Introduction

The statistical character of physical states and Hamilton’s formalism of classical mechanics form a fundament of quantum theory. We begin our discussion from the description of states with an example of the phenomenon of the polarization of light. Let the light waves, together with separate photons of this light beam, possess a particular polarization. We pass such a beam through a plate of tourmaline; on passing this crystal through unpolarized light, on the back of the plate we discover waves having the electric-field vector parallel to the optic axis of the crystal. If the electric-field vector in our beam is perpendicular to the optic axis, then as a result the entire absorption becomes observable. If the light is polarized at angle α to the axis, only a fraction equal to cos2 α from the initial beam passes through the crystal. From the point of view of classical optics, these facts are trivial. The question arises, however, in the case of separate photons, whether each photon is polarized at angle α to the axis. The answer is simple: if we pass photons one by one from our beam, we discover that one photon is entirely transmitted, whereas another is entirely absorbed; the probability of observing a particular photon from the beam is equal to cos2 α, and the probability of its absorption is sin2α. As a principle of quantum theory, one might thus apply the next device. Each photon can be represented in a state with polarization that is parallel to the axis or perpendicular to the axis. A particular superposition of these states produces the necessary state for the beam with polarization. In the result of an experiment, photons jump from an uncertain state to a state with a concrete polarization – those that pass and those that become absorbed.

The same condition occurs for the interference of photons. If an initial beam becomes split into two components, each photon with a particular weight enters partly into each component beam. As we have observed, however, that a particular photon is entirely in one component, it is at once precluded from being in the other component beam. A priori we may characterize a physical system with states of a particular number that have a statistical character. Quantum mechanics requires that each photon interferes only with itself during the interference of the two components. An electromagnetic wave and a photon are two descriptions of light. The same condition, as we see further, applies for physical particles with which one might also associate individual wave fields. In this sense, the individuality emphasizes the stability of all material – electrons, protons and so on.

Let us generalize the facts above. What should we understand about the state of the system – a motion, a rest, an interaction? These concepts exist in classical mechanics. Something similar holds in quantum mechanics, but it is less determinate. What is the meaning therein? If the system is presumably in one state, we must consider that it is partly in another state, so that its real state represents the superposition of all possible states that have non-zero probabilities. As a classical analogue of the expression of this principle, one might apply a wave packet, for which a complicated wave motion is resolvable into Fourier components; through this analogy, quantum mechanics is generally called wave mechanics. As a result, this principal idea yields a new theory – a theory of probabilities or amplitudes of physical states.

For states in quantum mechanics, as far as practicable, we use Dirac’s notation. In this case, to each state we ascribe a ket vector |〉, inside of which might appear letters, words, numbers and other symbols. Keep in mind that in classical mechanics a vector is also applied to describe motion, but it is Euclidian there, whereas here Hilbert’s type prevails. Vectors |A1〉, |A2〉,… that belong to a Hilbert space might be added together and might be multiplied by arbitrary complex numbers c1, c2,…, as a result of which we obtain another vector

This vector, which is expressible in a form of linear combination of others, is linearly dependent on them. Like a Euclidean space, the systems of linearly independent vectors are therefore of special interest. Each physical state of interest is expressible as an expansion in terms of these system vectors. Conversely, any such state might describe a concrete state of a physical system. It is important that a procedure of multiplying the vector by the number gives no new state; for instance, |A〉 and −|A〉 describe one and the same state. The principle of superposition in quantum mechanics has an important significance; considering the concrete physical problems, we generally appeal to this postulate.

Let us now consider Hamilton’s formalism, which we will review briefly with regard to methods of classical mechanics. It is remarkable that the equations of the old theory can be borrowed with a somewhat altered meaning to construct the new mechanics. Lagrange’s function of a mechanical system represents a function of generalized coordinates qi, their temporal derivatives (generalized velocities) and time t:

By definition, the momentum is and the force is Fi=∂ℑ/∂qi. The energy of the system equals

Lagrange’s function ℑ is such that integral has a minimum; this condition leads to the Euler–Lagrange equation

if ∂ℑ/∂qi=0, pi is a constant of motion and qi is a cyclic coordinate.

There exists, however, an alternative method to describe a mechanical system that employs the language of coordinates and momenta. To convert to variables qi and pi, we apply a Legendre transformation:

Consequently,

Here, H is Hamilton’s function; this description is called a Hamiltonian formalism. One sees that this method possesses great symmetry. Moreover, it is convenient that H represents the total energy of the system. For instance, for interacting particles, the energy comprises kinetic and potential contributions:

in which mi is the mass of particle i and V is the potential energy of interaction of the particles. In this case, Lagrange’s function has a form

In Hamilton’s formalism, physical quantity f is represented as a function of the coordinates, momenta and time: f(qi,pi,t). Its total derivative with respect to time has a form

Here,

is a Poisson bracket. For instance, and Poisson brackets play an important role not only in classical mechanics but also in quantum theory; they therefore deserve special attention.

As an example, we consider the Hamiltonian of a particle in an external electromagnetic field, which is determined by vector potential A and scalar potential U. The energy of this particle with charge e′ and velocity v in such a field is given with this expression

in which c is the speed of light. For Lagrange’s function, we thus have

The momentum is

By definition, we write expression for Hamiltonian H:

However, v→(p−e′A/c)/m, so that finally

One sees that, to proceed from the Hamiltonian of the freely moving particle to the Hamiltonian describing the motion in the external field, one must perform a replacement p→p−e′A/c and add a trivial static energy e′U. Elsewhere in what follows, classical mechanics in Hamilton’s form becomes the initial point of our research and prompts the correct form of initial equations.