What Exactly Is "Quantum Thinking"?

I. An Ineffable Admonition

When discussing quantum physics, we are frequently admonished: "Do not look at this problem with classical thinking."

This phrase appears with extremely high frequency—in classrooms, in academic reports, and in forum debates. Whenever someone tries to追问 (probe) the physical image behind a quantum phenomenon, or whenever someone feels that a certain interpretation of quantum mechanics is unreasonable, this phrase is brought out like a magical talisman to terminate the discussion.

But the question remains: If we shouldn't use classical thinking, then what kind of thinking should we use? What is "quantum thinking"?

If you ask the person issuing the admonition, they likely cannot articulate it. They will say, "The quantum world is counter-intuitive," "You just have to accept it," or "Don't try to understand it using everyday experience." But these are not answers; they are merely repetitions of the admonition itself. If a mode of thinking cannot even be clearly defined by its own advocates, is it truly a way of thinking, or is it merely an avoidance tactic?

This article attempts to do one simple thing: to clarify what is meant by so-called "quantum thinking."

II. So-Called "Quantum Thinking" Is Actually Algebraic Thinking

Although "quantum thinking" has never been formally defined, if we carefully observe its usage in actual physical discussions, its connotation becomes clear. So-called "quantum thinking" is actually algebraic thinking—more precisely, it is the thinking mode of operator algebra within an abstract Hilbert space.

Its mode of operation is as follows: Do not think about what the physical object is; do not ask what it looks like, how it moves, or why it behaves that way. You only need to know that there is a state vector ∣ψ⟩∣ψ⟩ residing in a Hilbert space, upon which a set of operators act. Measurement is simply an eigenvalue problem of operators, and evolution is a unitary transformation. All physical problems are translated into algebraic problems. Once the translation is complete, the physics ends.

The key point here is that in this mode of thinking, physical images are not only unnecessary but forbidden. You should not ask "What is the electron doing?"; you should only ask "What are the eigenvalues of the operator?" You should not ask "What is spinning in 'spin'?"; you should only ask "What are the eigenvalues of S^zS^z ( ±ℏ/2±ℏ/2 )?" To追问 (probe) the physical image is to engage in "classical thinking," which is deemed incorrect.

In other words, the dividing line between so-called "quantum thinking" and "classical thinking" is not a difference between two types of physical intuition, but rather the difference between having a physical image and having no physical image.

III. The Case of Spin: How a Concept Is Hollowed Out

Spin is the example that best illustrates this point.

In classical physics, the meaning of angular momentum is clear: it is the quantitative description of a physical process—rotation—where an object with a mass distribution rotates around an axis. The speed of rotation and the mass distribution determine the magnitude and direction of the angular momentum.

Electrons also possess angular momentum. Initially, people naturally guessed that this was the angular momentum of the electron spinning on its own axis—hence the name "spin." The physical image implied by this name was clear: an electron of finite size rotating.

Then, "quantum thinking" intervened.

First, it was discovered that if one treats the electron as a classical sphere, the surface speed required to produce the experimentally observed magnetic moment would exceed the speed of light. What is the conclusion? A person with "classical thinking" would say: "This simply indicates that the electron is not a uniform rigid sphere; its mass-charge distribution is far more complex." This is entirely reasonable—there is no experimental evidence suggesting the electron is a uniform rigid sphere.

However, the approach of "quantum thinking" is completely different. It states: Spin is not rotation. Spin is not any kind of classical motion. Spin is a "purely quantum mechanical intrinsic property." It manifests mathematically as angular momentum—satisfying the angular momentum commutation relations [Si,Sj]=iℏϵijkSk[Si,Sj]=iℏϵijkSk —but it does not correspond to any physical rotation process.

Note what has happened here. A concept with a clear physical meaning—rotational angular momentum—has been replaced by a purely algebraic structure—an operator satisfying specific commutation relations. The name remains "spin," and the mathematical form remains that of angular momentum, but the physical content has been completely hollowed out. All that remains is an abstract two-dimensional Hilbert space C2C2 , a set of Pauli matrices σiσi , and the two eigenvalues of S^zS^z : ±ℏ/2±ℏ/2 .

You ask, "What is spin?" The answer is: "Spin is the Pauli matrices."
You ask, "What is the electron doing?" The answer is: "This question is meaningless."
You ask, "Why is the spin quantum number 1/2 and not something else?" The answer is: "Because that is how the irreducible representations of SU(2)SU(2) are classified."

This is "quantum thinking": replacing physical images with algebraic structures, and then declaring physical images to be illegal.

IV. The Cost of Algebraic Thinking

It must be admitted that the algebraic method is extremely successful in terms of calculation. The mathematical framework of quantum mechanics—Hilbert spaces, operator algebra, and unitary evolution—is a precise computational machine capable of predicting experimental results with astonishing accuracy. This is undeniable.

However, computational success does not equal success in understanding. Algebraic thinking trades physical images for computational efficiency, and this transaction brings about several deep-seated problems.

First, the hollowing out of concepts. When "spin" no longer means rotation, "orbit" no longer means a trajectory, and "transition" no longer means moving from one place to another, these words become mere labels for algebraic operations. Physics degenerates from a science about nature into a technology about calculation rules. Students learn not what the electron is doing, but how to manipulate Pauli matrices. The distinction between these two things is intentionally erased by "quantum thinking."

Second, the institutionalization of paradoxes. Wave-particle duality, the measurement problem, non-local entanglement, Schrödinger's cat—these so-called quantum paradoxes are unsolvable within the algebraic framework because the framework provides no language to discuss physical reality. Consequently, paradoxes are redefined as "essential features of the quantum world," and incomprehensibility is elevated to an intrinsic property of physical laws. This is not solving the problem; it is renaming the problem as a feature.

Third, the termination of exploration. The most serious consequence is that algebraic thinking cuts off the path to asking "why." If spin is just Pauli matrices with no deeper physical content, then there is nothing left to explore. If quantum phenomena are "essentially counter-intuitive," then there is no need to seek intuition. The admonition of "quantum thinking" is essentially a prohibition: Do not ask. In science, any form of "do not ask" should raise alarms.

V. Looking at Spin from a Different Perspective

If we refuse to accept the admonition of "do not ask" and return to the experimental facts themselves, what aspect does the problem of spin present?

Experiment tells us: Electrons possess a magnetic moment, and this magnetic moment is proportional to angular momentum. This is the core fact. The question lies in how to interpret this fact.

• The Interpretation of "Quantum Thinking": Spin is a pure quantum attribute with no classical counterpart. The spin quantum number is s=1/2s=1/2 , and the angular momentum is ℏ/2ℏ/2 .

• The Interpretation of NQT (Non-Quantum Thinking/Realist approaches): The electron is a particle of finite size (on the order of the Compton wavelength). It is genuinely rotating and possesses a real physical angular momentum of 1ℏ1ℏ . The reason the apparent spin manifests as 1/21/2 in all atomic system experiments is that relativistic effects, such as Thomas precession, halve the apparent contribution during atomic coupling. Here, 1/21/2 is not the value of the electron's intrinsic spin, but the apparent effective value of the electron's spin within an atom.

Both interpretations yield identical experimental predictions for atomic systems—because in atoms, the apparent spin is indeed 1/21/2 . However, their physical images are drastically different: one considers spin to be merely an algebraic structure with no physical content; the other considers spin to be genuine rotation with clear physical content, where the apparent value is merely modified by the atomic environment.

Which is closer to "thinking"? The answer is self-evident. Algebraic manipulation is calculation, not thinking. The inquiry into physical reality is thinking.

VI. The Trap of Function Thinking

Another aspect of "quantum thinking" is function thinking—more accurately, the mode of thinking that treats the wave function as fundamental reality.

In classical physics, we describe the position and velocity of particles, and the distribution of fields—these are direct representations of physical reality. However, in standard quantum mechanics, the fundamental object becomes the wave function ψ(x,t)ψ(x,t) —a complex-valued function defined on configuration space. The wave function does not reside in three-dimensional physical space (the wave function of a multi-particle system resides in a 3N3N -dimensional configuration space). Its squared modulus gives probability density, but what the function itself is—is it a real wave? Is it an encoding of knowledge? Is it some ontological intermediary?—remains a subject of debate with no consensus after more than eighty years.

This is a natural extension of algebraic thinking: once you strip away the physical image, all that remains are mathematical functions. You are trained to manipulate these functions—solving the Schrödinger equation, calculating inner products, performing Fourier transforms—rather than to think about what physical processes these functions describe. You become a proficient function operator, but your understanding of the physical world may be no greater than before you started operating.

VII. Redefining "Quantum Thinking"

By this point, my conclusion is clear: what is currently called "quantum thinking" is not a mode of thinking, but rather a method of calculation plus a prohibition. The method of calculation is operator algebra and wave function evolution in Hilbert space; the prohibition is "do not追问 (probe) the physical image."

True "quantum thinking," if the term is worth using, should be this: While acknowledging all experimental facts of quantum phenomena, persist in questioning the physical image, persist in seeking intuitive understanding, and persist in reducing mathematical structures to physical processes.

This means: Do not give up on experimental facts (particles have size, fields are real, angular momentum is physical angular momentum); do not give up on mathematical tools (Hilbert spaces and operator algebra remain valid within their applicable scope); but also, do not give up on the basic pursuit of a physicist—to understand nature.

When someone tells you "Do not use classical thinking," ask them in return: "Can you tell me what 'quantum thinking' is?" If their answer is "Accept the algebraic structure and abandon the physical image," then what they are giving you is not a mode of thinking, but an abandonment of thinking.

The vitality of physics lies in questioning. What abandons questioning is not a new way of thinking; it is merely an old dogma.