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- 1. Chapter 10 introduction to quantum mechanics David Morin, Morin@physics.edu This chapter gives a brief introduction to quantum mechanics. Quantum mechanics can be thought of roughly as the study of physics on very small length scale, although there is also certain macroscopic system it directly applies to the description “quantum” arises because in contrast with classical mechanics, certain quantities take on only discrete value. However, some quantities still take on continuous values, as we see. In quantum mechanics, particles have wavelike properties, and a particular wave equation, the Schrödinger equation, governs how these waves behave. The Schrödinger equation governs how these waves behave. The Schoninger equation 9 is different in a few ways from the other wave equations we have seen in this book. But these differences would not keep from applying all of our usual strategies or solving a wave equation and dealing with the resulting solutions. In some respect, quantum mechanics is just another example of a system governed by a wave equation. In fact, we will find below that some quantum mechanical system has exact analogies to systems we have already studied in this book. So, the results can be carried over, with no modifications whatsoever needed. However, although it is fairly straightforward to deal with the actual waves, there are many things about quantum mechanics that combination of subtle, perplexing and bizarre. To name a fer: the measurement problem, hidden variables along with Bell theorem, and wave particle duality. You shall learn all about these in an actual course on quantum mechanics. Even though there are many things that are highly confusing about quantum mechanics, then I use thing is that it's relatively easy to apply quantum mechanics to a physical system to figure out how it behaves. There is fortunately no need to understand all of the subtleties about quantum mechanics in order to use it. Of course, in most cases this is not the best to strategy to take: it's usually not a good idea you blindly forge ahead with something if you did not understand what you are actually working with. But this lack of understanding can be forgiven in the case of quantum mechanics, because no one really understands it. (Well may be a couple people do, but they are few and far between.) If the world waited to use quantum mechanics until it understood it, then we had been stuck back in the 1920s. The bottom line is that quantum mechanics can be used to make prediction that are consistent with the experiment. It has not failed us yet. So, it would be foolish not to use it. The main purpose of this chapter is to demonstrate how similar certain results in quantum mechanics are to earlier results we have derived in the book. You are we actually know a good deal of quantum mechanics already, whether you realize it or not. The outline of this chapter is as follows. In Section 10.1 we gave a brief history of the development of quantum mechanics. In section 10.2 we write down, after some motivation, the Schrödinger wave equation, both the time in the important thing to take away from this section is section 10.3. We discuss a number of examples. The most important thing to take away this section is that all of the examples we discuss have exact analogies in the string/spring system earlier in the book. So, we technically would not have to solve anything new
- 2. Here. All the work has been done before. The only thing new that we shall have to do is interpret the old results. In section 10.4 we discuss the uncertainty principle. As in section 10.3 we will find that we already did the necessary work earlier in the book. The uncertainty principle turns out to be a direct consequence of a result from the Fourier analysis. But the interpretation of this result as an uncertainty principle has profound implications in the quantum mechanics. 10.1 A brief history Before discussing the Schrödinger wave equation, let's take a brief (and by no means comprehensive) look at the historical timeline of how quantum mechanics came about. The actual history is of course never as clean as an outline like this suggests, but we can at least get a general idea of how things proceed. 1900 (PLANK): Max Planck supposed that light with the frequency as emitted and the quantized lumps of energy that come in integral multiple of this quantity. E=HV=HW Where h~6.63.10’-34J.s is plank`s constant, and h=h2£=1.06*10’-34J.s. The frequency we of light is generally very large (on the order of 1015 per second for the visible spectrum), But the smallness of “h” wins out, the HV unit of energy is very small (at least on an everyday energy scale). The energy is therefore essentially continuous for most purpose. However, a puzzle in late 19th -century physical was the blackbody radiation problem. In a nutshell, the issue was that the classical (continuous) theory of light predicted that certain objects would radiate an infinite amount of energy, which of course cannot be cracked. Plank's hypothesis of quantized meditation not only got rid of the problem of the Infinity, but also correctly predicted the shape of the power curve as a function of temperature. The results that we derived from the electromagnetic wave is chapter 8 are still true. In particular, the energy flux information is given by the pouting vector in Eq.8.47 and E=PC for a light. Plank’s hypothesis simply odds the information of how many lumps of energy a wave content. Although strictly speaking, Plank initially thought that the quantization was only a function of the mission process and not inherent to the life itself. 1905 (EINSTEIN): Albert Einstein stated that the quantization was in fact inherent to the light and that the lamps can be interpreted as particles, which we now call “photons”. This proposal was a result of his work on the photoelectric effect, which deals with the absorption of light and emission of electron from material. We know from Chapter 8 that E=PC for a light wave. (This relation also follows from Einstein’s 1905 work on the relativity, where he showed that E=PC for any massless particle, and example of which is a photon.) And we also know that W=cK for a light wave. So, Plank’s E=HW relation becomes, E=HW~~PC=h(ck)~~p=hk This result relates the momentum of a photon to the wavenumber of the wave is it is associated with.
- 3. 10.2. A BRIEF HISTORY 1913 (BOHR): Niels Bohr stated that electrons in atom have wavelike properties. This correctly explained a few things about hydrogen, in particular the quantized energy levels that were known. 1924(DE BROGLIE): Louis de Broglie propose that all particles are associated with waves, where the frequency and wave number of the waves are driven by the same relations we found above for photons, namely E=hw and P=hk. The larger E and P are, the larger W&K are. Even for small E and p that are typical of a photon, % and k are very large because h is so small. So, any everyday-sized particle with large (in comparison) energy and momentum value will have extremely large w and k values. This (among other reasons) makes it virtually impossible to observe the wave nature of microscopic amount of matter. This proposal (that E =hw and P=hk also hold for massive particles was a big step. Because many things that are true for photons are not true for massive and (nonrelativistic) particles. For example, E=pc (and hence w=ck) holds only for massless particles (we shall see below how w and k are related for massive that the resulting prediction agree with experiments. The fact that any particle has a wave associated with it leads to the so-called wave particle duality. Are things particles, or waves, or both? Well, it depends what you are doing with them. Sometimes things behave like waves, sometimes they behave like particles. A vaguely true statement is that things behave like waves until a measurement takes place, at which point they behave like particles. However, approximately 1,000,000 things are left unaddressed in that sentence. The waves particle duality is one of the things that few people, if any, understand about quantum mechanics. 1925 (HEISENBERG): Werner Heisenberg formulated a version of quantum mechanics that was based on matrix mechanics. We would not the deal with those matrix formulation (it's rather difficult). But instead with the following wave formulation due to the Schrödinger (this is the waves book, after all). 1926 (SCHRODINGER): Erwin Schrodinger formulated a version of quantum mechanics that was based on wave. He wrote down a wave equation (the so-called Schrödinger equation) that governs how the waves evolves from space and time. We shall deal with this equation with depth below. Even though the equation is correct, the correct, the correct interpretation of that the wave actually mean was still missing. Initially Schrödinger thought (incorrectly) that the wave represented the charge density. 1926 (BORN): Max born correctly interpreted Schrödinger waves as a probability amplitude. By “amplitude” we mean that the wave must be squared to obtain the desired probability. More precisely, since the wave (as we shall see) as in general complex, we need to square its absolute value. This yields the probability of the finding a particle at a given location (assuming that the written as a function of x). This probability is not a consequence of ignorance, as it the case with the virtually every other example of probability you are familiar with. For example, in a toss, if you know everything about the initial motion of the coin (velocity, angular velocity) along with all external influences (Air current nature of the floor it lands on, etc.) Then you can predict which side will land facing up. Quantum mechanical probabilities are not like this. They are not a consequence of missing information. The probabilities are truly random. And there is no further information (so called “hidden variable”)
- 4. that will make things un-random. The topic of hidden variables includes various there’s (such as Bell’s theorem) and experiments results that you will learn about in a quantum mechanics course. 10.3. EXAMPLES It turns out that energies and states again discrete and can be labeled by an integer n, just as in the infinite well case. However, the energies do not take the simple from an equation (18), although they approximately to fit the well in deep. Figure.6 shows the five states for a well of a particular depth V°’. We have drawn each wave relative to line that represents the energy E. Both ¥ and ¥ continuous at x=+a and ¥ goes to 0 at x=+••. We’ve chosen that various parameter (one of which is the depth) so that there are exactly 5 states (see problem 10.2 for the details on this) the deeper the well, the more states there are consistent with equation ¥ is indeed oscillatory inside well (that is, the curvature is not towards x axis). Increase (so the wiggles inside the well have shorter wavelength). and also, that decreases (so the exponentials decay is slower). These facts are evident in figure.6. the exact details of the waves depend on various parameters, but the number of bumps equals n. Explanation of the quantized energy levels The most important things to note about these states is that they are discrete. In the infinite-well case this discreteness was clear because an integral number of half wavelengths needed to fit into the well (because ¥=0 at the boundaries). The discreteness is not too obvious in the finite well case (because ¥=0 at boundaries), but it is still reasonably easy to see. There are two ways to understand it first, the not-so-enlightening way as to note that we initially had 7 equations 7 unknowns. So, all the unknown, including E, are determined. There may be different discrete solutions, but at least we know that we can't just take a random value for E and expert, it to work. Now pick a point x• to in the left region, and imagine marching rightward on the x axis. We claim that all the subsequent values of ¥ are completely determined, all the way up to =+••. This is clear in the left region. Because we know that the function is. But it is also true in the middle and right regions, because we know that values of ¥, ¥ and ¥ at any given points, so we can recursively find these three values at the “next” point as we march along. More precisely; (a) from the definition of the derivative, the value of ¥ and ¥ at the given point yield the value of the next point, (b) likewise, the values of the ¥ and ¥ at a given point yield the value of ¥ that at the next point, and (c) finally, the value of ¥ at a given the point yields the value of ¥” at that point, via the Schrödinger equation (9). So, we can recursively march all the way up to x=+••., and the entire function is determined. There is no freedom whatsoever in what the function turns out to be. A particular choice of E and A, might yield the first (top) function shown in the fig. It has the correct exponential and oscillatory nature in the left and middle region, respectively. But in the right region it apparently has an exponentially growing piece. Because it diverges at x=+•• this function is not an allowable one. So, it is impossible for the energy to take on the value that we chose. We can try to remedy that divergence by increasing at x=+•• by increasing the value of E. This will make ¥ oscillate quicker inside the well, So, that it encounters the x=a boundary with a negative slope.
- 5. We then might end up with the second function shown in the figure.7. We still have the divergence at x=+••, so again we have an invalid ¥. If we increase E a little more, by precisely the right amount, then we shall end up with the third function and shown in the figure.