# Quantum Sandbox: Part 2

Quantum Sandbox: Part 1
What is a quantum state? How is an ordinary state different from a quantum one? At what point does the ordinary state become quantum, and what will happen if we remove the quantum properties from it? Will it still be quantum or will it turn into ordinary? It was just quantum. It must have become confused, and the cat also became confused.

In this article we will try to answer these questions and understand the essence of quantum mechanics.
Purpose: to write a simple program that imitates quantum evolution, so that you can finally feel these qubits with pens.

• Part I: Classic State
• Part II: Quantum State
• Part III: The Cat
• Part IV: Roy

What is a normal “state”? This term is used so often that it began to be perceived completely intuitively.

# Part I: Classic State

Question No. 1: “Given particle P, which can be observed along a segment . What is the state of a particle P? ”
Answer: The classical state of a particle P is a number from a segment .

The attentive reader will be attracted by the word “observe” - how is this generally understood?
It turned out that all this time on the site there were some kind of “detectors” that “observed”, but why didn’t we say anything about them? And how many of them are there at all?

We said that the state of a particle is a number from a segment . The power of the set is equal to the continuum - between our "borders" A and B there are infinitely many numbers, and they are located infinitely close to each other - so we need an infinitely many detectors for each point? Sounds pretty expensive, right?

But, asserting that state is a number, it turns out that we mean exactly that. It is precisely the fact that we have infinitely many detectors.But this is not so. And this can not be in principle.

In practice, we would split the segment into a finite number of segments, and put detectors at the intersections, and each detector would be able to approximately tell whether there is a particle in its vicinity or not.

What has been done above is called quantization . In this case, we quantized the segment into segments. Quantum is an indivisible portion of something within the framework of the model used, an abstract term.

The most interesting phenomena begin precisely because the state of the particle is now no longer just a number.

# Part II: Quantum State

Question No. 2: “A particle P is given, which can be observed only in the vicinity of a certain number of detectors in a segment . What is the state of the particle P? ”

Consider an example:

Given a segment and two detectors located at points A and B.

Each detector shows a certain number, according to which we can determine how far the particle is from this detector.

A - the first detector, - its readings ( = 1, if the particle hit directly to A)
B - the second detector, - its readings ( = 1, if the particle hit directly to B) We will make an

assumption about the particle to somehow limit our circle Research:

Assumption: There is only one particle; it cannot just take and clone itself.

From this assumption it follows that if a particle is in A, then it cannot be in B, and vice versa.

Or, what is the same if= 1, then = 0 and vice versa.

Now consider the “movement” of a particle from detector A to detector B. The particle was in A ( = 1, = 0), then it began to fly toward B. The readings of detector A began to decrease ( <1), and the readings of detector B began to increase ( > 0). Then the particle reached detector B and its readings are = 1, and detector A notifies us that there is no particle in it = 0.

Thus, we describe the state of the particle using the detectors themselves and their readings.

This entry means that configuration X includes a detector A showing us the number c1 and a detector B showing us the number c2.

Question 2: “Given particle P, which can be observed only in the vicinity of detectors located at points A and B, which are a quantization of a segment into one segment . What is the state of a particle P? ”
Assumption: There is only one particle; it cannot take and just clone itself.
Answer: The quantum state of a particle P is a vector of a two-dimensional Hilbert space with basis vectors A = {1, 0} and B = {0, 1}. Moreover, this vector is normalized to unity ( ), and the basis vectors A and B are classical states from Question 1. Such particles are also called qubits due to the two-dimensionality of the basis. When the basis is three-dimensional, the particles are called cutrites , etc.

Question No. 2 (generalized): “A particle P is given, which can be observed only in the vicinity of a finite number of detectors located at points that are a quantization of a segment into an N - 1 segment . What is the state of a particle P? ”
Assumption: There is only one particle; it cannot take and just clone itself.
Answer: The quantum state of a particle P is a vector of an N-dimensional Hilbert space with basis vectors . Moreover, this vector is normalized to unity , and the basis vectors are classical states from question 1.

# Part III: The Cat

We came close to the most interesting manifestations of quantum mechanics. Without a doubt, each of the readers, at least out of their ears, has heard of such terms as “quantum superposition” or “quantum entanglement” - these effects and other similar magic begin at the very moment when you will not make those conclusions that are not required .

We have two definitions of the state.

Definition No. 1: The classical state of a particle P is a number from a segment ...
Assumption: There is only one particle, it cannot take and just clone itself.
Definition No. 2 : The quantum state of a particle P is a vector of two-dimensional Hilbert space ...

Usually, corollaries are derived from some definitions, but here we will be interested in what does not follow from the definition, but we will still call it consequences for harmony.

Corollary No. 1: It does not follow from the definition of a quantum state that a particle is at one point in a segment. In general, from nowhere it should not.

That is, a particle can be located at two points at once! For example, for a particle that is in a quantum state it does not follow that it is at one point. Yes, it may be somewhere in the middle, at some point M between A and B, but asserting this, we will show unreasonable liberty.

Corollary No. 2: It does not follow from the definition of a quantum state that a particle is divided into small pieces, some pieces flew there, and others here.

How to understand this? How can a particle be located at two points at once and at the same time remain indivisible ? We are used to the fact that Schrödinger's cat is both alive and dead at the same time, which means that the particle is also here and there at the same time. But she is indivisible . What did she stretch out?

We introduce the concept of a swarm and an instance of virtual particles.

# Part IV: Roy

Definition No. 3: A particle instance is a virtual object that corresponds to a position in space at a given time, the trajectory of movement over time, and also a complex number (called amplitude), which has a module and an argument for which all algebraic rules hold:
Definition No. 4 : Roy is a collection of instances.
Definition No. 5: Particle - swarm (when performing the operation of quantization of space).

Imagine an instance as a ball inside which there is an arrow corresponding to a complex number in the complex plane. It is important to understand that the ball can have one direction of movement , and the arrow inside it is another , that is, these directions are different .

But why different? The fact is that the processes inside an elementary particle are so difficult to describe that the influence of these processes on the movement of the particle itself cannot be predicted at a fundamental level, therefore there is no connection between the arrow inside the ball and the direction of movement of the ball itself.

The verbal manipulations that we have now committed are useless unless the laws of change in the quantities r , φ andthe law of motion , because everything rests on them.

The law of variation of the argument: φ constantly increases uniformly by dφ as the instance moves.

In other words, our integrated arrows are constantly spinning in the same direction. Why is this needed? So that the system does not cease to evolve under any circumstances.

The law of addition and multiplication: As you move along a single trajectory, the amplitudes multiply. Amplitudes along all kinds of trajectories add up.

This law is also known as the "principle of superposition in quantum mechanics."

The law of motion of instances in space: Let a particle be given in a quantum state. A copy is given that is located in some cell of space (over which the quantization operation on the cells was performed). Around this cell of space there are neighboring cells.
1. This instance clones itself as many times as there are neighboring points around it.
2. Each clone moves to that neighboring point that corresponds to it.
3. This father instance moves to an arbitrary point
The process is repeated for each instance.

 1. Inside each ball is the same complex arrow that rotates at an angle dφ after each movement of an instance from one cell to another. 2. So we have a huge dynamic system that constantly clones itself. 3. The direction of movement of the very first specimen, in general, determines the movement of the swarm, but the swarm nonetheless spreads in all directions. If you trace the movement of any single instance (without paying attention to the clones), then it will move along an absolutely random trajectory.

We do not forget that inside each ball there is a complex arrow, which has its own direction and length . How to predict which resulting arrow will appear in an arbitrary cell in space at a given moment in time? Obviously, for this you need to know what happened to the whole system at the previous moment in time. We get a differential equation (it is called the Schrödinger equation in honor of Schrödinger, who discovered it).

The law of motion of instances in space: Let be the quantum state of a particle, a column vector in which amplitudes in all cells of space are written one after another. - An energy operator that defines the way interactions between instances. Then the swarm moves according to the following law: .

The formation of the energy operator “in pieces with handles” will be considered in the next article.

Structurally, we figured out the following concepts:
1. The classical state of a particle as a number (and not like anything else)
2. The quantum state of a particle as a vector (and not as “something that is in several places at once”)
3. A particle as a swarm (when performing a quantization operation)
4. The principle of swarm superposition , according to which the amplitudes are multiplied along one trajectory, and added along all kinds of trajectories
5. The law of movement of specimens

In the next article we will consider the most interesting - systems with an arbitrary number of particles. We will analyze what tensors are, entangled states, and finally, we will write a program that can "imitate" quantum evolution and conveniently draw it.

Since the topic of quantum mechanics has been actively popularized recently (starting from the relevant magazines, ending with entire exhibitions devoted to “quantum entanglement”), it seems to me that there is a need to monitor the current state so that you can go in and check “what’s with us quanta? ”Perhaps this information will be useful - pleaded.ru