Physical quantities becoming operators is the same in classical mechanics as well.

The idea of "making physical quantities into operators" is not unique to quantum mechanics. Behind it lies the universal way of thinking known as the "generator of state transformation." And even within the scope of canonical classical mechanics, this can be clearly observed.

In the canonical theory of classical mechanics, a very useful tool called the Poisson bracket is defined for physical quantities A and B, which are functions of a particle's momentum p and its position coordinate q.

The physical quantity known as energy is regarded in canonical theory as a function of the two variables p and q, and is called the Hamiltonian H(p,q).

Using this, the canonical equations, which are the fundamental equations of motion for a system, can be written simply as follows.

By determining the values of p and q at a certain time and solving these equations, the future (and past) p(t) and q(t) are uniquely determined.

Even in this classical mechanics, it is very natural to consider operators for physical quantities. For example, the differential operator known as the Hamiltonian operator below can be associated with energy.

Then, the aforementioned canonical equations of motion can be easily rewritten as follows. Note that on the right-hand side, we first treat p and q as independent variables to calculate the derivatives within the operator, and then substitute the time-dependent functions p(t) and q(t) into p and q.

Not only for energy, but for any physical quantity A(p,q) that is a function of p and q, a similar operator can be defined as follows.

For example, the momentum operator and the position coordinate operator are as follows.

Furthermore, when the momentum operator is multiplied by a constant and placed in the exponent of an exponential function, that operator acts on the probability distribution function ρ(p,q) regarding the particle's position and momentum by adding a constant to the value of the position coordinate, as shown below. In other words, the momentum operator is a translation operator for position coordinates even in classical mechanics. Similarly, an operator created from the position coordinate operator acts as a translation operator for momentum values. The position operator and momentum operator can be understood as generators of such transformations.

In this way, even in classical mechanics, through the exponential function exp(・), it can be understood that these physical quantity operators are "generators" that produce state changes. Also, from the perspective of generators of state changes, the famous "Noether's theorem" regarding symmetry can be proven. That is, when there is symmetry, a conservation law automatically emerges stating that the physical quantity of that generator does not change over time. By using the notation of operators, this relationship between generators and symmetry can be discussed in a universal form for classical mechanics, quantum mechanics, and even general probability theories that differ from both.

When learning quantum mechanics, the fact that physical quantities become "operators" is often overemphasized, which might lead learners to feel a certain sense of mystery about it. However, as stated here, this is actually something that happens normally in classical mechanics as well, and it is not particularly strange or mysterious at all.

Furthermore, based on this framework, I have written the following textbook that treats quantum mechanics correctly as an operational information theory. I hope you find it helpful.