—Why We Should Call It the "Probability Amplitude Function"

Introduction: The Confusion Behind the Term

Among all concepts in quantum mechanics, the "wavefunction" is perhaps the most fundamental yet also the most misleading. This terminology, which has been in use for nearly a century, has actually been misguiding our understanding of the quantum world's essence. Today, it is time to reexamine this concept.

I. Is the Wavefunction Really a "Wave"?

When we say "sound wave," we refer to periodic changes in the displacement of air molecules; when we say "electromagnetic wave," we refer to oscillations in electric and magnetic field intensities. These are all genuine physical quantities undulating in space.

But what about the "wavefunction"?

Let's look at what the value of the wavefunction ψ represents: probability amplitude.

Here lies the problem—probability amplitude is not any observable physical quantity. You cannot directly measure probability amplitude with an instrument, just as you cannot directly "see" probability. What we can measure is only the squared modulus of the probability amplitude |ψ|², which gives the probability density of finding a particle at a certain location.

A Simple Comparison

Wave Type	Undulating Physical Quantity	Directly Measurable?	Physical Dimension
Sound Wave	Pressure/Displacement	Yes	Pascal/Meter
Water Wave	Water Surface Height	Yes	Meter
Electromagnetic Wave	Electric Field Intensity	Yes	Volt/Meter
"Wave" Function	Probability Amplitude	No	?

This comparison clearly shows: the wavefunction is fundamentally not a wave in the traditional sense.

II. The Dimensional Mystery: Does the Wavefunction Have Dimensions?

There exists an interesting divergence in the physics community on this question.

The Textbook Account

Most quantum mechanics textbooks will tell you: the wavefunction has dimensions. The reasoning comes from the normalization condition:

∫|ψ(x)|²dx = 1

Since the integral element dx has the dimension of length [L], and the integral result is the dimensionless 1, |ψ|² must have the dimension [L⁻¹], therefore ψ has the dimension [L⁻¹/²]. In three-dimensional space, it would be [L⁻³/²].

A Deeper Consideration

But let's ask a more fundamental question: Does probability have dimensions?

The answer is obviously: No. Probability is simply a pure number between 0 and 1.

Then, as the square root of probability, why would probability amplitude have dimensions? Isn't this strange?

In fact, the so-called "dimension" is merely a product of mathematical formalism. When we write |ψ(x)|²dx, what truly has physical meaning is this entirety—it represents the probability of finding a particle within the range dx, and this probability is naturally dimensionless.

Analogy with Classical Probability

In classical probability theory, we often use probability density functions ρ(x). Formally, ρ(x) appears to have the dimension [L⁻¹], but no one would say that probability density is a "physical quantity." We all understand that it is merely a mathematical tool used to calculate probability:

P = ∫ρ(x)dx

The same reasoning should apply to the wavefunction. The so-called "dimension" of the wavefunction is nothing more than a mathematical formality brought about by normalization conventions and does not reflect any physical essence.

III. Why Is This Distinction So Important?

1. Eliminating Mysticism

The term "wavefunction" makes quantum mechanics appear mysterious and unfathomable: How can a particle simultaneously be a wave? What is this wave composed of? In what medium does it propagate?

If we call it the "probability amplitude function," everything becomes clear: it's just a mathematical tool for calculating probabilities, like the probability distribution in weather forecasts—nothing mysterious at all.

2. Correctly Understanding the Measurement Problem

The so-called "wavefunction collapse" sounds like some physical process—as if some entity really "collapses." But if understood as "updating of probability distribution," this becomes an entirely ordinary statistical concept: when we obtain new information, the probability distribution naturally updates.

3. Clarifying Ontological Confusion

For over a century, physicists have been debating: Is the wavefunction physical reality?

If we recognize that it is merely a probability amplitude function—a statistical descriptive tool—this debate loses its meaning. Just as no one would ask, "Is the precipitation probability in weather forecasts physical reality?"

IV. A Proposal: The Probability Amplitude Function

Based on the above analysis, I propose using a more accurate term in quantum mechanics:

Probability Amplitude Function

The advantages of this name are:

• Accurate Description: Clearly indicates that the function value is probability amplitude

• Avoids Misleading: Won't make people mistakenly think it's some kind of physical undulation

• Highlights Essence: Emphasizes its nature as a statistical tool

• Preserves Uniqueness: The term "probability amplitude" preserves the unique features of quantum mechanics (complex values, interference effects, etc.)

V. Conclusion: Language Shapes Thought

The history of physics tells us that inappropriate terminology can severely hinder our understanding. "Caloric" in the caloric theory, "ether" in ether theory—these erroneous concepts were all given unwarranted substantiality due to their naming.

The term "wavefunction" may be the greatest terminological trap in quantum mechanics. It has led generation after generation of students and researchers into unnecessary conceptual confusion and has provided fertile ground for various forms of quantum mysticism.

It is time to make a change. When we begin calling it the "probability amplitude function," we take an important step toward truly understanding the essence of quantum mechanics: quantum mechanics is not about mysterious matter waves, but about the statistical laws of nature.

This change may seem small, yet it could trigger a profound transformation in our understanding of the quantum world. After all, correct language is the prerequisite for clear thinking.