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Many of the predictions of quantum physics appear to be contrary to our intuitive perceptions, and the student will learn how it comes about that microscopic objects particles behave in unusual ways that are called quantum effects, what we mean by quantum, and where this idea came from. The textbook is supplemented with Problems and Solutions in Quantum Physics, which contains a wide range of tutorial problems from simple confidence builders to fairly challenging problems that provide adequate understanding of the basic concepts of quantum physics.
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We can continue adding up to two electrons to each energy state until all have been accommodated. We now apply this process to atoms, first considering helium, which has two electrons. If we initially ignore the fact that the electrons exert a repulsive electrostatic force on each other, we can calculate the quantum states in the same way as we did for hydrogen, but allowing for the fact that the nuclear charge is doubled.
This doubling means that all the energy levels are considerably reduced i. The lowest energy state therefore has both electrons with opposite spin in the lowest. In the case of lithium with three electrons, two of these will be in the lowest state, while the third must be in the next higher energy state.
The latter state can actually contain a total of six electrons: two of these occupy a state of spherical symmetry while the others fill three separate non-spherical states. Thus lithium has one electron outside a closed shell, as does sodium with eleven electrons — i. The whole struc- ture of the periodic table can be understood in terms of the atomic shell structure, which in turn is a consequence of the properties of the quantum waves associated with the electrons.
Although the above considerations allow us to describe the electronic structure of atoms in some detail, it is much harder to make a precise calculation of the energy levels.
Moreover, the exact agreement between the energies of spheri- cal and non-spherical states only holds in the case of hydrogen, so the spectra of the other atoms are generally much more complex. However, modern computational techniques have largely taken over where traditional mathematics has failed. When applied to any atom, these produce values for the allowed energy levels and numerical representations of the wave functions which are in good agreement with experiment.
All the evidence points to quantum physics providing a complete description of the physical properties of matter at the atomic scale. Summary This chapter has introduced the main ideas of quantum physics, which will be developed and applied to various physical situa- tions in the chapters to come. Readers would be well advised to ensure that they understand these basic principles, which are summarized below. They are all typified by a frequency, which determines how many times per second any point on the wave vibrates, and a wavelength, which measures the repeat distance along the wave at any time.
This is exemplified in the notes produced by musical instruments. Because an electron can be in one of two spin states, this means that each standing wave can contain up to two electrons.
Both are adjusted in many instruments: e. However, this oscillating phase plays little, if any, part in determining the properties we shall be discussing. The magnitude of the wave function can also vary, but only in circumstances where the energy of the particle is not well defined, and I shall not be discussing these in this book.
However, the size of this spread is similar to dp as defined above. The basic result that an electron can be in one of two spin states is also a consequence of this. As soon as humankind discovered fire and how to use it, quantum physics was directly involved in energy production and this is still true for many of the forms of energy generation that play an essential role in modern life.
We burn petrol in our cars and use gas or oil to heat our homes. Much of the power we use reaches our homes in the form of electricity, although it is important to remember that this is not in itself a source of power, but only a method of transferring energy from one place to another. Of all these, only wind and wave power do not depend directly on quantum physics. Chemical fuels A fuel such as wood, paper, oil or gas contains many hydro- carbons, which are compounds consisting mainly of hydrogen and carbon atoms.
When these are heated in air, the hydrogen and carbon combine with oxygen from the air to make water and carbon dioxide, respectively. To see how this depends on quantum physics, we start with the simplest example of chemical combination, which is two hydrogen atoms coming together to form a hydrogen molecule — see Figure 3.
The hydrogen atom contains one electron attracted to a proton whose charge is equal and opposite to that of the electron, so a hydrogen molecule is composed of two protons and two electrons. Now consider how the total energy of the system will be affected if we bring two hydrogen atoms towards each other.
We first consider the potential energy which changes in three ways. First, it increases because of the electrostatic repulsion between the two positively charged protons; secondly, it decreases because each electron is now subject to attraction by both protons; thirdly it increases because of the repulsion between the two negatively charged electrons.
In addition, the kinetic energy of the electrons decreases because the electrons are able to move around and between the two nuclei, so the size of the effective box confin- ing them is increased.
We saw in Chapter 2, when we discussed the quantum behaviour of a particle in a box, that the larger the box, the lower is the kinetic energy of the ground state. We also note that the Pauli exclusion principle allows both electrons to occupy the ground state, provided they have opposite spin.
The net effect of all these changes depends on how far the atoms are apart: when they are widely separated, there is little change in the total energy, and when they are very close, the electrostatic repulsion between the nuclei dominates. The graph b shows the variation of the energy of the system as the separation between the hydrogen atoms changes.
The final state of the molecule corresponds to the point of lowest energy marked P. Figure 3. At this point, the differ- ence between the energy of the molecule and that of the widely separated hydrogen atoms equals about one third of the ground- state energy of the hydrogen atom.
Where does this surplus energy end up? The answer is that some of it goes into the kinetic energy of the moving molecule, while the rest is given off in the form of photons. Both are effectively forms of heat, so the overall effect is a rise in temperature, which is just what we expect from a fuel. The above example illustrates the principle of how energy can be released by bringing atoms together to form molecules, but the particular case of hydrogen is not in practice a useful source of energy, because any hydrogen gas we have on Earth is already composed of molecules.
A more practical example is the combination of hydrogen and oxygen to make water: the ground state energy of the water molecule is less than the total ground state energies of the single oxygen and two hydrogen atoms that are its constituents.
However, like hydrogen, oxygen gas is also composed of diatomic molecules and if we simply mix hydrogen and oxygen together at room temperature, nothing happens. This is because, before they can combine to form water, the oxygen and hydrogen molecules must first be split into their constituent atoms, which requires an input of energy from an external source. However, once a few water molecules have formed, the energy released in this process is more than sufficient to split apart some more hydrogen and oxygen molecules, and the process very quickly becomes self-sustaining.
An example of this is lighting a gas flame in a laboratory or kitchen, using a match: the high temperature produced by the match splits some of the nearby hydrogen and oxygen molecules and the resulting atoms combine to form water molecules, with a release of energy that heats more of the gas to the point where it too can ignite.
The process is then self-sustaining and the heat produced can be used to warm a house, boil a kettle, etc.
A hydrocarbon fuel, such as oil or gas, contains molecules composed primarily of carbon and hydrogen, which have remained stable for a long time — perhaps millions of years. This stability is maintained even when the compounds are exposed to air at room temperature, but once energy is supplied to split the molecules, the atoms rearrange themselves into a mixture of water and carbon dioxide with the release of energy. The princi- ples involved are those of quantum physics: the total energy of the quantum ground states of the water and carbon dioxide molecules is less than that of the initial hydrocarbon molecules.
However, to initiate this change, energy must be supplied; once the mixture has been heated to a sufficiently high temperature, the process becomes self-sustaining and unless the process is extinguished energy continues to be released until the fuel is exhausted. Nuclear fuels The principles of nuclear power are remarkably similar to those underlying the burning of chemical fuels, although the amounts of energy involved in the nuclear processes are very much greater.
As we saw in Chapter 1, the nucleus of an atom is made up of a number of protons and neutrons bound together by the strong nuclear force. The structure of the nucleus is also subject to the rules of quantum physics, although the details are rather more complex than the atomic case discussed in Chapter 2.
This is because the latter is dominated by the attraction of the electrons to the heavier nucleus, whereas the interactions between the protons and neutron inside the nucleus are all of similar mass. As discussed in Chapter 1, deuterium is an isotope of hydrogen whose nucleus is composed of a proton and a neutron and which makes up about 0. As the neutron carries no charge, the extra positive charge must go somewhere and it is actually carried off by the emission of a positron which is the same as an electron but with a positive charge and a neutrino a very small neutral particle.
The ground state energy of the deuterium nucleus is consider- ably lower than that of two protons, so we might have expected that all the protons in the universe would have been fused into deuterium nuclei many years ago, in the same way that practi- cally all hydrogen atoms have formed hydrogen molecules.
The fact that this has not happened is due to the electrostatic repul- sion between the two positively charged protons.
As the protons come together, the electrostatic repulsion increases to a very large value before the nuclear force kicks in, creating a potential barrier as illustrated in Figure 3. Classically, this barrier would completely prevent the protons ever combining, but, in princi- ple, they can penetrate it by quantum tunnelling see Chapter 2.
Detailed calculations show that the probability of this happening is very low unless the protons are moving towards each other at very high speed, in which case the effective tunnelling barrier is both lower and narrower — see Figure 3.
However, before this can occur, they must tunnel through the potential barrier created by the electrostatic repulsion. The probability of this happening is very low, unless the protons approach each other at high speed and therefore high kinetic energy, which makes the effective barrier lower and narrower. Note: the scale of this diagram is about two thousand times larger than that in Figure 3.
The energy obtained as a result of the fusion process is also high: that released from the fusion of two protons is about ten million times that associated with the formation of a hydrogen molecule from two hydrogen atoms. One place where temperatures as high as a million degrees occur naturally is the sun, and indeed nuclear fusion is the process that keeps the sun shining. Many other fusion processes besides that of two protons to form deuterium take place there and the end point is the most stable nucleus of all — that of iron.
In this case, the ignition is achieved from a nuclear explosion generated using atomic fission, which will be discussed shortly. This heats the material to a temperature high enough for fusion to start, after which it is self-sustaining and an enormous explosion results.
The gener- ation of controlled fusion power that could be used for peaceful purposes has been an aim of nuclear researchers for over fifty years. International collaborations such as the JET2 project have been formed to pursue this and it is now believed that a machine capable of producing significant amounts of fusion power will be built during the first half of the twenty-first century.
This work has been largely discredited, but some efforts continue in this direction. Beyond this, the trend is reversed and splitting a nucleus of a heavier element into pieces may well result in a lowering of the total ground-state energy and the release of energy.
To understand this in a little more detail, consider an example where a heavy nucleus splits into two equal-sized fragments Figure 3. Before this occurs, the two parts are held together by the strong nuclear force, but once they are separated the electrostatic repulsion between the positively charged fragments takes over, pushing them further apart into a yet lower energy state and releasing the surplus energy.
We can think of this as the reverse of the process of fusing two deuterons to make an alpha particle, except that in the present case the energy of the widely separated state is less than that of the ground state of the united nucleus so energy is released when the nucleus splits — i. However, an initial energy barrier has to be surmounted before fission can occur, which would seem to imply a need to inject energy into the system.
To do so would be even more imprac- tical than in the case of fusion, so a different approach is needed. The key to the initiation of fission lies in some of the detailed properties of nuclear structure. When this is analysed, taking full account of the nuclear forces and the electrostatic repulsion, it is found that stable bound states occur only for a limited number of particular combinations of protons and neutrons.
In contrast, a nucleus formed with one fewer nucleon i. U is relatively stable and a small amount a little less than one per cent occurs in natural uranium. The released neutrons can cause fission in other U nuclei, which can produce a chain reaction. As a neutron carries no charge, there is no energy barrier preventing it from entering a U nucleus, so converting it to U, which then undergoes fission.
We note that the neutron does not have to possess any extra energy for this to happen; indeed, if it is moving too fast, it is likely to pass by the U nucleus without interacting with it.
We do not have to ignite a fission process with energy, but to start it off we do have to supply neutrons. In particular, radiation in the form of high- energy alpha particles i. He4 nuclei is produced as well as some free neutrons. These are then available to induce fission in other U nuclei that may be nearby.
To start it off, we seem to need to seed it with some neutrons, but in fact there is a small probability of U undergoing spontaneous fission and producing a few neutrons. Some of these may strike other uranium nuclei, inducing them to split also. If the piece of uranium is small in size, many of the neutrons will escape from it and the process will not be self-sustaining, but in the case of a large sample the process will multiply and form a chain reaction.
In an atomic bomb, this is achieved by bringing two or more smaller masses together very quickly using a conventional explosive; in contrast, a nuclear reactor is designed so that the fission process is controlled in order that the energy released can be used to gener- ate electricity. Neither process can be realized without materials that contain a sufficiently high concentration of U No doubt fortunately, this enrichment process is difficult and expen- sive and constitutes one of the major technological barriers to the use of nuclear energy, particularly in the weapons field.
There are several different designs of nuclear reactor. As a result, the water is heated up and the slowed-down neutrons cause further fission. Rods made of neutron-absorbing material are raised or lowered to control the rate of the reaction.
The high-pressure hot water is used to create steam, which in turn generates electricity. Rods of enriched uranium are held in a vessel along with water under high pressure. It turns out that slow neutrons are considerably more efficient at inducing fission in uranium nuclei than are neutrons of higher energy, so the moderation leads to an increase in the efficiency of the fission process.
Because the water in the reactor is at very high pressure, it can be heated to a high temperature without boiling. Rods of a material such as boron that absorbs neutrons passing into it can be lowered into the water; this reduces the number of neutrons available to induce further fission and so allows the rate of the reaction to be controlled. Both fission and fusion produce energy by inducing transi- tions from quantum states of high energy to other more stable lower energy states.
The laws of quantum physics and the wave properties of the neutrons and protons in turn determine the energies of these states. Once again the practical importance of quantum physics is demonstrated.
The reason for this is a very common misunderstanding. However, the mass loss is not the cause of the energy change which is the strong nuclear force and the electrostatic repulsion but an inevitable consequence of it.
Thus, when we burn hydrogen and oxygen to form water, the mass of a resulting water molecule is a little less than the total masses of the hydrogen and oxygen atoms it is composed of. However, in the combustion case these changes are extremely small and difficult to measure typically, less than one part in a hundred million whereas in the case of nuclear energy they are much more significant: e. However, this does not change the fact that it is the energy change that causes the mass change rather than the other way around.
Some initial concerns were focused on nuclear energy, where the inevitable radiation accompanying nuclear processes and the disposal of radioactive waste products constitute hazards, which some feared could not be controlled. This was exacer- bated by a small number of quite major nuclear accidents, partic- ularly that in Chernobyl in the Ukraine, which released a considerable amount of radioactive material across Europe and beyond. More recently, however, the long-term consequences of more traditional methods of energy production have become clear.
There is even the possibility of a runaway process in which heating would result in more heating until the Earth became completely uninhabitable. The greenhouse effect is so named because it mimics the processes that control the behaviour of a glass greenhouse of the type found in many gardens.
Sunlight passes through the transparent glass without being absorbed and strikes the earth and other contents of the greenhouse, warming them up. The warmed objects then try to cool down by emitting heat radiation, but this has a much longer wavelength than that of light and cannot easily pass through the glass, which reflects most of the heat back into the greenhouse cf. This process continues until the glass has warmed up to the point where it radiates as much power outwards as that of the sunlight coming into it.
The latter process is assisted by convection: air near the bottom of the greenhouse is heated, becoming less dense and rising to the top of the green-house, where it helps warm the glass as it cools and then falls back downwards.
This is where quantum physics plays an important role. As I pointed out in Chapter 2 and have noted several times in this chapter, when electrons are confined within an atom or molecule, wave—parti- cle duality ensures that the energy of the system must have one of a set of quantized values. Moreover, the excitation of such a system from its ground state can be caused by the absorption of a photon, but only if its energy matches the difference between the energies of the levels.
The wavelength of this radiation is much longer than that of the sunlight and cannot readily pass through the glass, so there is a net heating. A photon that strikes one of these molecules can be absorbed, leaving the molecule in an excited state.
It quickly returns to the ground state by emitting a photon, but this can be in any direction and it is just as likely to return towards the Earth as it is to be lost to outer space. As mentioned above, water is also an effective greenhouse gas, but the amount of water vapour in the atmos- phere is determined by a balance between the evaporation of liquid water on the Earth, notably the surface of the oceans, and its re-condensation.
This is controlled by the temperature of the Earth and its atmosphere and remains largely unchanged. However, in the short term at least, our concern is not with water vapour but with other gases, such as methane and especially carbon dioxide.
Such an increase is today caused by human activity, particularly the burning of fossil fuels. The concentra- tion of carbon dioxide in the atmosphere is estimated to have increased by about thirty per cent since industrial activity began in about , and is currently increasing by about 0. We saw in Chapter 2 that the energy required to excite an electron from the ground state of a typical atom corresponds to that of a photon associated with visible light. The key point here is that in a molecule the atomic nuclei can be made to vibrate relative to each other.
When we discussed the formation of molecules earlier in this chapter, we found that the distance between the nuclei corresponds to the point where the various contributions to the energy add up to the smallest possible total Figure 3. This means that if we were able to move the nuclei a little away from this equilibrium position, the energy would be raised, so that if we now released them the nuclei would move back towards the equilibrium point, converting the excess energy into kinetic energy associated with their motion.
They would then overshoot the equilibrium point, slow down and return, and this vibrational motion would continue indefinitely unless the energy were lost in some way.
In this sense a molecule behaves as if the nuclei were point masses connected by springs, under- going oscillation as the springs stretch and contract. The above applies just as much to nitrogen and oxygen as it does to carbon dioxide and water, so we still have to understand why heat radiation can induce vibration in the latter two gases but not the former two. To address this, we first recall the discussion of the wave function of the electrons in an atom in Chapter 2, where we noted that as long as the atom stayed in its ground state, we could think of the electronic charge as being diffused over the volume of the atom, with a concentration at any point that is proportional to the square of the wave function at that point.
A similar situation applies to the ground state of a molecule; to a first approximation the charge distribution has the form of over-lapping spherical clouds, as indicated in Figure 3.
Because the two hydrogen atoms are identical, this molecule is symmetric and the two overlapping charge clouds are also identical. However, this is not true in the case of more complex molecules. Considering the lowest energy state of carbon dioxide in particular, it turns out that the total charge in the cloud surrounding the central carbon atom is a little less than six electronic charges and so does not fully balance the charge on the carbon nucleus, while the charge surrounding each oxygen atom is a little more than that corres- ponding to the total of eight associated with a free oxygen atom.
The net effect of this is that, although the total electronic charge on the molecule balances the total nuclear charge, each oxygen atom carries a small net negative charge, and a balancing positive charge is associated with the carbon atom.
We now consider what happens when the molecule is subjected to an electric field directed along its length. Returning to Figure 3. The atoms in a molecule can move as if they were connected by springs as illustrated in b.
In the carbon dioxide molecule, the central carbon carries a net positive electric charge and the two outer oxygen atoms are negatively charged. When an electric field is applied to the molecule, oppositely directed forces are applied to the oxygen and carbon atoms, which then respond as shown in c , so exciting the vibra- tion illustrated in b.
A similar process occurs when an applied field is perpendicular to the line of the molecule: the carbon atom moves in one direction and the two oxygens in the other, an effect that in this case causes the molecule to bend.
This also leads to a greenhouse effect for radiation of the appropriate frequency. Why then does a similar effect not arise in a molecule like oxygen or nitrogen? The reason is that such a molecule contains two identical atoms, which must therefore either be neutral or carry the same net charge.
In either case, they cannot be pushed in opposite directions by an electric field, so a vibration cannot be set up by an electromagnetic wave and such a gas cannot contribute to the greenhouse effect. If quantum physics plays a role in creating the greenhouse effect and its associated problems, can it also help us avoid and resolve them? We already know that nuclear reactions are governed by the laws of quantum physics and produce no carbon dioxide or other greenhouse gases. Thus, the generation of nuclear energy both fission and fusion makes no contribu- tion to the greenhouse effect.
We have seen that nuclear energy may have problems of its own and it certainly has had a bad press since about However, some environmentalists have been revising their opinions in recent years. Although air and water are composed of atoms, which in turn depend on quantum physics for their existence and properties, the motion of wind and waves is governed by classical physics and is independent of the internal structure of the atoms, so, as we noted in Chapter 1, we do not classify this as a manifestation of quantum physics.
Solar energy comes in two main forms. This is discussed in the next two chapters. We return to the operating principles of photovoltaic cells towards the end of Chapter 5. Summary This chapter has discussed the role of quantum physics in the production of the energy we use to power our civilization. This principle underlies all energy production by chemical combustion.
Fusion is the process that fuels energy production by the sun and the stars and the explosion of a hydrogen bomb.
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