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2-4: The Uncertainty Relation

A wave function is a function spread out spatially. If the position of a particle, x, is measured at a time t in this state , the resultant value would show the extent of the dispersion of the probability density . Namely, if we measure the position x many times under the same condition, we would have different value each time, and the data would disperse like the probability density. It can be considered that, if we intend to make a state in which the position of a particle is concentrated around one point in the space, we would have to localize the wave function at that point as well as possible. Such a wave is sometimes called a wave packet. We will discuss the property of a wave packet below.

[Wave Packet]
Let the wave function at a time t be in a one-dimensional space. In order to make a wave localized in the vicinity of x = 0. We use the Fourier transformation, which is well known in mathematics, and we can write

This mathematical theorem says that an arbitrary wave function can be expressed by a superposition of plane waves with various wave numbers (or wavelengths).

Let the wave number distribution C(k) be a square-box type function shown in the above Fig. (A). The Fourier transformation in this case is easily calculated and the corresponding probability density becomes that in Fig. (B).
It should be noted in these figures, Figs. (A) and (B), that the wider the wave number distribution becomes, the narrower the width of the probability density distribution is. Namely, if we intend to make a wave function localize in a narrow space, we would have to make the wave number distribution wider.

[Gaussian Wave Packet]
Let us take Gaussian wave packet as a typical example. In the Fourier transformation (1), we suppose the wave number distribution C(k) graphed in Fig. (C). The corresponding wave function is given by a Gaussian function as well. This just corresponds to the popular mathematical formula

and it is shown in Fig. (D).
The wave number distribution is shown in Fig. (C) below and the corresponding wave function is in Fig. (D). Similarly to Figs. (A) and (B), the width of the wave function, a , becomes narrower, as the width of the wave number distribution, 1/a, is wider.

[The Uncertainty Relation]
As seen in the above two examples of wave packets, if the width of a wave function is made narrower, the width of the wave number distribution included in it would in general be wider.
The wave number k and the momentum p are connected with the Einstein-de Broglie relation

Accordingly, when the position coordinate x and momentum p of a particle are simultaneously measured, the dispersion of the resultant data of momentum, , would be larger in the states with smaller dispersion of the measurement of position x .
For example, let us consider the case of Gaussian wave packet. If we intend to make the width of the wave function, a , smaller than , the corresponding dispersion of momentum, , should be

Then we have

The same relation is valid in the case of Figs. (A) and (B) as well.
Thus, in an arbitrary state , when we measure the position x and the momentum p of the particle simultaneously, there would necessarily be a more precise relation between the dispersions and

This relation (3) is exactly proved by quantum mechanics in general, but we omit it here. The relation (3) is called the uncertainty relation.
Namely, quantum mechanics asserts that, if we measure the position and the momentum of a particle simultaneously, we cannot improve the precision beyond the limit of the uncertainty relation.

[Is Quantum Mechanics an Incomplete Theory?]
Particles of matter possess the duality of the particle nature and the wave nature. The wave function represents the wave nature and the uncertainty relation is derived from the Einstein-de Broglie relations which connect these dual natures with each other. Therefore, the uncertainty relation comes from the essence of matter, i.e. the particle-wave duality. This means that we cannot carry out more precise simultaneous measurements of the coordinate and the momentum of a particle than the limit of the uncertainty relation. This looks very embarrassing from the viewpoint of the classical mechanics. This seems to mean that quantum mechanics might be an incomplete theory that cannot necessarily predict the coordinate and the momentum of a particle.
No, this is not so. We have to change our way of thinking about matter and existence. In classical mechanics, a matter particle has been considered "corpuscular", and this corpuscle moves to depict an orbit as a curved line. This is a common sense in classical physics. This is our habit of thought which we have been used to for a long time. However, in the microscopic world, we have to abandon this classical way of thinking. We have to shake ourselves free from the classical way of thinking to move to the quantum mechanical way of thinking based on the particle-wave duality.
Yes, let us agree to this. But there still remains a question. Is there a possibility that our precision of measurement goes beyond the limitation of the uncertainty relation? When we measure the position coordinate and the momentum of a particle simultaneously, if we have some method to be able to get more precise results than the limit of the uncertainty relation, then it would mean that quantum mechanics cannot completely describe the results. If so, quantum mechanics would be an incomplete theory which cannot describe the results of experiments.
This is just what we have learned on the page, Heisenberg's Uncertainty Principle, Bohr and Heisenberg repeated Gedanken experiments many times, and finally reached the conclusion that there must be no practical experiment which can exceed the limit of Heisenberg's uncertainty principle. Namely, quantum mechanics is able to describe all the experimental results we can obtain and, in this sense, quantum mechanics is a complete theory.

[The Criticism by Einstein]
Quantum mechanics is surely a complete theory in the sense that it can describe all the experimental data we can obtain. However, it does predict motions of matter only in a probabilistic way. Is this all right?
Einstein tried all possible means to think up counterexamples to the uncertainty principle to undermine the new theory, i.e. quantum mechanics, but he could not find any valid counterexample. Thus he finally admitted that quantum mechanics is enough applicable to the microscopic world. However, he could not agree that quantum mechanics is a fundamental theory of the microscopic world, because he believed that theory must be "deterministic", not probabilistic. He wrote in a private letter to Born, "God does not play dice".
Concerning this subject, Einstein had disputed with Bohr many times. This was famous as the Bohr-Einstein controversy. The keen criticisms by Einstein to the orthodox interpretation of quantum mechanics which was deliberated by Bohr, Heisenberg, Born and others had, beyond doubt, made immeasurable contributions to deepening the philosophy of quantum mechanics.
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