist.
I hope that these last few pages have given you a good idea of de Broglie's contribution to quantum mechanics. However, if you're still a little lost, the main thing you need to know is that all matter, everything that exists, has as its core characteristic the wave/particle duality. This dual nature of matter is probably the one significant thing that most quantum cosmology rests upon.
More Wave Mechanics
A beat has a very different meaning than what we're accustomed to when it's talked about in wave mechanics. In this case, it refers to combining two waves of different frequencies that produces an additional frequency equal to the difference between the two. You can think if it as another word for the auditory experience of harmony.
While de Broglie's theories offered a picture of what was going on inside of an atom, more was needed to explain the shifting patterns of the wave when it changed its energy and emitted light. Erwin Schrödinger, an Austrian physicist, found a mathematical equation that explained the changing wave patterns inside an atom. Don't worry—I'm not going to take you through the math this time. As a matter of fact, I just want to give you a brief overview of his wave mechanics because I want to spend more time on the really weird stuff that's just around the corner.
Schrödinger, like de Broglie, used the analogy of a vibrating violin string to explain his mathematical equation. The movement of an electron from one orbit to another lower energy orbit was a simple change in notes on a violin. As the violin string undergoes such a change, there is a moment when both harmonics (sound produced by the notes) can be heard. This results in the well-known experience of harmony, or as wave scientists call it, the phenomenon of beats. The beats between two notes are what we hear as harmony and this really produces a third sound. The vibrational pattern of the beats is determined by the difference in the frequencies of the two harmonics.
A function is a very useful mathematical tool because it relates two distinct quantities or qualities to each other. Temperature can be a function of time as when something is heated up. As more time goes by, the temperature of the object increases. You can also relate space and time together. As you already know, time changes space. So the space you're in now is not the space you were in five seconds ago. As you move through space, time goes by, or if you move through time, space changes.
This was just what was needed to explain the observed frequency of the light waves of photons emitted when an electron in the atom underwent a change from one orbit to the other. The light was a beat, a harmony, between the upper and lower harmonics of Schrödinger's and de Broglie's waves. When we see light emitted in this process, we are observing an atom singing in harmony.
So what's the big difference between de Broglie's and Schrödinger's waves? It essentially boils down to the following:
And after all was said and done, Schrödinger's equations became the preferred method for solving quantum interactions. And it opened the door for another cornerstone of quantum mechanics, probability theory.
Excerpted from The Complete Idiot's Guide to Theories of the Universe © 2001 by Gary F. Moring. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.