PHYSICS 107. Lecture 16 The Quantum

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1 PHYSICS 107 Lecture 16 The Quantum Wave Nature of Light With light, as with sound, we only perceive a portion of the whole range of frequencies, the whole range being referred to as the spectrum. Visible light is only in the wavelength range from m (blue-to-violet) to m (red). As far as we know, light can have any frequency and wavelength, so there is a vast range of electromagnetic waves. Electromagnetic waves are also called electromagnetic radiation, since electromagnetic waves emanate, radiate away, from a source. Here s a chart that gives just a little taste of all things that are electromagnetic waves. Different portions of the spectrum interact with matter in very different ways, so all portions of the spectrum have different uses. Your cell phone is actually just a radio transmitter and receiver. Cell phone frequencies are generally in the range

2 from 500 to 2000 MHz. In this range we have efficient transmitters and receivers, the atmosphere is relatively transparent, and the signals can be translated back and forth from and to audio signals. All these waves obey our basic rule that c = λ ν, so on the chart you can see that as the wavelength (top line) gets longer the frequency (second to bottom line) gets lower. In the 19 th century, physicists established to their own satisfaction that light was a wave. The main experiments that confirmed this were interference experiments. You see it in oil films, soap bubbles, or CDs in the different colors as the angle of viewing changes. This is due to the interference between the wave reflected off the front surface and the wave reflected off the back surface. (CDs have a thin transparent coating.) So there was a universal consensus that light was a wave. This by the way was in contradiction to Newton's opinion that light consisted of particles. Particle Nature of Light I have spent a lot of time convincing you that interference is a purely wave phenomenon. We have seen with our own eyes that light shows interference and have concluded that it must be a wave. But actually, around 1900, some difficulties began to crop up. The first experiment that seemed to be a problem was the spectrum of light emitted by a glowing object ( blackbody radiation ). Careful analysis of this spectrum showed that it could not be explained using the idea that light consists of waves. In a second class of experiments, electrons were knocked out of the surfaces of metals by shining light on the surfaces. Max Planck explained the results of the first class of experiments, and Einstein explained the results of the second class of experiments, both by making the hypothesis that the energy in a beam of light comes in packets. Each packet has an energy E proportional to the frequency: E = hν. h has forever after been called Planck's constant, and its value is h= J-s. A joule is a unit of energy: 1 J = 1 kg m 2 / s 2. If you look

3 back at the diagram you will see the energy on the bottom line, and it is proportional to the frequency. The experiments that Einstein was looking at were actually fairly simple. We shine a beam of light on a piece of metal. Inside the metal there are electrons, particles that carry a negative electric charge. They are free to move about inside the metal, which is why metals can carry electrical current from one end of a wire to the other end. It's for this reason that copper, a very nice metal, is used for wiring in the house. But there is a barrier that keeps them from escaping through the surface, which means that the piece of metal is essentially like a cage for electrons. If light is shone on a metal, then some electrons will be ejected from the metal by the light. This effect is called photoemission. It takes a certain amount of energy, called w, the work function, to get an electron out of the metal. So this energy w must be supplied by the light. You would expect, since light is a wave, that the more intense the light, the more electrons should be able to absorb the energy and escape. The frequency of the light shouldn't matter, only the intensity. But that is not how it works. Instead, the frequency plays a crucial role. If the frequency is below a threshold, given by ν T = w/h, then no electrons are ejected. Einstein suggested that the only way to explain this was to say that each electron gets a packet of energy equal to hν when it absorbs a particle of light. Only if the energy in the packet is sufficient to get the electron over the barrier can it escape from the metal. So somehow the size of the packets is proportional to the frequency. Planck worked out the magnitude of h from the experimental data on blackbody rasiation and Einstein then noted that the magnitude from the photoemission experiments agreed with Planck s value. Planck always regarded the energy packets as some sort of convenient fiction. In other words he regarded his formula as a mathematical way of getting a formula that agreed with the experiments but that the packets did not correspond to any real physical entity. Einstein with his usual boldness went further. He postulated that light did in fact somehow consist of a stream of wavepackets, each of which had energy equal to hν. He called the packet the light quantum, and they later became known as photons. The phot prefix of course means light, and the on suffix refers to a particle. So a photon is a particle of light.

4 It's a strange thing that Einstein's truly revolutionary theory of relativity was accepted very quickly by the scientific community and Einstein began to move up the professional ladder very quickly becoming a professor in Berlin, the highest academic position in the German-speaking world by This professional advance was due to many different important works in physics. Only his work on the photon was regarded as implausible and unacceptable. Only in 1923 when other indisputable evidence for the existence of the photon began to appear in experiments did people accept that they were wrong and Einstein was right. In the theory of relativity there was a seeming contradiction between the classical transformation laws and the constancy of the speed of light. Einstein was able to show that by focusing on events that we actually observe this contradiction could be resolved. He offered no such resolution for this new seeming contradiction, namely that light seemed to be a wave, and yet at the same time seemed to involve packets of energy, as if light was a stream of particles. But how can this be? We already proved to our satisfaction that light is a wave. How can it be a particle at the same time? Atoms Let's shelve this question for a moment, and instead shift our focus to atoms. By 1913, physicists had accepted that the basic building blocks of ordinary matter were atoms. The lightest atom is the hydrogen atom. It is also the simplest and so we will focus exclusively on hydrogen. It was known that the hydrogen atom consisted of two particles, a nucleus with a positive charge, and a much lighter electron with a negative charge. Since these particles attract each other according to Coulomb s law, and since that law is the same as the gravitational law, we would expect that the electrons should orbit in an ellipse about the nucleus. Just as for the planets in the solar system, all sizes and shapes of these ellipses should be possible. Each different ellipse would correspond to a different total energy for the system. Yet this was not what was observed. If one looked at the light coming from the hydrogen atom, only discrete frequencies are observed. If one converts these to energies to frequencies according to the known law E = h ν then we must conclude that only discrete energies are allowed for the atom. Energies in atoms are measured in electron volts, usually abbreviated as ev.

5 1 ev = J. There is the ground state with E 1 = ev, the first excited state with E 2 = -3.4 ev, and other states at higher energies. All of the energy levels (as they are called) are given by the formula E n = ev/n 2.. n = 1,2,3, n runs all the way up to n = infinity. But the absolutely key, and really revolutionary, point is that the possible energies of the hydrogen atom are discrete they are not drawn from a continuous set of numbers. This was the insight of Niels Bohr in We can understand this only if the electron is somehow also wavelike. Just as a guitar string has discrete frequencies, so also do atoms. Here's how it works. The electron goes around in a circle. It must go through an integral number of wavelengths, so 2 π r = n λ. Again n = 1,2,3, In addition we make a new rule for the electron, somewhat similar to Planck s formula for the photon, which is that λ = h /p = h / mv. λ is the wavelength and p = mv is the momentum, as usual. This is called the de Broglie relation. With a little algebra this leads to the previous formula for the energy levels. So suddenly we had light that was supposed to be a wave, and the electron that was supposed to be a particle, and now both seem to be both particle AND wave!