De Broglie's "matter waves" do not exist.

Like the caloric theory that was plausibly believed around the 18th century, or the ether that was believed to be the medium for electromagnetic waves before relativity, the existence of de Broglie's matter waves, also known as de Broglie waves, is denied in modern times. Atoms are not filled with matter waves as physical realities. Compared to the last century, our modern understanding of quantum mechanics has advanced.

For example, explaining the double-slit experiment of electrons as a physical reality called matter waves is incorrect.

During the rise of quantum mechanics at the beginning of the last century, conceptual confusion persisted. The wave function Ψ was initially thought to be a matter wave, which is a physical reality, but when considering multi-particle quantum mechanics, Ψ became a function in a multi-dimensional space, making it impossible to interpret as a physical wave in three-dimensional space. For example, the wave function of a two-particle system becomes the following complex function with a total of six arguments.

Therefore, it is impossible to interpret Ψ as a matter wave in three-dimensional space. Furthermore, it is now understood that the wave function is not such a physical reality, but rather an informational concept, as described in the article below.

If there were a statement saying, "It is not particles but matter waves ψ that are real and essential. However, the fact that ψ collapses at a speed that breaks causality when observed remains a mystery even today," it can be said that it is completely wrong. In modern quantum mechanics, such a "measurement problem" does not exist.

While it may be different in other fields such as chemistry, as part of physics education, early quantum theory that deals with "matter waves" is completely unnecessary today. It is sufficient to teach students about the failure of classical mechanics, and continuing to teach concepts like "matter waves," which are now known to be incorrect, is a source of confusion in understanding. Early quantum theory is very interesting as a historical drama, but there are too many errors in the story of its process from a modern perspective.

However, although "matter waves" have been denied, the so-called "de Broglie relations" remain in modern quantum mechanics and quantum field theory with a changed interpretation. These are the following relations that connect the wavenumber k and angular frequency ω of a plane wave function of a free particle to the particle's momentum p and energy E. De Broglie proposed these relations as properties of matter waves.

Even in modern quantum mechanics, an arbitrary wave function of a free particle can be constructed as follows by superimposing plane wave functions that satisfy these de Broglie relations with arbitrary weights.

Here, if the weight function F(p) is a constant function independent of momentum p, the wave function can be a delta function localized at x=0 at the initial time t=0. Also, if F(p) is a Gaussian function with a large width Δp, the wave function becomes a Gaussian wave packet with a small position width Δx. Conversely, if Δp is small, the wave packet width Δx becomes large. All these properties can be understood from the de Broglie relations and the property of superposition. In that sense, the de Broglie relations continue to be used by physicists today as a tool to provide intuition.

Furthermore, in particle quantum mechanics, explanations such as "a narrow matter wave packet can be created using quantum superposition, so that is the particle, and it was originally a wave" are sometimes seen, but that is not correct. In the low-energy region, an electron is strictly a "particle" itself, and never a "matter wave" or a wave packet thereof.

In quantum field theory, de Broglie-type plane waves appear as mode functions that expand field operators.

Although de Broglie's "matter waves" were denied as a reality, the "de Broglie relations" themselves, which were proposed based on that incorrect image, survive in modern physics in a meaningful way. Not all of de Broglie's achievements regarding "matter waves" have been denied.

The process of the birth of quantum mechanics in the 20th century was, metaphorically speaking, an "intrusion into truth through the back door." It was a process that chose a steep, winding path of errors based on determinism and reductionist realism. Today, it is possible to enter quantum mechanics through the front door. It is a route that follows the orthodox understanding of "quantum mechanics as information theory." The textbook below is an introductory book from that front entrance.