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Giáo trình Tiếng Anh nâng cao chuyên ngành Vật lý: Phần 2

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Giáo trình Tiếng Anh nâng cao chuyên ngành Vật lý: Phần 2 cung cấp cho người học những kiến thức về vật lý thống kê (Statistical physics). Phần này cũng hướng dẫn các bạn thực hành viết báo cáo khoa học và trình bày trước lớp. Mời các bạn cùng tham khảo.

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Tron Bo SGK: https://bookgiaokhoa.com CHAPTER 4 STATISTICAL PHYSICS PART 4.1: Reading, English-Vietnamese translation See 39. The Equilibrium Distribution o f M olecules in an ideal gas The subject o f statistical physics T he methods of quantum mechanics set out in the third chapter make it possible, in principle, to describe any assembly o f electrons, atoms, and molecules comprising a macroscopic body. In practice, however, even the problem o f an atom with two electrons presents such great mathematical difficulties that nobody, so far, has solved it completely. It is all the more impossible not only to solve but even to write down the wave equation for a macroscopic body consisting, for example, o f 102' atoms with their electrons. •k J Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 168 CHAPTER 4. STATISTICAL PHYSICS Yet in large systems, we encounter certain general laws of motion for which it is not necessary to know the wave function of the system to describe them. Let us give one very simple example of such a law. W e shall suppose that there is only one molecule contained in a large, completely empty vessel. If the motion of this molecule is not defined beforehand, the probability o f finding it in any half of the vessel is equal to 1/2. If there are two molecules in the same vessel, the probability of finding them in the same half of the vessel simultaneously is equal to ( l / 2 ) 2= l/4 . The probability o f finding all of a gas, consisting of N particles, in the same half of the vessel (if the vessel is filled with gas) is ( l / 2 ) N, i.e., an unimaginably small number. O n the average, there will always be an approximately equal number of molecules in each half of the vessel. The greater the number o f molecules forming the gas, the closer to unity will be the ratio of the num ber of molecules in both halves o f the vessel, no matter at what time they are observed. This approximate equality for the number of molecules in equal volumes of the same vessel gives an almost obvious example of a statistical law applicable only to a large assembly of objects. In addition to a spatial distribution, molecules possess a definite velocity distribution, which, however, can in no way Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com ADVANCED ENGLISH FOR PHYSICISTS 169 be uniform (if only because the probability of an infinitely large velocity is equal to zero). Statistical physics studies the laws governing the motion of large assemblies of electrons, atoms, quanta, molecules, etc. T he problem of the velocity distribution of gas molecules is one of the simplest that is solved by the methods of statistical physics. Statistical physics introduces a series of new quantities, which cannot be defined in terms o f single body dynamics or the dynamics o f a small number of bodies. An example of such a statistical quantity is temperature, which turns out to be closely related to the mean energy of a gas molecule. If a gas is confined only to one half o f a vessel, and the barrier dividing the vessel is then removed, the gas will itself uniformly fill both halves. Similarly, if the velocity distribution o f the molecules is disturbed in some way, then, as a result o f collisions between the molecules there will be established a very definite statistical distribution, which, for constant external conditions, will be maintained approximately for an indefinitely long time. This example involving collisions shows that regularity in statistical arises not only because a large assembly o f objects is taken, but also because they interact. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 170 CHAPTER 4. STATISTICAL PHYSICS The statistical law in quantum mechanics Quantum mechanics also describes statistical regularities, but relating to a separate object. Here, the statistical regularity manifests itself in a very large number of identical experiments with identical objects, and in no way related to the interaction of these objects. For example, the electrons in a diffraction experiment may pass through a crystal with arbitrary time intervals and nevertheless give exactly the same statistical picture for the blackening of a photographic plate as if they had passed through the crystal simultaneously. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com ADVANCED ENGLISH FOR PHYSICISTS 171 Regularities in alpha disintegration cannot be accounted for by the fact that there is a very large num ber of nuclei: since there is practically no interaction between nuclei inducing the process, the statistical character predicted by quantum mechanics is only manifested for a large number of identical objects; it is by no means due to their number. In this connection, a description of phenomena in quantum mechanics involves the concept of probability phase, similar to the concept of the phase of a light wave. In principle, the wave equation can also be applied to systems consisting o f a large num ber o f particles. T he solution of an equation represents a detailed quantum mechanical description of the state of the system. Let us suppose that as a result of the solution of the wave equation we have obtained a certain spectrum of energy given values of the system $=&0,&\,&2, ... (39.1) in states with wave functions Vo»Vi.V2 »-»Vn. - Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 172 CHAPTER 4. STATISTICAL PHYSICS Then the wave function for any state, as was shown in See. 30, can be represented in the form of a sum of -functions of states with definite energy values: (39.2) n The square of the modulus .2 — c n| gives the probability (when the energy of a system in the state vj/is measured) that the result will be the nth value. The expansion (39.2) makes it possible to determine not only amplitudes, but also relative probability phases corresponding to a detailed quantum mechanical description of the system. The methods of statistical physics make it possible straightway to determine approximately the quantities (on = |c n |2 , i.e., the probabilities themselves omitting their phases. For this reason, the wave function of the Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com ADVANCED ENGLISH FOR PHYSICISTS 173 system cannot he determined from them, although it is possible to find the practically important mean values of quantities that characterize macroscopic bodies (for example, their mean energy). In this section we shall consider how to calculate the probabilitycon as applied to an ideal gas. Ideal gases An ideal gas is a system of particles whose interaction can be neglected. The interaction resulting from collisions between molecules is essential only when the statistical distribution con is in the process of being established. W hen this distribution becomes established the effect of interaction is very slight. As regards condensed (i.e., solid and liquid) bodies, the molecules are all the time in vigorous interaction, so that the statistical distribution depends essentially upon the forces acting between the bodies. But even in a gas the particles must not be regarded as absolutely independent. For example, Pauli's principle imposes essential limitations on the possible quantum states of a gas. W e shall take these limitations into account when calculating probabilities. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 174 CHAPTER 4. STA TISTICAL PHYSICS The states o f separate particles o f a gas In order to distinguish the states of separate particles from the state of the gas as a whole we shall denote their energies by the letter e and the energy of the whole gas by &. Thus, for example, if the gas is contained in a rectangular potential well (see Sec. 25), then the energy values for each particle are calculated according to equation (25.19). Let e take on the following series ofvalues: e =e0,e1,e2.... en,... (39.4) where there are no particles in the state with energy e 0 and in general there are nk particles wit energy ek in the gas. Then the total energy o f the gas is ® = £ nkek (39.5) k By giving different combinations of numbers m, we will obtain the total energy values forming the series ( 39. 1 ). Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com ADVANCED ENGLISH FOR PHYSICISTS 175 W e have repeatedly seen that the energy value ek does not yet define the particle states. For example, the energy of a hydrogen atom depends only upon the principal quantum number n, so that the atom can have 2 tf states for a given energy [see (33.1)]. This number, In1, is called the weight of the state with energy cn . But it is also possible to place the system under such conditions that the energy' value defines the energy in principle uniquely. W e note, first of all, that in all atoms except hydrogen the energy depends not only on n, but on / also, i.e., on the azimuthally quantum number. Further, account of the interaction between spin and orbit shows that there is a dependence of the energy upon the total angular m om entum j and, finally, if the atom is placed in an external magnetic field, the energy also depends upon the projection of angular m om entum on the magnetic field. Thus, one energy value mutually corresponds uniquely to one state of the atom. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 176 CHAPTER 4. S TATISTICAL PHYSICS In a magnetic field all the 2tr states with the same principal quantum number are split. W e now consider how the states of a gas in a closed vessel are split. W e shall suppose that the vessel is of the form of a box with incommensurable squares of the sides a ^ a i a ] . Then, in accordance with equation (25.19), the energy o f the particles is proportional to the quantity 2 2 2 where n\, m, tij are positive integers. Any combination o f these *1 a2 a3 integers gives one and only one number for the incommensurable values a 2 ,a 2 ,a 2 . Therefore, specification of the energy defines all three integers Hi, Hz, n 3. If the particles possess an intrinsic angular momentum , we can, so to speak, remove the degeneracy by placing the gas in a magnetic field (an energy eigen value to which there correspond several states of a system is termed degenerate). We shall first consider only completely removed degeneracy. i Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com A l)\ ANCED ENGLISH FOR PI IYS1CISTS 177 States o f an ideally closed system We shall now consider the energy spectrum of a gas completely isolated from possible external influences and consisting of absolutely noninteracting particles. For simplicity, we shall assume that one value of energy corresponds to each state of the system as a whole, and, conversely, one sate corresponds to each energy value. This assumption is true if all the energy eigenvalues for each particle are incommensurable numbers. W e shall call these numbers s k .Then, if there are m particles in the klh state, the total energy of the gas is equal to t > = ^ n ks k . But, for incommensurable ck , it is possible in principle to k determine all ik from this equation, provided & is precisely specified. It is clear, however, that the energy of a gas consisting of a sufficiently large number of particles must be specified with truly exceptional accuracy for it to be possible to really find all /n from £>. It is not a question o f determining the state of an individual particle from its energy &, but of finding the state of the whole gas from the sum of the energies of all of its particles. Every interval of values d&, even very small (though not infinitely small), will include very many eigenvalues &. Each of them corresponds to its own set o f values m, i.e., to a definite state of the system as a whole. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 178 CHAPTER 4. STATISTICAL PI IYSICS States o f a nonideally closed system Energy is an cxact integral of motion only in an ideally closed system. The state of such a system is maintained for an indefinitely long time, and the conservation of the quantity & provides for the constancy of all m. but nature does not (and cannot) have ideally closed systems. Every system interacts in some way with the surrounding medium. W e will regard this interaction as weak and will determine how it affects the behaviour of the system. Let us assume that the interaction with the medium does not noticeably disturb the quantum levels of separate particles. Nevertheless, every level ck ceases being a precise num ber and receives a small, though finite, width Af.k . This is sufficient for the meaning of the equation 6 = ^ n ke k to change in a k most radical way: in a system consisting of a large num ber of particles, the equation containing approximate quantities ek no longer defines the num ber m. In other words, an interaction with the surrounding medium, no matter how weak, makes impossible an accurate determination o f the state from the total energy £>. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com ADVANCED ENGLISH FOR PHYSICISTS 179 Transitions between close-encrgy states In an ideally closed system, transitions were forbidden for all states corresponding to an energy interval d& because the energy conservation law held strictly. For weak interaction with the medium, all transitions that do not change the total energy of the gas as a whole are possible to an accuracy which is, in general, compatible with the determination of the energy of a nonideally closed system. Let us suppose that the interaction with the medium is so weak that, for some small interval ol time, it is possible, in principle, to determine all the quantities m and thus to give the total energy of the gas & = nkek . But over a k large interval of time the state of a gas can now vary within the limits o f that interval of total energy which is given by the inaccuracy in the energy of separate Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 180 CHAPTER 4. STATISTICAL PHYSICS states Aek . All transitions will occur that arc compatible with the approximate equation & = ^ T n k(ek ± Aek ). Naturally, the state in which all Aek are of one k sign is extremely improbable; this is why the double symbol ± is written. W e must find the state that is formed as a result of all possible transitions in the interval d&. The probabilities o f direct and reverse transitions A very important relation exists between the probabilities for a direct and reverse transition. Let us first of all consider this relation on the basis of equation (34.29), which is obtained as a first approximation in perturbation theory. Let there be two states A and B in the system, with wave functions vj/ A and \|/j3- The same value of energy corresponds to these states, within the limits of inaccuracy rfo given by the interaction of the system with the medium. In the interval d&, both states may be regarded as belonging to a continuous spectrum. Then, from (34.29), the probability of a transition from A and B in unit time is equal to — b f An |2 gn, and from Bto A, — I^ b a I2 i’A 'w^ ere h e X AB = JV b ^ V AdV » #B A = jH 'A ^ I 'B d V » Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com ADVANCED ENGLISH FOR PHYSICISTS 181 (the weight of the states are denoted by gA g B ). But, if gA = gB, then the probabilities for direct and reverse transitions, which we shall call WAB and \V[?A, are equal because | ^ a b * = I^baT • Naturally, the transition is only possible because the energies and 6» are not defined with complete accuracy, and a small interval d& is given in which the energy spectrum is continuous. In an ideally closed system we would have &b . T he relationship we have found only holds to a first approximation in the perturbation method. However, there is also an accurate relationship that can be deduced from the general principles of quantum mechanics. In accordance with the accurate relationship, the probabilities for transitions from A to B and Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 182__________________________________________ CHAPTER 4. STATISTICAL PHYSICS B to A’ are equal; here A’ and B’ differ from A and B in the signs of all the linear- and angular-momentum components. The equal probability fo r states with the same energy Wc have seen that, due to interaction with the medium, transitions will occur, in the system, between all kinds of states A, B, C ,..., belonging to the same energy interval d&. If we wait long enough the system will pass equal interval of time in the states A, B, C ,... This is most easily proven indirectly, supposing first of all that the probabilities for direct an reverse transitions are simply equal: WA]j = WBA. The refinement WAB = does not make any essential change. For simplicity we shall consider only two states such that WAB = WBA. Wc at first assume that U is greater than so that the system will more frequently changc from A to B than from B to A. But this cannot continue over an indefinitely long time, because if the ratio increases, the system will finally be constantly in A despite the fact that a transition is possible from A to B. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com ADVANCED ENGLISH FOR PHYSICISTS 183 Only the equality U = tp can hold for an indefinite time (on the average) on account of the fact that direct and reverse transitions occur on the average with equal frequency. T he same argument shows that if there are many states tor which direct and reverse transitions are equally probable, then over a sufficiently long period of time the system will, on the average, spend the same time in each state. We can suppose that t A = t . , because the states A and A' differ only in the signs of all linear and angular momenta (and also the sign of the external magnetic field, which must also be changed so that the magnetic energy of all particles is the same in states A and A'). If we proceed from the natural assumption that i A = t A* , then all the preceding argument can be extended to the case when WAB = .. We have thus seen that the system spends the same time in all states (with the same weight) that belong to the same total energy interval d&. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 184 CHAPTER 4. STATISTICAL PHYSICS The probability o f a separate state We will call the limit of the ratio t j t , when t increases indefinitely, the probability of the state C[a . The equality of all U implies that corresponding states are equally probable. But this allows us to define the probability o f each state p p directly. Indeed, if there are P states, then ^ q A = 1, because t A = t. But A=1 A=i since the states are, as proven, equiprobable, we find that q,\ = \/P. Similarly, the probability that a tossed coin will fall heads is equal to 1/2, since the occurrence of heads or tails is practically equiprobable. Hence, the problem o f finding probability is reduced to that of combinatorial analysis. But in order to use this analysis we must determine which states of the system can be regarded as physically different. When computing the total number P we must take each such state once. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com ADVANCED EN G LISII FOR I1! IYSICISTS 185 Specification o f gas states in statistics If a gas consists of identical particles, for example, electrons, helium atom, etc., then its state is precisely given if we know how many particles occur in each one o f the states. It is not meaningful to ask which particles occur in a certain state, since identical particles cannot, in principle, be distinguished from one another. If the spin of the particles is half-integral then Pauli's exclusion principle m ust hold and in each state there will occur either one particle or none at all. As an illustration for calculating the number of states of a system as a whole, let us suppose that there are only two particles and that each particle can have only two states a and b ( ea = eb ), each with weight unity. In all, there different states of the system are conceivable: 1) both particles in state a-, state b is unoccupied; 2) the same in state b; a is unoccupied; 3) one particle in each state. In view o f the indistinguishability of the particles, state (3) must be counted once because the interchange of identical particles between states does not have meaning. If, in addition, the particles are subject to the exclusion principle, then only the third state is possible. Download Ebook Tai: https://downloadsachmienphi.com
Tron Bo SGK: https://bookgiaokhoa.com 186 C H A PTER 4 . STATISTICAL PHYSICS Thus, if the exclusion principle is applicable to the particles then the system can have only one state and, if it is not, then three states are possible. Pauli exclusion greatly reduces the number of possible states of a system. A system of two different particles, for example, an electron and proton, would have four states because these particles can obviously be distinguished. Let us further consider the example of three particles occupying three states. If Pauli exclusion is operative; then one, and only one, state of the system as a whole is possible; one particle occurs in each separate state. If there is no exclusion, then the indistinguishable particles can be distributed thus: one in each quantum state, or two particles in one state and the third in one o f the two remaining states (this gives six states for the system as a whole), and all three particles in any quantum state. Thus we have obtained 1 + 3 + 6 = 10 states for the system as a whole. Download Ebook Tai: https://downloadsachmienphi.com

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