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THE MANY-BODY PROBLEM IN QUANTUM MECHANICS
V. A. Fock, Leningrad
1. In the lectures we have heard, a number of fundamental questions of modern quantum mechanics were touched upon. S. I. Vavilov spoke about light quanta, D. S. Rozhdestvenskii about the structure of the atom, and I. E. Tamm about attempts to construct a theory of the atomic nucleus.
From what has been said here, it is already sufficiently clear that each of the modern physical theories, and perhaps every theory in general, has its own limited domain of applicability. We already know quite well the laws that determine, for example, the structure of the electronic shell of the atom. The laws of interaction of atoms with one another, the laws of formation of molecules, are known. All these laws are known to us in principle, and the difficulties here consist only in deriving the mathematical consequences from these laws.
On the other hand, the laws relating to light quanta and to the interaction of light with matter are known to us with a much lesser degree of certainty.
Finally, as for the laws operating inside the atomic nucleus, we can only make conjectures. Only now is the experimental material being accumulated that will subsequently make it possible to formulate these laws.
The theory about which I wish to speak—the so-called quantum electrodynamics—includes the laws of interaction of charged material particles, of their interaction with one another and with the electromagnetic field, i.e. with light quanta. This theory, of course, also has its limited domain of applicability. It does not claim universality. The domain of applicability of this theory can first of all be characterized by the fact that, within its framework, we may legitimately disregard the structure of individual particles—electrons and nuclei—and may regard them as certain charged material points. On the other hand, the applicability of this theory already becomes doubtful in those cases when one has to deal with light quanta possessing very great energy—of the order of tens of millions of volts. But, despite these limitations, the domain of applicability of the theory is nevertheless sufficiently broad. This theory embraces, first, all ordinary quantum mechanics, i.e. the laws of interaction of electrons and nuclei in atoms and molecules. Secondly, it gives the laws of radiation, i.e. of the interaction of charged
particles with light quanta. Since quantum electrodynamics gives the interaction between charged particles, it also makes it possible to formulate the quantum many-body problem, i.e., to establish the fundamental equations that serve to describe a system consisting of many charged particles. But alongside this fundamental part—the establishment of the fundamental equations—the many-body problem also has its applied part. Indeed, in order to derive any concrete consequences from the theory, it is not enough to have a system of equations; one must also have methods for their actual, even if only approximate, solution.
My report will concern both parts of the many-body problem. I shall not attempt to give a survey of earlier investigations on this question—I shall mention only that the main part was done by Dirac, Heisenberg, and Pauli—but shall rely chiefly on my own work.
- In classical electrodynamics the electromagnetic field can be described by means of the scalar potential \(\Phi\) and the vector potential \(\mathbf A\), which are certain functions of coordinates and time. As a preliminary stage, we shall retain this classical method of describing the field, while for matter we shall use the quantum method of description by means of the wave function (which is customarily denoted by the letter \(\psi\)). The physical meaning of the wave function is that it represents a record of the information about a particle, or a system of particles, obtained as the result of a definite experiment performed on them. Knowledge of the wave function makes it possible to calculate the probability that, when a quantity pertaining to the given system is measured, one or another of its values will be obtained. The measured quantity may be, for example, the energy or the momentum of a particle.
The wave function depends on variables that correspond to the degrees of freedom of the system and, in addition, also on time. The dependence of the wave function on time has the physical meaning that it makes it possible to relate information obtained as the result of an experiment referring to a definite instant of time with the probability distribution for the results of subsequent experiments.
Let us first consider one material particle, for example an electron.
The degrees of freedom of the electron correspond to its three coordinates \(x, y, z\), which determine its position in space, and to one additional variable, corresponding, as it were, to its orientation and taking only two values. This additional variable is customarily called spin. Besides spin, according to Dirac’s theory the electron has one further degree of freedom, not observed directly in experiment but playing a major role in the theory of positrons, i.e., particles similar to the electron but with positive charge. This new degree of freedom of the electron corresponds to a possible change in the sign of its kinetic energy.
In quantum mechanics, definite mathematical operators are associated with all mechanical quantities. The following operators are associated with the kinetic energy of a particle \(T\) and its momentum \(\mathbf P\):
\[ \left. \begin{aligned} T &= i\hbar \frac{\partial}{\partial t} - e\Phi \\[4pt] P_x &= -i\hbar \frac{\partial}{\partial x} - \frac{e}{c} A_x \\[4pt] P_y &= -i\hbar \frac{\partial}{\partial y} - \frac{e}{c} A_y \\[4pt] P_z &= -i\hbar \frac{\partial}{\partial z} - \frac{e}{c} A_z \end{aligned} \right\}, \tag{1} \]
where \(e\) is the charge of the electron, \(c\) is the speed of light, and \(\hbar\) is Planck’s constant divided by \(2\pi\).
In classical mechanics the kinetic energy of a particle is expressed in terms of its momentum. If \(m\) is the mass of the particle, then in ordinary nonrelativistic mechanics
\[ T = \frac{1}{2m} P^2 \tag{2} \]
and in the mechanics of the theory of relativity
\[ T = c\sqrt{m^2c^2 + P^2}. \tag{3} \]
How is this relation to be transferred to operators? It is evident that there can be no identical relation between the operators \(T\) and \(\mathbf P\) similar to the classical one, since \(T\) contains differentiation with respect to time, whereas \(\mathbf P\) contains differentiation with respect to coordinates. But we can subject the wave function \(\psi\) to a special condition, namely: require that the result of applying to it the kinetic-energy operator be equal to the result of applying the operators appearing on the right-hand sides of the above equations and expressed in terms of \(\mathbf P\). Thus in the nonrelativistic case we arrive at the equation
\[ T\psi = \frac{1}{2m} P^2\psi . \tag{4} \]
In the relativistic case, however, the square root in the expression for the kinetic energy is extracted with the aid of special operators \(\alpha_1, \alpha_2, \alpha_3, \alpha_4\), and we obtain for \(\psi\) the equation
\[ T\psi = c(mc\alpha_4 + \alpha_1P_x + \alpha_2P_y + \alpha_3P_z)\psi . \tag{5} \]
The wave function of a particle depends, in addition to time, on three coordinates and on one or two additional variables, about which—
These operators will not obey the usual rules of multiplication, so that, for example,
\[ E_x A_x \ne A_x E_x . \tag{10} \]
For them special rules of multiplication will hold. In addition to these rules of multiplication, the potentials must satisfy the equations
\[ \Delta \Phi - \frac{1}{c^2}\frac{\partial^2 \Phi}{\partial t^2}=0;\qquad \Delta \mathbf{A} - \frac{1}{c^2}\frac{\partial^2 \mathbf{A}}{\partial t^2}=0. \tag{11} \]
Despite the presence of matter, these equations have the same form as for the field in vacuum. The action of matter on the field manifests itself only in an additional condition, which corresponds to the classical equation
\[ \operatorname{div}\mathbf{E}=4\pi\rho \tag{12} \]
and in our theory has the form
\[ C(x,y,z,t)\psi=0, \tag{13} \]
where \(C\) is the operator
\[ C=-\operatorname{div}\mathbf{A}+\frac{1}{c}\frac{\partial\Phi}{\partial t}-\sum_s e_s V_s, \tag{14} \]
with \(V_s\) being known functions of the coordinates and times of the particles and quanta.
These equations, together with the multiplication rules for the operators, give a complete formulation of the quantum problem of many bodies. The basic idea, as I have already said, is that particles interact with one another only through the field. A characteristic feature of the formulation is that, for each particle and for the light quanta, its own separate time is introduced, and that all equations, with the exception of one, have formally the same form as for free particles and for the field in vacuum.
- In the formulation proposed by Dirac and by me, electrostatic forces of the Coulomb type are not introduced explicitly. Nevertheless this formulation proves to be equivalent to the usual one, in which the Coulomb interaction is introduced explicitly. As I showed in another of my papers, this is connected with the fact that the system of equations for the wave function admits a certain kind of separation of variables.
If the field and the potentials are expanded in plane waves, then for each value of the wave vector \(\mathbf{k}\) the field can be characterized by four complex amplitudes of the scalar and vector potentials:
\[ a_x,\ a_y,\ a_z,\ \varphi . \tag{15} \]
Accordingly one may say that for each value of the wave vector four kinds of light quanta are possible. But
instead of these four amplitudes we can introduce four of their combinations, namely: the two components of the vector potential perpendicular to the wave vector:
\[ b_1,\ b_2\ (a_{\perp}) \tag{16} \]
and then the component parallel to it and the amplitude of the scalar potential:
\[ a_{\parallel},\ \varphi . \tag{17} \]
This corresponds to the division of light quanta into two types. To the first type belong transverse light quanta, characterized by the quantities \(b_1\) and \(b_2\). These are light quanta in the proper sense: their number—two—corresponds to the two possible states of polarization of light. To the second type belong the so-called longitudinal light quanta with amplitudes \(a_{\parallel}\) and \(\varphi\). It turns out that it is precisely they that transmit electrostatic interactions. The wave function \(\psi\), satisfying the system of equations written above, contains all four quantities (16) and (17), or more precisely the four quantities:
\[ b_1,\ b_2,\ a_{\parallel},\ \varphi . \tag{18} \]
But with the aid of the two equations for the amplitudes of the quantity \(C\), which follow from the supplementary condition (13), one can find the general form of the dependence of \(\psi\) on \(a_{\parallel}\) and \(\varphi\), namely:
\[ \psi=\psi_0(a_{\parallel},\varphi)\,\Omega(b_1,b_2), \tag{19} \]
where \(\psi_0\) is a known function. For any form of the function \(\Omega\), the supplementary condition (13) will be satisfied identically. As for the remaining equations (8), from them one can pass to equations for the function \(\Omega\) itself. Each of these new equations will be of the form
\[ i\hbar\frac{\partial\Omega}{\partial t_s}=H_s\Omega, \tag{20} \]
where \(H_s\) is an operator containing terms of Coulomb type.
The physical meaning of this transformation is that we have eliminated the variables pertaining to longitudinal light quanta and, in place of them, have obtained Coulomb electrostatic forces.
It now remains to take the last step and set all the times of the particles and light quanta equal:
\[ t_1=t_2=\ldots=t_n=t=T . \tag{21} \]
In view of the fact that
\[ i\hbar\frac{\partial\Omega}{\partial T} = i\hbar\left( \frac{\partial\Omega}{\partial t} + \frac{\partial\Omega}{\partial t_1} +\ldots+ \frac{\partial\Omega}{\partial t_n} \right)_T , \tag{22} \]
now we obtain for \(\Omega\) only one equation, into which the sum of the right-hand sides of equations (20) will enter. This equation may be written in the form
\[ H\Omega - i\hbar \frac{\partial \Omega}{\partial t} = L\Omega . \tag{23} \]
Here \(H\) is the usual operator for the total energy of a system of \(n\) particles without light quanta. In the nonrelativistic approximation it has the form
\[ H=\sum_{s=1}^{n}\frac{1}{2m_s}p_s^2+\sum_{u>v}\frac{e_u e_v}{|r_u-r_v|}+U, \tag{24} \]
where \(U\) is the potential energy of the particles in an external field. The first sum represents the kinetic energy, and the second—the Coulomb electrostatic energy of interaction of our particles. The operator \(L\) on the right-hand side will refer to the interaction with light quanta: it will give the reaction of the radiation on our material system. In many problems this reaction of the radiation, and hence also the operator \(L\), may be neglected, and then the ordinary Schrödinger equation for many bodies is obtained.
Let us now summarize the course of our reasoning. We proceeded from the notion that particles interact directly only with light quanta. Therefore we did not explicitly introduce the electrostatic Coulomb interaction. This interaction was obtained by us automatically, by eliminating longitudinal light quanta.
Thus the old question of how to reconcile the conception of light quanta with electrostatics is finally resolved here. The Coulomb interaction between charged particles turns out not only not to contradict, but to be a direct consequence of the conception of light quanta. At the same time it has been clarified how to reconcile the nonrelativistic character of Coulomb forces with the invariant character of the theory. Our initial equations are invariant. The separation of quanta into longitudinal and transverse ones, however, is not invariant. Therefore, although the Coulomb terms obtained under this separation do not have an invariant character, their noninvariance is compensated by the presence in our equations of the operator \(L\), which pertains to light quanta in the proper sense.
- Quantum electrodynamics, whose basic ideas I have attempted to set forth, constitutes a complete scheme, including both quantum mechanics in the narrower sense and the interaction of particles with radiation. In particular, this theory makes it possible to determine the natural width of spectral lines. This scheme can be extended to the case of a variable number of material particles, to the case of the creation and annihilation of electron and positron pairs. Despite all this, modern quantum electrodynamics suffers from certain fundamental shortcomings, which
formally manifest themselves in the fact that a rigorous solution of its equations proves in most cases impossible because of the divergence of certain integrals, and so on. These shortcomings were emphasized with particular sharpness in the work of the Kharkov physicist Landau (together with Peierls), who analyzed their physical cause. Landau even believed that quantum electrodynamics gives nothing essentially new in comparison with classical electrodynamics. This extreme view was subsequently refuted by Bohr and Rosenfeld, who investigated the limits of applicability of quantum electrodynamics. In their profound study Bohr and Rosenfeld showed that the applicability of quantum electrodynamics ends where the atomic structure of measuring instruments begins to become essential. In the general opinion of all physicists, the extension of these limits and the establishment of a new, more general theory uniting quantum mechanics and the theory of relativity require substantially new ideas, connected with an even greater abandonment of classical visualizable representations than is the case in quantum mechanics.
It is appropriate to mention here that the same idea which underlies Dirac’s work, my work, and Podolsky’s—namely, the idea of the transmission of interaction by means of quanta—was successfully applied to gravitation by the Leningrad physicist Bronstein. Using the mathematical apparatus developed in my work, he showed that, by introducing gravitational quanta, one can obtain the Newtonian interaction between bodies, and moreover it has the correct sign, so that two masses will always attract each other, whereas in electrodynamics two like charges, as is well known, repel each other.
In popular literature the question of so-called action at a distance and action by contact is still discussed from time to time. At present, although this question has been resolved in favor of action by contact, it has at the same time lost its topicality and even, in part, its meaning. Indeed, on the one hand, the theory of relativity in principle admits no instantaneous actions at a distance; on the other hand, we can no longer conceive action by contact quite so literally and quite so naïvely mechanistically as was done 300 years ago when this old dispute arose. For according to quantum mechanics, light and gravitational quanta cannot even be strictly localized in space and time, so that there is no question of the transmission of interaction from point to point through an intermediate medium (the ether). Nevertheless, if we wish to use the old terms “action at a distance” and “action by contact,” giving them, when necessary, a new, generalized meaning, we may do so. We can assess the theory I have set forth from the point of view of these old concepts. We must then say that our theory of interaction between particles is a consistent implementation of the idea of action by contact, and that from this idea it derives the laws of Coulomb and Newton without resorting to the mythical conception of the ether.
*
- I pass to the exposition of the main idea of my works—the approximate method for solving the quantum many-body problem.
Suppose that we have an atom with \(n\) electrons. If one neglects relativistic corrections and the radiation reaction on the atom, then the above formulation of the many-body problem leads to the ordinary Schrödinger equation, with the energy operator \(H\) written above. In order to find the stationary states of the atom, i.e. states with definite energy, one must solve the Schrödinger equation:
\[ H\psi = E\psi, \]
where \(E\) is the energy parameter. The function \(\psi\), satisfying this equation, depends on \(3n\) coordinates
\[ x_1,\ y_1,\ z_1;\ \ldots;\ x_n,\ y_n,\ z_n \tag{26} \]
and on \(n\) variables (spins)
\[ \sigma_1,\ \sigma_2,\ \ldots,\ \sigma_n. \tag{27} \]
Each of these latter takes only two values, according to the two possible orientations of the electron. This function must change sign under an interchange of the variables corresponding to two different electrons. For example:
\[ \psi(x_1,\ y_1,\ z_1,\ \sigma_1;\ x_2,\ y_2,\ z_2,\ \sigma_2;\ldots)= \]
\[ = -\psi(x_2,\ y_2,\ z_2,\ \sigma_2;\ x_1,\ y_1,\ z_1,\ \sigma_1;\ldots). \tag{28} \]
This requirement is called the Pauli principle.
Instead of regarding \(\sigma_1\) and \(\sigma_2\) as independent variables, we may consider them as indices of functions. Then we may say that it is required to find \(2^n\) functions, each of \(3n\) variables. In order to obtain an idea of the degree of complexity of this problem, let us consider a sodium atom, containing 11 electrons, and a copper atom with 29 electrons. For a sodium atom it is required to find \(2^{11}=2048\) functions of 33 variables, and for copper \(2^{29}\), i.e. more than half a billion functions, each of 87 variables.
It is quite clear that an exact solution of such a problem is practically impossible. It is therefore necessary to resort to approximate methods.
The idea of the method I proposed is as follows. I formulate the problem of solving the Schrödinger equation in the form of a variational problem of finding the minimum of the integral
\[ W=\int \psi H\psi\,d\tau, \tag{29} \]
representing the energy of the atom. The wave function \(\psi\), however, I represent in the form of a linear combination of functions \(\psi_1,\psi_2,\ldots,\psi_n\), each of which depends only on the variables of one electron:
\[ \psi=\frac{1}{\sqrt{n!}}\sum_{a_1\ldots a_n}\pm \psi_{a_1}(x_1,\sigma_1)\ldots\psi_{a_n}(x_n,\sigma_n). \tag{30} \]
Such a combination, instead of the simple product
\[ \psi=\psi_1(x_1,\sigma_1)\ldots\psi_n(x_n,\sigma_n) \tag{31} \]
has to be introduced in order to satisfy the Pauli principle. I introduce this combination into the expression for the energy \(W\) and then find the functions \(\psi_1,\ldots,\psi_n\) from the condition that it be a minimum. For these functions one obtains equations of the form
\[ \left(\frac{1}{2m}P_s^2+U_s\right)\psi_s-A_s\psi=E_s\psi_s. \tag{32} \]
Thus we have, as it were, returned to equations for separate electrons.
In these equations \(U_s\) is the potential energy arising from the nucleus and from the remaining electrons, while \(A_s\) is a certain integral operator. In my work I also showed that if, instead of a linear combination of wave functions satisfying the Pauli principle, we take the simple product of functions (31), then the variational principle will lead us to equations of the form (32), but without the integral operator \(A_s\). These simplified equations were first proposed by the English mathematician Hartree and were named by him the equations of the self-consistent field.
The characteristic difference between my equations and Hartree’s equations lies in the term with the integral operator \(A_s\). Let us dwell in more detail on its interpretation. Since it is added to the energy of the electron, it itself represents some kind of energy. But this kind of energy is unknown to classical mechanics and constitutes a purely quantum phenomenon. The term \(A_s\) arose from the fact that, instead of the product of the functions of the individual electrons, we introduced a linear combination of such products satisfying the Pauli principle. Thus it is connected with the Pauli principle. But the Pauli principle is an expression of the identity of electrons, an expression of the fact that nothing changes if two electrons exchange places. Therefore the energy associated with this is customarily called the energy of quantum exchange.
The concept of the energy of quantum exchange was first introduced by Heisenberg and Heitler. In my work I showed how it can be taken into account for atoms by introducing the integral operator \(A_s\). The equations I derived, which take quantum exchange into account, were first solved by my collaborators at the Optical Institute in Leningrad for the atoms of sodium and lithium. These calculations showed,
that the energy of quantum exchange constitutes a significant fraction of the energy of the valence electron in the atom. If it is taken into account, the computed energy level of the atom differs from the observed one by 1 or 2%; if it is neglected, the error rises to 20 or 30%.
The role of quantum exchange is even more significant in the quantum-chemical theory of molecules. It turns out that the very existence of molecules composed of two identical atoms is due to the presence of this energy, whereas according to classical mechanics molecules of this kind could not exist at all.
The method described above for the approximate solution of the many-body problem with allowance for the energy of quantum exchange, which I developed for atoms, has found numerous applications. Thus, it was applied by Tamm and Brillouin in the theory of solids, by Dirac in the theory of positrons, and finally, in recent times, by Heisenberg in the theory of the nucleus.
This method gives fairly accurate results for atoms, but requires comparatively complicated calculations. I should therefore like to mention another method I have proposed, about which I reported here in March of last year. This new method is based on the properties of the wave functions of hydrogen-type atoms. Its essential interest lies in the fact that it uses the symmetry of hydrogen-like atoms, which, as I have shown, coincides with the symmetry of a sphere in a space of four dimensions. This symmetry makes it possible to formulate, in an extremely simple way, the properties of atoms with closed electron shells. At present there is already numerical material for the atoms of sodium, aluminum, copper, and zinc, which leads to the conclusion that, for all its simplicity, the new method is capable of giving results quite satisfactory in accuracy.