Fractions in Arithmetic and Physical Quantity Operators in Quantum Mechanics

In quantum mechanics taught at the university level, the fact that physical quantities are written as Hermitian operators is one of the reasons students stumble in their learning. Moreover, because these operators are used by multiplying them with wave functions that take complex values, rather than waves that take real values, it can lead students to misunderstand them as something mystical and profoundly deep.

On the other hand, it is said that many students stumble when they learn fractions in arithmetic. Kuruma Takahira, from the comedy duo "Reiwa Roman," is one such person who says he stopped understanding arithmetic because of fractions. While the difficulty of fractions and operators might seem different, there may be a common structure in the barriers that hinder their understanding.

Kuruma, who was an elementary school student at the time, said he could not understand fractions because, unlike natural numbers such as 1, 2, and 3, he felt that fractions "don't exist!" I have speculated on why he felt that way as follows. For example, a fraction like 2/3 is defined by the procedure, or "operation," of "dividing 2 by 3." A number like "2/3" expresses the volume of two pieces when one round cake is divided into three equal parts as a ratio or proportion. While natural numbers have a realistic image of "the number of things" that is intuitively easy to understand, fractions are abstract. It is no wonder that many elementary school students intuitively feel that they are different from "numbers" like natural numbers.

Similarly, Hermitian operators in quantum mechanics do not represent physical quantities as realities themselves, but rather represent the "operations" that define physical quantities. Physical quantities in quantum mechanics are defined by first applying a unitary operation specified by the eigenvector of a Hermitian operator to the target system, then performing an operation called measurement, and assigning the eigenvalues of that Hermitian operator to each piece of experimental data that emerges.

The eigenvectors u and eigenvalues a of a Hermitian operator A are things learned in university-level linear algebra, referring to those that satisfy Au=au for A. A unitary operation is a physical operation described by a unitary operator U, which is also learned in linear algebra. We define an operator U† using an operation called the Hermitian conjugate, which simultaneously performs a transposition operation that swaps the rows and columns of U as a matrix and an operation that takes the complex conjugate of each component. A unitary operator is defined as a U where the product of U† and the original U, U†U, results in the identity operator I. I is called the identity operator, which is an operator that "does nothing" to whatever it is multiplied by, and if expressed as a matrix, it corresponds to a matrix where all diagonal components are 1 and non-diagonal components are 0.

Just as the fractional notation 2/3 refers to the operation of "dividing 2 by 3," Hermitian operators are also a "list of operations" or "instructions for a procedure." They are not things that "are there" before measurement, like physical quantities in classical mechanics. This abstraction may also be linked to the difficulty of studying quantum mechanics. This point is likely one of the reasons why it is hard to fully grasp quantum mechanics.

In quantum mechanics lectures, forcing the idea that "Hermitian operators are physical quantities!" as an axiom is, educationally speaking, not good. After all, it is difficult for anyone to understand that "quantities" are replaced by abstract operators, and in reality, the operators themselves do not point to existing physical quantities. From that perspective as well, I believe it is better to teach that "Hermitian operators are notes representing a series of operations that define the corresponding physical quantities."