Quantum mechanics deals with objects of the microworld, with the most elementary components of matter. Their behavior is determined by probabilistic laws, manifested in the form of wave-particle duality - dualism. In addition, such a fundamental quantity as a physical action plays an important role in their description. The natural unit defining the quantization scale of this quantity is the Planck constant. It also controls one of the fundamental physical principles - the uncertainty relation. This simple-looking inequality reflects the natural limit to which nature can simultaneously answer some of our questions.
Preconditions for deriving the uncertainty relation
The probabilistic interpretation of the wave nature of particles, introduced into science by M. Born in 1926, clearly indicated that classical notions of motion are not applicable to phenomena on the scales of atoms and electrons. At the same time, some aspects of matrix mechanics, created by V. Heisenberg as a method of mathematical description of quantum objects, required clarification of their physical meaning. So, this method operates with discrete sets of observable quantities, presented in the form of special tables - matrices, and their multiplication has the property of non-commutability, in other words, A × B ≠ B × A.
With regard to the world of microparticles, this can be interpreted as follows: the result of operations to measure parameters A and B depends on the order in which they are carried out. In addition, inequality means that these parameters cannot be measured simultaneously. Heisenberg investigated the relationship of measurement with the state of a micro-object, setting up a thought experiment to achieve the limit of accuracy for simultaneous measurement of particle parameters such as momentum and coordinate (such variables are called canonically conjugate).
Uncertainty Principle
The result of Heisenberg's efforts was the conclusion in 1927 of the following restriction on the applicability of classical concepts to quantum objects: with increasing accuracy in determining the coordinate, the accuracy with which the momentum can be known decreases. The converse is also true. Mathematically, this restriction was expressed in the uncertainty relation: Δx ∙ Δp ≈ h. Here x is the coordinate, p is the momentum, and h is the Planck constant. Heisenberg later specified the relation: Δx ∙ Δp ≥ h. The product of "deltas" - the scatter in the value of the coordinate and momentum - having the dimension of action, cannot be less than the "smallest portion" of this value - the Planck constant. As a rule, the reduced Planck constant ħ = h / 2π is used in the formulas.
The above ratio is generalized. It must be borne in mind that it is valid only for each coordinate pair - component (projection) of the pulse on the corresponding axis:
- Δx ∙ Δp x ≥ ħ.
- Δy ∙ Δp y ≥ ħ.
- Δz ∙ Δp z ≥ ħ.
Briefly, the Heisenberg uncertainty relation can be expressed as follows: the smaller the region of space in which the particle moves, the more uncertain is its momentum.
Mental experience with a gamma microscope
As an illustration of the principle he discovered, Heisenberg considered an imaginary device that allows you to measure the position and speed (and through it the momentum) of an electron as accurately as possible by scattering a photon on it: after all, any measurement comes down to the act of particle interaction, without it, a particle cannot be detected at all.
To increase the accuracy of coordinate measurement, a shorter-wave photon is needed, which means that it will have a large momentum, a significant part of which will be transmitted to the electron during scattering. This part cannot be determined, since the photon scatters randomly on the particle (moreover, the momentum is a vector quantity). If the photon is characterized by a small pulse, then it has a large wavelength, therefore, the coordinate of the electron will be measured with a significant error.
The fundamental nature of the uncertainty relationship
In quantum mechanics, the Planck constant, as noted above, plays a special role. This fundamental constant is included in almost all the equations of this section of physics. Its presence in the Heisenberg uncertainty relation formula, firstly, indicates the scale at which these uncertainties appear, and, secondly, suggests that this phenomenon is associated not with the imperfection of measurement tools and methods, but with the properties of matter itself and is universal in nature.
It may seem that in reality the particle still has specific values of speed and coordinate at the same time, and the act of measurement introduces unrecoverable interference into their establishment. However, it is not. The movement of a quantum particle is associated with the propagation of a wave whose amplitude (more precisely, the square of its absolute value) indicates the probability of being at a particular point. This means that the quantum object has no trajectory in the classical sense. We can say that it has a set of trajectories, and all of them, according to their probabilities, are realized when moving (this is confirmed, for example, by experiments on the interference of an electron wave).
The absence of a classical trajectory is equivalent to the absence of a particle in such states in which the momentum and coordinates would be characterized by exact values simultaneously. In fact, it makes no sense to speak of a “wavelength at some point”, and since the momentum is related to the de Broglie ratio p = h / λ, the particle having a specific momentum does not have a specific coordinate. Accordingly, if a microobject has an exact coordinate, the impulse becomes completely uncertain.
Uncertainty and action in the micro and macro world
The physical action of the particle is expressed through the phase of the probability wave with the coefficient ħ = h / 2π. Consequently, the action, as the phase controlling the wave amplitude, is associated with all probable trajectories, and the probabilistic uncertainty with respect to the parameters forming the trajectory is fundamentally unrecoverable.
The action is proportional to the coordinate and momentum. This value can also be represented as the difference between the kinetic and potential energy, integrated over time. In short, an action is a measure of how a particle’s movement changes over time, and it depends, in particular, on its mass.
If the action significantly exceeds the Planck constant, the trajectory determined by the probability amplitude corresponding to the smallest action becomes the most probable. The Heisenberg uncertainty relation briefly expresses the same if it is modified taking into account that the momentum is equal to the product of mass m and velocity v: Δx ∙ Δv x ≥ ħ / m. It immediately becomes apparent that as the mass of the object increases, the uncertainties become less and less, and classical mechanics is quite applicable in describing the motion of macroscopic bodies.
Energy and time
The uncertainty principle is also valid for other conjugate quantities representing the dynamic characteristics of particles. These, in particular, are energy and time. They, as already noted, determine the action.
The energy – time uncertainty relation takes the form ΔE ∙ Δt ≥ ħ and shows how the accuracy of the particle energy ΔE is related to the time interval Δt during which this energy needs to be estimated. Thus, it cannot be argued that a particle can possess a strictly defined energy at some exact moment in time. The shorter period Δt we will consider, the greater will be the fluctuation of the particle energy.
Electron in atom
It can be estimated, using the uncertainty relation, the width of the energy level, for example, a hydrogen atom, that is, the scatter of the values of the electron energy in it. In the ground state, when the electron is at a lower level, the atom can exist indefinitely, in other words, Δt → ∞ and, accordingly, ΔE assumes a zero value. In the excited state, the atom only remains for some finite time of the order of 10 -8 s, which means it has an energy uncertainty ΔE = ħ / Δt ≈ (1.05 ∙ 10 -34 J ∙ s) / (10 -8 s) ≈ 10 - 26 J, which is about 7 ∙ 10 -8 eV. The consequence of this is the uncertainty in the frequency of the emitted photon Δν = ΔE / ħ, which manifests itself as the presence of spectral lines of some blur and the so-called natural width.
We can also, through simple calculations, using the uncertainty relation, estimate the width of the spread of the coordinates of the electron passing through the hole in the obstacle, and the minimum size of the atom, and the magnitude of its lower energy level. The ratio, deduced by V. Heisenberg, helps in solving many problems.
Philosophical Understanding of the Principle of Uncertainty
The presence of uncertainties is often mistakenly interpreted as evidence of complete chaos, allegedly reigning in the microworld. But their ratio tells us something completely different: always speaking in pairs, they seem to impose a completely logical restriction on each other.
The relationship, mutually linking the uncertainties of dynamic parameters, is a natural consequence of the dual - particle-wave - nature of matter. Therefore, it served as the basis for the idea put forward by N. Bohr with the aim of interpreting the formalism of quantum mechanics - the principle of complementarity. We can get all the information about the behavior of quantum objects only through macroscopic instruments, and we are inevitably forced to use the conceptual apparatus developed in the framework of classical physics. Thus, we have the opportunity to study either the wave properties of such objects, or corpuscular, but never both at the same time. Due to this circumstance, we should not consider them as contradictory, but as complementary to each other. A simple formula of the uncertainty relation indicates the boundaries near which it is necessary to connect the principle of complementarity for an adequate description of quantum-mechanical reality.