Chapter 2: Popular science to Hilbert space

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In the text, the golden finger used by our protagonist Wang Qi for the second time is the Hilbert space from the great mathematician David Hilbert of the earth.
Since I don't want to count the number of words in the text, I will post the science of this mathematical method here! Interested book friends may wish to come in and take a look ~
Albert space does not really exist, but an abstract tool for calculation, namely phase space.
Every friend who has read middle school mathematics should have established a two-dimensional Cartesian plane: draw an x-axis and a y-axis perpendicular to it, and add arrows and scales [also known as the plane rectangular coordinate system] . In such a planar system, each point can be represented by a coordinate (x, y) containing two variables, such as (1,2) or (4.3, 5.4), these two numbers represent the point Projection on the x-axis and y-axis. Of course, it is not necessary to use a rectangular coordinate system, and polar coordinates or other coordinate systems can also be used to describe a point, but in any case, for a two-dimensional plane, two numbers can uniquely indicate a point. If we want to describe a point in three-dimensional space, then we have three numbers in our coordinates, such as (1,2,3), these three numbers represent the projection of the point in three mutually perpendicular dimensions.
Let us expand our thinking: If there is a point in a four-dimensional space, how should we describe it? Obviously we need to use coordinates with 4 variables, such as (1,2,3,4), if we use the rectangular coordinate system, then these 4 numbers represent the projection of the point in 4 mutually perpendicular dimensions , Extended to n dimensions, the same situation. You do n’t have to worry about conceiving in your mind how 4-dimensional or 11-dimensional spaces are perpendicular to each other in 4 or even 11 directions. In fact, this is just a hypothetical system that we construct mathematically.
What we care about is: a point in n-dimensional space can be uniquely described by n variables, and conversely, n variables can also be covered by a point in n-dimensional space.
Now let us return to the physical world, how do we describe an ordinary particle? At each time t, it should have a definite position coordinate (q1, q2, q3), and also have a definite momentum p. Momentum, which is velocity multiplied by mass, is a vector that has components in each dimension, so to describe momentum p, you must use 3 numbers: p1, p2, and p3, which represent its speed in three directions. In summary, to fully describe the state of a physical particle at time t, we need to use a total of 6 variables. As we have seen before, these six variables can be summarized by a point in a 6-dimensional space, so using a point in a 6-dimensional space, we can describe the classic behavior of an ordinary physical particle. The deliberately constructed high-dimensional space is the phase space of the system.
If a system consists of two particles, then at each time t this system must be described by 12 variables. But again, we can replace it with a point in a 12-dimensional space. For some macro objects, such as a cat, it may contain too many particles. Suppose there are n, but this is not an essential problem. We can still describe it with a particle in a 6n-dimensional phase space. In this way, the activity of a cat in any period of time can actually be equivalent to the movement of a point in the 6n space (assuming that the number of particles constituting the cat is unchanged). We do this not because we are too full to eat, but because mathematically, it is more convenient to describe the movement of a point, even a point in a 6n-dimensional space, than to describe a cat in an ordinary space. . In classical physics, for such a point in the phase space that represents the entire system, we can use the so-called Hamiltonian equation to describe and draw many useful conclusions.
Partly selected from Cao Tianyuan's "The History of Quantum Physics"
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