Physics Modeling

Wave Mechanics

In quantum mechanics, a particle is represented by a wave packet. So if m is the mass of this particle, p=m v, its momentum, its total energy E, is given by

In turn, any wave packet can be considered simply as the superposition (or the result of wave interference) of an infinite number of traveling waves, whose amplitudes of each i-th component wave can be represented in the form

Illustratively given the algebraic sum

has its Cartesian representation in the figure below as a result of a superposition of wave functions, where the region of maximum interference is implied

The representation in the form of a wave packet is associated with the following statements:

1️⃣ The particle is considered “located” in the region of maximum interference of the wave packet components

2️⃣ The frequency and wavelength at the center of the wave packet obey the following relationships:

3️⃣ The group velocity of the wave packet as a whole is given by

and it is equal to the velocity of the particle represented by the wave packet.

This last statement can be deduced by differentiating the expression of the total energy

for a free particle the value of dV is zero, so we have

As far as we are discussing modeling, after all, what are we fundamentally dealing with, with a particle described by a mass, an energy, a momentum, or with a wave described by an amplitude, frequency and wavelength?

Thus, it is postulated that the dual nature of the wave-particle is only apparent. Both features do not appear simultaneously in any observation of nature.

What may appear as wavelike or corpuscular depends on the means used to observe it and also on the questions planned in the experiment. The issue of duality is raised here mainly by the simplistic nature of the mathematical treatment used and becomes much less apparent in a more sophisticated quantum theory that presents itself.

So it is very important here to also present another characteristic of the microscopic world, strictly related to the problem of duality: it is Hisenberg’s uncertainty principle.

This principle establishes that there are pairs of variables, referring to a microscopic system, which cannot be known simultaneously with infinite precision. For example, with respect to an electron, its position (x,y) and its linear momentum p are known only with a certain accuracy of measurements, such that:

👉 if Δx is the uncertainty in the position measurement

👉 Δp is the uncertainty in the measurement of momentum, and

👉 h is the value of Planck’s constant so

It follows that if the position x is perfectly known, then it follows that nothing is known about the magnitude of the momentum p, and vice versa is true. And as a consequence of this same inequality, the relation of will be maintained for the pair of variable energy and time, this relative to any given event or states, that is,

The uncertainty principle arises from that we are forced to model a particle via a wave packet in which the infinite number of monochromatic waves forming the packet have an effective frequency bandwidth in the range

The modeled particle will be somewhere within the Δx region of the packet, and the uncertainty in the magnitude of its momentum will be in variational form:

This uncertainty is inherent to the very nature of the systems under study and represents a last quantitative limit of information that is known about them. It has nothing to do with technical difficulties encountered in the actual construction of more accurate measuring instruments.