Quantum Mechanics - Page 1 of 3

Calculus

Quantum Mechanics is one of the most monumental and strange theories ever devised. It strangeness will become more apparent as we progress into this chapter. However, despite its apparent strangeness, nature in its most fundamental level is operated by quantum physics. Theoretical physicists working on Grand Unified Theories, or GUT's, would like to unify Einstein's relativity with quantum theory into a theory of quantum gravity. However, we are still trying to understand the mysteries of general relativity, let alone the even greater mysteries imposed by quantum theory. In 1965, Richard P. Feynmann, a physicist who worked with quantum mechanics most of his life said in his "Character of Physical Law:" "There was a time when the newspapers said that only twelve men understood the theory of relativity. I don not believe there ever was such a time. There might have been a time when only one mad did because he was the only guy who caught on, before he wrote his paper. But after people red the paper a lot of people understood the theory of relativity in one way or another, certainly more then twelve. On the other hand, I can safely say that nobody understands quantum mechanics.

Origins

Quantum Mechanics had aroused from a strange and puzzling problem. Physicists experienced this problem at the end of the nineteenth century, and has its roots in oven radiation! Imagine that you have a perfectly insulated oven, you put some food in the over and set it for 300 degrees Fahrenheit. When the oven is heated up, waves of radiation are generated in its interior. This radiation is caused, of course, by the oven's internal heat, just like the electromagnetic radiation emitted by the surface of the sun. What is the problem, then? Electromagnetic radiation, as you probably already know, carries energy. By the end of the nineteenth century and the beginning of the twentieth, physicists had calculated the total energy carried by all electromagnetic energy inside an oven at any chosen temperature. they used a well-known calculation procedure, yet they came up with an inexplicable answer: For any chosen temperature, the total energy in the over is infinite!

This appeared as pure nonsense. An oven certainly has energy in it's interior emitted by electromagnetic waves, however, certainly not an infinite amount. Max Planck came to the scene with a powerful, but weird idea. This idea led to what we know as Quantum Mechanics.

What was the idea he had that would solve the infinite energy problem? He introduced the concept of the quantum, as a result of physicists' research into blackbody radiation during the end of the nineteenth century. A black body is an ideal surface or body that absorbs all radiant energy without any reflection. A body at a sufficiently high temperature would emit "red heat," meaning that it gives out its radiation in the low frequency regions of the electromagnetic scale, such as red and infrared. A body at a higher temperature would emit "white heat," meaning it gives it's radiation at higher frequencies such as yellow, green or blue. Physicists, by the time of Planck, had conducted detailed studies of these phenomena and showed their results in a series of curves and graphs. The classical, pre-quantum, theory (the one that had infinite answers) did not match up to the curves. Therefore, both the idea of the infinities were both experimentally and theoretical nonsense  Planck devised a mathematical formula to describe these curves exactly. He then developed a physical hypothesis. Energy is radiated only in quanta of energy, meaning energy comes in discrete units. Also, when an electron, for example, drops from one energy level to the next, it emits a quanta of energy (as later would be seen in Bohr's model of the atom). His hypothesis was that energy is radiated only in quanta of energy, hv, where h is Planck's constant (the constant came as a mathematical solution, and it was later confirmed experimentally by American physicist Robert Millikan in 1916), and v is the frequency of the wave. Planck's constant has a value equal to 1.05 * 10-27 grams-cm/sec. It is a fundamental parameter in quantum mechanics. It determines the size of discrete unites that energy is partitioned by in the microscopic world.

What is the usage of Planck's constant (donated h bar)? Planck found that by inserting this constant into the calculations, he could calculate the total value of electromagnetic energy found in the oven for any chosen temperature. You can understand how this new constant resolved the old dilemma if you know how is it obtained. It's the proportionality factor between the frequency of the wave (electromagnetic radiation in the oven, and in general) and the minimum amount of energy that the wave can possess.  According to Planck, the size of these discrete units of energy (quanta) grows as the frequencies of the waves get higher and higher (meaning wavelengths getting shorter and shorter). This can be put in the following simple equation: Emin=hf, where Emin is the lowest possible energy carried by the wave, h is Planck's constant, and f is the frequency of the wave. He estimated that constant using experiments showing the curves of blackbody radiation. The idea of discrete units of energy come when Planck asserts that the energy of any molecular vibration could be only some whole number multiplied by hf: E = nhf, n = 1, 2, 3, 4 and so on.

This is known as Planck's quantum hypothesis. By the way, the word "quantum" means fixed amount. Planck's quantum hypothesis, stated in the last equation means that E = hf, or E = 2hf, or E = 3hf, and so on. However, there can't be vibrations whose energy lies between these values. The smallest amount of energy possible is hf and is known as a quantum of energy.

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