## Table of contents

Why is there such a question? Surprisingly, even professionals are confused because textbooks do not explicitly define quantum.

**Quantum undefined**

As a specialized course on quantum, quantum mechanics should obviously first define what a quantum is. However, the strange thing is that after you have seen the concepts of double-slit interference experiment, uncertainty relation, operator, commutation relation, observable quantity, Schrödinger equation, and more calculation and treating methods, you still cannot find the definition of quantum. At least most textbooks have never defined quantum from beginning to end.

However, some popular science textbooks, or even popular science books, will start to tell the story of quantum. The story begins with the double-slit interference experiment, followed by the discussion of waves and particles, or one of the “two dark clouds” in physics in the late 19th and early 20th centuries – the blackbody radiation problem, how Planck made assumptions to get the correct formulation for blackbody radiation energy spectrum, thus leading to the concept of energy quantum.

As a general idea, we obviously want to know the specific properties of energy quantum, its behavior in various situations, and so on. However, after textbooks or popular science books introduce energy quantum and say that radiation is in pieces, they stop abruptly, and there is no more talk of energy quantum.

Then there may be Bohr’s atomic model, Schrödinger equation, one-dimensional potential well, etc. Various problems and calculations will be introduced, and the readers’ doubts will no longer be taken care of. In Bohr’s atomic model, the quantization of orbital angular momentum will be mentioned. However, when it is mentioned that the electron’s spin is half-integer, it will only be said that it is a fermion and must comply with the Pauli exclusion principle. What is spin? It is an intrinsic property of the electron. What does it mean? Do not ask. Intrinsic properties are the final answer. Then there are matrices, operators, Hermitian operators, eigenstates, eigenvalues, and many concepts and methods to learn.

Have you forgotten the question of what quantum is? This is exactly the mental journey of most students studying quantum mechanics. Before studying quantum mechanics, students were enthusiastic about figuring out the working principles of the microscopic world and had many questions to ask. After finishing the course, they found they had no satisfactory answers to the questions they wanted to know before, but they could no longer ask questions. If a student persists and must ask, the teacher will tell you meaningfully: “No one understands quantum mechanics.”

So, it should be clear what quantum is. What is energy quantum? Does angular momentum have quantum? You cannot ask these questions. Look at how many application domains quantum mechanics has, and they have all been successful. Atomic spectra, periodic tables, chemical bonds, nuclear structures, nuclear reactions, solid physics, energy bands, semiconductors, etc., all have methods to deal with them. If you don’t use quantum mechanics, there is no solution. Do you believe it?

“I have learned the methods you mentioned and can use them. But what is a quantum?”

“We will not test this kind of question.”

“Okay.”

If we must be ignorant students and ask questions to the end, we can summarize all popular science books, textbooks, and references and get the following concepts and questions:

**What we know and do not know about quantum**

The atomic theory of Leucippus and Democritus in ancient Greece states that all matter is composed of indivisible and indestructible basic elements, namely atoms and the void. The void can correspond to the vacuum in modern physics. Atoms do not correspond to any concept in modern physics but are similar to classical particles. Ancient Indian scholars also had similar views.

It should be noted that even at that time, the atomic theory of ancient Greece was only the view of a few people. Aristotle, who was more influential in academia, did not recognize the atomic theory. He believed that matter was continuous, like a fluid.

Planck’s energy quantum is considered to be the beginning of modern quantum theory. Planck himself is, therefore, called the father of quantum. However, what is energy quantum is a question. By definition, energy quantum is Planck’s constant multiplied by the frequency of radiation. However, the frequency is variable, from zero to infinity. This does not conform to the concept of “minimum unit”. In other words, there are countless energy quanta of any size, and even energy can be infinite. At that time, relativity had not yet appeared. If relativity was considered, there would be the problem that the same energy quantum has different energies in different reference frames. By the way, Planck energy has nothing to do with energy quantum. Planck energy should be seen as an energy scale, not a unit. The value of Planck energy is too large and is not in the energy range of “quantum” at all.

Shortly after Planck proposed energy quantum (1900), Einstein proposed the photon concept in 1905 when discussing the photoelectric effect and believed that the photon was Planck’s energy quantum. This should have been a happy thing for everyone, but Planck disagreed with the statement that photon is energy quantum. Energy quantum is pure energy in the electromagnetic radiation field. However, isn’t photon radiation? But photon also has momentum and spin, which is “not pure”. Except for calculating blackbody radiation, there is no statement of energy quantum.

So, what is a photon? Although Einstein won the Nobel Prize in Physics for this, he was baffled in his later years. If the photon is a point particle, how can we explain its wave behavior? If not, how about a gamma photon? The same photon will appear in different inertial reference frames with different energy levels.

Planck’s energy quantization is different from Bohr’s later quantization. Bohr’s atomic model is the result of spectral measurement. They only found the law of discrete energy spectrum and did not say that angular momentum is quantized. Angular momentum quantization will not appear until the Schrödinger equation is proposed and the atomic energy level is directly solved.

So, is there an angular momentum quantum? Is it a half-integer or an integer (angular momentum unit h/2pi)? We know that the angular momentum of different eigenstates is an integer or a half-integer, but can we say that angular momentum is quantized? How do we calculate the superposition state?

What other quanta are there? Is time and space quantized? Some people think so, and more people do not express their opinions.

In all academic documents, in the description involving the definition of quantum, the statements of Leucippus, Planck, Einstein, and Bohr will be faithfully quoted and presented without judgment, inference, or conclusion. Such a description is rigorous and neutral, but there is still a strong hint or implication that quantum is the basis of everything in the world. Beginners generally cannot understand the author’s understatement, but they can accept hints and the fundamental nature of a quantum but still do not know the exact definition of quantum.

Quotation is neither proof nor definition. For the original proposer, the concept of quantum is just a conjecture. A conjecture, quoted many times, is still a conjecture. It still needs rigorous argumentation if it is to be used as a principle.

What is quantum thinking? When it cannot be understood with normal logic, forcefully accept something that cannot be explained by logic? Conforming to logic is the most basic requirement. The freedom of not conforming to logic is too large, and there is not only a “quantum” way. Saying that something is not what we generally define as a concept, that is, it is not a classical concept, but the concept should also be clearly defined. We should not say that a thing is neither a particle nor a wave nor is it both a particle and a wave. This is rhetoric, not science.

Also read: Clock behavior at high speeds

**Quantum in the Copenhagen Interpretation**

The Copenhagen Interpretation does not directly define quantum. However, we can summarize the definition of quantum under the Copenhagen Interpretation based on the basic concepts and formula system of quantum mechanics. I made this summary, and it can not be found in textbooks or literature.

The de Broglie matter wave hypothesis is the most essential assumption of quantum mechanics. It assumes that all matter is quantum and all waves can undergo double-slit interference, which must be described by the Schrödinger equation. Being quantized is wave nature, which is not clearly stated in textbooks.

Therefore, any matter is quantum and has a de Broglie wavelength, from elementary particles to planets and galaxies.

Of course, we are also a quantum and work according to quantum principles. Therefore, the statement that “people in a classical environment cannot understand quantum behavior” is untenable.

Quantum has wave-particle duality, so each quantum is also a particle and can randomly collapse to a point in its wave function. If macroscopic objects have size and shape, the changes caused by collapse cannot be seen because the de Broglie wavelength is very short.

The complete description of a quantum is its wave function, and the wave function is a probability (amplitude) set. So, each quantum is a probability set. When you observe it, you get the probability of all different values. *Quantum* is a probability set, which means complete randomness. In the Copenhagen image, everything in the world is random.

*Quantum* is a wave, which implicitly assumes that each quantum is infinite because the wave function fills the entire space unless it is in an infinitely deep potential well, and infinitely deep potential wells do not exist.

For the wave function of a single quantum, all probabilities must add up to 1, that is, normalization. If you look at it from a different angle, that is, via presentation transformation, because it is still the same particle, the sum of the transformed probabilities must also be 1, which requires the transformation process, such as the transformation matrix, to be unitary. So, you will often see the term unitary transformation.

The lack of a clear definition of quantum is essential for its mystery. For adults, the mysterious quantum is Santa Claus or Monkey King.

**Quantum in global approximation interpretation**

In Global Approximation Interpretation of Quantum Mechanics,

A quantum is any global pattern. The basis is quantum field theory.

Quantum is a relatively stable perturbation mode of space-time field, generally an intrinsic perturbation mode, including composite modes, such as protons, neutrons, atoms, molecules, and larger structures. The only special one is the photon because it has no rest mass and a very wide energy range. Other particles have rest mass.

Under certain boundary conditions, the global mode is generally discrete, that is, “quantized”.

Large structures also have larger-scale global modes under larger environments and boundary conditions.

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