Chapter 37. Lasers, a Model Atom and Zero Point Energy

Transcription

1 Chapter 37 Lasers, a Model Atom and Zero Point Energy CHAPTER 37 LASERS, A MOEL ATOM AN ZERO POINT ENERGY Once at the end of a colloquium I heard ebye saying something like: Schrödinger, you are not working right now on very important problems...why don t you tell us some time about that thesis of de Broglie, which seems to have attracted some attention? So in one of the next colloquia, Schrödinger gave a beautifully clear account of how de Broglie associated a wave with a particle, and how he could obtain the quantization rules... by demanding that an integer number of waves should be fitted along a stationary orbit. When he had finished, Chapter ebye 6 casually remarked that he thought this way Mass of talking was rather childish... To deal properly with waves, one had to have a wave equation. FELIX BLOCK, in an address to the American Physical Society in 976. Schrödinger took ebye s advice, and in the following months devised a wave equation for the electron wave, an equation from which one could calculate the electron energy levels. The structure of the hydrogen atom was a prediction of the equation without arbitrary assumptions like those needed for the Bohr theory. The wave nature of the electron turned out to be the key to the new mechanics that was to replace Newtonian mechanics as the fundamental theory. In the next chapter we will take a look at some of the electron wave patterns determined by Schrödinger s equation, and see how these patterns, when combined with the Pauli exclusion principle and the concept of electron spin, begin to explain the chemical properties of atoms and the structure of the periodic table. The problem one encounters when discussing the application of Schrödinger s equation to the hydrogen atom, is that relatively complex mathematical steps are required in order to obtain the solutions. These steps are usually beyond the mathematical level of most introductory physics and chemistry texts, with the result that students must simply be shown the solutions without being told how to get them. We will have to do the same in the next chapter. In this chapter we will study a model atom, one in which we can see how the particle-wave nature of the electron leads directly to quantized energy levels and atomic spectra. The basic idea, which we illustrate with the model atom, is that whenever you have a wave confined to some region of space, there will be a set of allowed standing wave patterns for that wave. Whether the patterns are complex or simple depends upon the way the wave is confined. If the wave is also a particle, like an electron or photon, you can then use the particle wave nature to calculate the energy of the particle in each of the allowed standing wave patterns. These energy values are the quantized energy levels of the particle. An example of a set of simple standing waves that are easily analyzed is found in the laser. It is essentially the laser standing wave patterns that we use for our model atom. For this reason we begin the chapter with a discussion of the laser and how the photon standing waves are established. In the model atom the photon standing waves of the laser are replaced by electron standing waves. An analysis of the model atom shows why any particle, when confined to some region of space, must have a non zero kinetic energy. The smaller the region of space, the greater this so called zero point kinetic energy. When these ideas are applied to the atoms in liquid helium, we see why helium does not freeze even at absolute zero. We also see why the entropy definition of temperature must be used at these low temperatures.

2 37- Lasers, a Model Atom and Zero Point Energy THE LASER AN STANING LIGHT WAVES The laser, the device that is at the heart of your C player and fiber optics communications, provides a common example of a standing light wave. In most cases a laser consists of two parallel mirrors with standing light waves trapped between the mirrors as illustrated in Figure (). The light comes from radiation emitted by excited atoms that are located within the standing wave. How the light radiated by the excited atoms ends up in a standing wave is a story in itself. An atom excited to a high energy level can drop down to a lower level by emitting a photon whose energy is the difference in energy of the two levels. This photon will have the wavelength of the spectral line associated with those two levels. Spectral lines are not absolutely sharp. For example, due to the oppler effect, thermal motion slightly shifts the wavelength of the emitted radiation. If the atom is moving toward you when it radiates, the wavelength is shifted slightly towards the blue. If moving away, the shift is toward the red. In addition the photons are radiated in all directions, and waves from different photons have different phases. Even in a sharp spectral line the light is a jumble of directions and phases, giving what is called incoherent light. In contrast the light in a laser beam travels in one direction, the phases of the waves are lined up and there is almost no spread in photon energies. This is the beam of coherent light which made it so easy for us to study interference effects like those we saw in the two slit and multiple slit diffraction patterns. These patterns would be much more difficult to observe if we had to use incoherent light. optically flat mirror excited gas atoms standing light wave partially reflecting optically flat mirror Figure Laser consisting of two parallel mirrors with standing light waves trapped between the mirrors laser beam The purity of the light in a laser beam depends upon the standing light wave pattern created by the two mirrors, and upon a quantum mechanical effect discovered by Einstein in 95. Einstein found that there were two distinct ways an excited atom could radiate light, either by spontaneous emission or stimulated emission. An example of spontaneous emission is when an excited atom is all by itself and eventually drops down to a lower energy level. The emitted photon can come out in any direction and can be oppler shifted. If, however, a photon with the right energy passes by the excited atom, there is some chance that the atom will emit a photon exactly like the one passing by. This is called stimulated emission. (The energy of the passing photon has to be close to the energy the atom would naturally radiate.) It is the process of stimulated emission that can lead to a laser beam. Suppose we have a gas of excited atoms located between parallel mirrors. At first the atoms radiate spontaneously in all directions. (We assume that there is some mechanism to excite the atoms). After a while one of the photons hits a mirror straight on and starts reflecting back and forth between the parallel mirrors. As the photon moves back and forth, it passes by an excited atom, stimulating that atom to emit an identical photon. Now there are two identical photons bouncing back and forth. Each is likely to stimulate another atom to emit an identical photon, and we have four identical photons, etc. Soon there are so many identical photons moving through the excited atoms that there is little chance that an atom can radiate spontaneously. All the radiation is stimulated and all the photons are identical to the one that started bouncing back and forth between the mirrors. The mirrors on the ends of the laser are not perfect reflectors, a few percent of the photons striking the mirror pass through, forming the beam produced by the laser. The photons lost to the laser beam are continually replaced by new identical photons being emitted by stimulated emission. One of the tricky technical parts of constructing a laser is to maintain a continuous supply of excited atoms. There are various ways of doing this that we need not discuss here.

3 37-3 Photon Standing Waves The photons bouncing back and forth between the mirrors in a laser are in an allowed standing wave pattern. Back in Chapter 5 in our discussion of standing waves on a guitar string, we saw that only certain standing wave patterns were allowed, those shown in Figure (5-5) reproduced here which had an integral or half integral number of wavelengths between the ends of the string. For photons trapped between two mirrors, the allowed standing wave patterns are also those with an integral or half integral number of wavelengths between the mirrors, as indicated schematically in Figure (). Because of the simple geometry, we do not need to solve a wave equation to determine the shape of these standing light waves. The waves are sinusoidal, and the allowed wavelength are given by the same formula as for the allowed waves on a guitar string, namely λ n = wavelength of the n nth standing wave where is the separation between the mirrors. mirror optically flat surface with reflective coating λ mirror () bridge string nut standing light wave first harmonic or fundamental λ λ = second harmonic λ λ = λ = / third harmonic 3 λ λ = 3 3 fourth harmonic 4 λ λ = 4 4 λ = / Figure 5-5 (reproduced) On a guitar string only certain standing wave patterns which have an integral or half integral number of wavelengths between the ends of the string are allowed. λ = /3 Figure Three longest wavelength standing wave patterns for a light beam trapped between two mirrors.

4 37-4 Lasers, a Model Atom and Zero Point Energy Photon Energy Levels The special feature of the standing light wave is that the light has both a wave and a particle nature. Equation, which tells us the allowed wavelengths, is all we need to know about the wave nature of the light. The particle nature is described by Einstein s photoelectric effect formula E = hf = hc λ. Applying this formula to the photons in the standing wave, we find that a photon with an allowed wavelength λ n has a corresponding energy E n given by E n = hc () λ n Because only certain wavelengths λ n are allowed, only certain energy photons, those with an energy E n are allowed between the mirrors. We can say that the photon energies are quantized. If the separation of the E n= hc/λ n E = 4(hc/) 4 mirrors is, then from Equation ( λ n =n n), and Equation ( E n = hc λ n ), we find that the quantized values of E n are E n = hc λ n = E n =n hc hc n (3) From Equation 3 we can construct an energy level diagram for the photons trapped between the mirrors. In contrast to the energy level diagram for the hydrogen atom, the photon energies start at zero because there is no potential energy. We see that the levels are equally spaced, a distance hc/ apart. Exercise If you could have two mirrors A o apart (the size of a hydrogen atom) what would be the energy, in ev, of the lowest 5 energy levels for a photon trapped between the mirrors? E = 3(hc/) 3 E = (hc/) A MOEL ATOM Now imagine that we replace the photons trapped between two mirrors with an electron between parallel walls located a distance apart, as shown in Figure (4). For this model, the allowed standing wave patterns are again similar to the guitar string standing waves. The allowed electron wavelengths are E = (hc/) E = 0 λ n = allowed wavelength n of an electron trapped (a) between two walls The difference between having a photon trapped between mirrors and an electron between walls, is the formula for the energy of the particle. If the energy of the electron is non relativistic, then the formula for its ;; ; ; photon energy levels Figure 3 Energy level diagram for a photon ;; electron trapped between two mirrors. Figure 4 Electron trapped between two walls

5 37-5 kinetic energy is / mv, not the Einstein formula E = hc/λ that applies to photons. The difference arises because the electron has a rest mass while the photon does not. For the electron trapped between walls, there is no electric potential energy like there was in the hydrogen atom. Thus we can take / mv as the formula for the electron s total energy, ignoring the electron s rest mass energy as we usually do in non relativistic calculations. To relate the kinetic energy to the electron s allowed wavelength λ n, we use de Broglie s formula p=h/λ. The easy way to do this is to express the energy / mv in terms of the electron s momentum p = mv. We get E = / mv = m mv (4) m Next use the de Broglie formula p=h/λ to give us E= p E n = h/λ n m = h (5) mλ n as the formula for the energy of an electron of wavelength λ n. Finally use Equation, λ n =/n, for allowed electron wavelengths to get E n = h m n If the electron is in one of the higher levels and falls to a lower one, it will get rid of its energy by emitting a photon whose energy is equal to the difference in the energy of the two levels. Thus the trapped electron should emit a spectrum of radiation with sharp spectral lines, where the lines correspond to energy jumps between levels just as in the hydrogen atom. Thus the electron trapped between plates is effectively a model atom, complete with an energy level diagram and spectral lines. n E = n E E = 6E 4 E = 9E 3 E = 4E E n =n h 8m (6) This is our equation for the energy levels of an electron trapped between two plates separated by a distance. The corresponding energy level diagram is shown in Figure (5). The energy levels go up as n instead of being equally spaced as they were in the case of a photon trapped between two mirrors. E = E = 0 h 8m electron energy levels Figure 5 Energy level diagram for an electron trapped between two walls.

6 37-6 Lasers, a Model Atom and Zero Point Energy Our model atom is not just a fantasy. With the techniques used to fabricate microchips, it has been possible to construct tiny boxes, the order of a few angstroms across, and trap electrons inside. An electron microscope photograph of these quantum dots as they are called, is shown in Figure (6). The allowed standing wave patterns are reasonably well represented by the sine wave patterns of Figure (5), where is the smallest dimension of the box. Thus we predict that electrons trapped in these boxes should have allowed energies En close to those given by Equation (6), and emit discrete line spectra like an atom. This is precisely what they do. (Some of the low energy jumps are shown in Figure 7.) In calculating with the model atom we have not fudged in any way by modifying Newtonian mechanics or even picturing a wave chasing itself around in a circle. We see a spectrum resulting purely from a combination of the particle nature and the wave nature of electrons and photons, where the connection between the two points of view is de Broglie s formula p = h λ. Exercise Assume that an electron is trapped between two walls a distance apart. The distance has been adjusted so that the lowest energy level is E = 0.375eV. (a) What is? (b) What are the energies, in ev, of the photons in the six longest wavelength spectral lines radiated by this system? raw the energy level diagram for this system and show the electron jumps corresponding to each spectral line. (c) What are the corresponding wavelengths, in cm, of these six spectral lines? (d) Where in the electromagnetic spectrum (infra red, visible, or ultra violet) do each of these spectral lines lie? If any of these lines are visible, what color are they? (Partial answer: the photon energies are.5,.875,.65, 3.00, 3.375, and 4.5 ev) Exercise 3 Explain why an electron, confined in a box, cannot sit at rest. This is an important result whose consequences will be discussed next. Try to answer it now. E n = n E E4 = 6E E3 = 9E E = 4E Figure 6 Grid of quantum dots. These cells are made on a silicon wafer with the same technology used in making electronic chips. An electron trapped in one of these cells has energy levels similar to those of our model atom. (See Scientific American, Jan. 993, p8.) E = h 8m E=0 some electron transitions Figure 7 When an electron falls from one energy level to another, the energy of the photon it emits equals the energy lost by the photon.

7 37-7 ZERO POINT ENERGY One of the immediate consequences of the particlewave nature of the electron is that a confined electron can never be at rest. The smaller the confinement, the greater the kinetic energy the electron must have. This follows from the fact that at least half a wavelength of the electron s wave must fit within the confining region. If is the length of the smallest dimension of the confining region, then the electron s wavelength cannot be greater than. But the smaller is, the shorter the electron s wavelength is, the greater its kinetic energy. The de Broglie wavelength formula λ =h/p applies not only to photons and electrons, but to any particle, even an entire atom. As a result, an atom confined to a region of size should have a wavelength no greater than λ =, and thus a minimum kinetic energy E min = h 8m atom (7) where we simply replaced the electron's mass by the atom' mass in Equation 7. Equation 7 is somewhat approximate if the atom is confined on all sides in a three dimensional box, but it is reasonably accurate if is the smallest dimension of the box. An atom in a solid or a liquid is an example of a particle confined in a box. The atom is confined by its neighboring atoms as illustrated in Figure (8). We may think of its neighbors as forming a box of size where is the average spacing between atoms. Thus atoms in solids or liquids have a minimum kinetic energy given by Equation 7, and the atoms must be in continual motion no matter how low the temperature! Cooling the solid cannot get rid of this so-called zero point energy. Figure 8 A helium atom in liquid helium is confined by its neighbors. As a result it has a zero point energy like an electron confined between walls. our atom neighboring atoms Exercise 4 In liquid helium, the helium atoms are about 3A o apart and the atoms have a mass essentially equal to 4 times the mass of a proton. (a) what is the zero point energy, in ergs, of helium atoms in liquid helium? (b) at what temperature T is the helium atom's thermal kinetic energy 3/ kt equal to the zero point energy calculated in part (a)? [Answer: (a) ergs, (b) 4.4 kelvin.] Helium is an especially interesting substance to study at low temperatures because it is the only substance that remains a liquid all the way down to absolute zero. The only way you can freeze helium is to take it down to very low temperatures, and then squeeze it at relatively high pressure. In all other substances, at low enough temperatures the atoms settle down to a solid array. To melt the solid, you have to add enough thermal energy to disrupt the molecular bonds that hold the atoms in a more or less fixed array. Why can't helium atoms be cooled to the point where molecular forces dominate and the atoms form a solid array? Part of the answer is that the molecular forces between helium atoms are very weak, the weakest there is between any atoms. Consequently you have to go to very low temperatures before helium gas even becomes a liquid. At atmospheric pressure, helium becomes a liquid at 4.5 kelvins. To turn liquid helium into a solid you should have to go to still lower temperatures. From Exercise 4, you saw that, in one sense, you cannot get helium to a lower temperature, at least as far as the kinetic energy of the atoms is concerned. The zero point energy of the atoms is as big as the thermal energy that the atoms would have at a few kelvin kelvin by our rough estimate in Exercise 4. As a result, cooling the helium further cannot remove enough kinetic energy to allow the helium liquid to freeze. Helium thus remains a liquid all the way down to absolute zero.

8 37-8 Lasers, a Model Atom and Zero Point Energy efinition of Temperature This discussion raises interesting questions about the very concept of temperature. Our initial experimental definition of temperature was the ideal gas thermometer, which, as we saw from the derivation of the ideal gas law, is based on the thermal kinetic energy of the particles. The simple idea of absolute zero was the point where all the thermal kinetic energy was gone and the atoms were at rest. Now we see that no matter how much thermal kinetic energy we try to remove, zero point or "quantum kinetic energy" remains. This is not a problem at ordinary temperatures, but it can significantly affect the behavior of matter at temperatures close to absolute zero. At low temperatures, the ideal gas thermometer is not adequate, and a new definition of temperature is needed. That new definition is provided by the efficiency of Carnot's heat engine. As we suggested in Chapter 7, this gives us a definition of temperature based, not on the kinetic energy of the molecules, but upon the degree of randomness or disorder. A system at absolute zero is as perfectly ordered as it can be. If zero point energy is required by the particle wave nature of the atoms, if it cannot be removed, then the most organized, least disordered state of the system must include this zero point energy. Helium can go to its most ordered state at absolute zero, retain its zero point energy, and remain a liquid. TWO IMENSIONAL STANING WAVES In our discussion of percussion instruments in Chapter 6, we saw that a drumhead has a set of allowed standing wave patterns or normal modes, in some ways like the standing waves or normal modes on a guitar string. On a guitar string we have one dimensional waves, while the drumhead has the two dimensional wave patterns. The six lowest frequency patterns are shown in Figure (6-4) repeated here. We could excite and observe individual standing waves using the apparatus shown in Figure (6-40) also shown again here. That we get the same kind of standing wave patterns on an atomic scale is seen in Figure (9), which is a recent tunneling microscope image of an electron standing wave on the surface of a copper crystal. The standing wave, which is formed inside a corral of 48 iron atoms, has the same shape as one of the allowed standing waves on a drumhead. (This particular standing wave pattern is excited because the average wavelength in the standing wave is closest to the wavelength of the conduction electrons at the surface of the copper.) A colleague Geoff Nunes, who works with scanning microscopes, describes the image: The incredible power of today s personal computers has been made possible by our ability to make smaller and smaller transistors. The smallest transistor one could imagine building would be made up of single atoms. In a dramatic series of experiments at IBM, on Eigler and his co-workers have shown how to use a tunneling microscope to move and arrange single atoms. Figure 6-4 (reproduced) Standing waves on a drumhead.

9 37-9 This picture (Figure 9) shows a ridge of 48 iron atoms arranged in a circle on the surface of a copper crystal. Electrons in the copper are reflected from these iron atoms much as the waves on the surface of a pond are reflected from anything at the surface: rocks, weeds, the shoreline. Inside the ring, the electron waves form a beautifully symmetric pattern. This pattern occurs often in the physical world. For example it is the shape that the head of a drum forms when struck. You can easily observe a similar pattern by gently skidding the base of a Styrofoam cup full of coffee across the surface of a table. plywood frame rubber sheet speaker strobe light Figure 6-40 (reproduced) Exciting and observing the standing waves on a drumhead. Figure 9 Conduction electrons on the surface of a copper crystal, forming a standing wave inside a corral of 48 iron atoms. The shape is the same as one of the symmetric standing waves on a drumhead. (Photo credits: Crommie and Eigler/IBM.)

10 37-0 Lasers, a Model Atom and Zero Point Energy Index A Allowed standing wave patterns 37- Atoms Model atom 37-4 ebye, on electron waves 37- E Energy Kinetic energy In model atom 37-5 Zero point energy 37-7 Chapter on 37- Energy, kinetic, in terms of momentum 37-5 Energy level diagram Model atom 37-4 Photon in laser 37-4 K Kinetic energy In model atom 37-5 In terms of momentum 37-5 L Laser Chapter on 37- Standing light waves 37- Light Lasers, chapter on 37- M Model Atom 37-4 Chapter on 37- Energy levels in 37-4 Momentum Kinetic energy in terms of momentum 37-5 P Particle-wave nature Energy level diagrams resulting from 37-4 Photon Energy Energy levels in laser 37-4 Standing waves 37-3 Q Quantum mechanics Model atom 37-4 Zero point energy 37-7 S Schrödinger wave equation Felix Block story on 37- Schrödinger, Erwin 37- Standing waves Allowed standing waves in hydrogen 37- Light waves in laser 37- Photons in laser 37-3 Two dimensional 37-8 Electrons on copper crystal 37-9 On drumhead 37-8 T Temperature And zero point energy 37-8 Two dimensional standing waves 37-8 W Wave Standing waves Allowed 37- Two dimensional 37-8 Wave equation Schrödinger's iscovery of 37- X x-ch37 Exercise 37-4 Exercise 37-6 Exercise Exercise Z Zero point energy 37-7 And temperature 37-8 Chapter on 37-