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The original title: "Why is the principle of uncertainty uncertain?" It was understood by Fourier again. "

Heisenberg I often hear people say that it is because the momentum of photons is affected by the interaction between photons and electrons that leads to the Heisenberg uncertainty principle.

The observer must observe the electron by affecting its momentum (or some quantum states), which may be true, but this is not the real cause of the uncertainty principle!

Before we get into this topic, let's define the Heisenberg uncertainty principle (Heisenberg's uncertainty principle).

In quantum mechanics, there are a series of inequalities about conjugate physical quantities (such as position and momentum), which limit the accuracy of measuring these pairs of physical quantities at the same time. Any of these inequalities can be called the uncertainty principle (or Heisenberg uncertainty principle).

-Wikipedia

A common expression is that you cannot accurately measure the momentum and position of particles at any given point in time.

This uncertainty does not depend on the quality of the equipment, nor is it because it is difficult to eliminate measurement errors. No matter how well we do, we can't measure these two quantities (such as momentum and energy) accurately at the same time.

First of all, there are many kinds of uncertainty principles, many of which can be seen in the macro world. Even if you are not aware of their existence, you have been dealing with these phenomena.

Secondly, the Heisenberg uncertainty principle is closely related to mathematics.

All waves and matter (conjugate variables) must follow a series of principles of uncertainty, which are really derived from a mathematical fact (detailed later).

Music, radar technology, energy technology and light also have "principles of uncertainty" that must be followed, and as we will soon see, it is mathematics that determines all this.

Bo, it all comes down to a very simple thing. No matter how complex the signal or function is, it is actually the superposition of sine waves. Sine waves are waves with specific wavelengths and amplitudes.

Superposition simply means that all waves interact with each other, and the sum of all waves (called interference) is the superposition of more complex signals.

In other words, we can decompose a function into its simpler parts (sine waves). This is almost all we have to do when calculating the Fourier coefficients of Fourier series. It is worth mentioning that this method is also applicable to aperiodic functions.

This effect is well known in music. For example, overtones in guitarists interfere with the main wave (the frequency of strings). In other words, the sound of a guitar (and any other instrument, including your voice) is made up of sine waves with different frequencies and amplitudes.

When we describe such a complex signal, we have two equivalent ways to choose. In other words, we can choose two different units to describe it.

We can choose to use time to describe how all the waves that produce the interference pattern interact at the same time, or we can choose to use the frequency of the sine wave that makes up the interference pattern to describe it.

Events that can be described in two equivalent ways are called dual relationship.

It would be great if we could find a mathematical tool to describe the dual relationship between time signal and frequency signal. In fact, we did find such a tool.

Fourier transform the tool I mentioned above to describe this dual relationship is called Fourier transform (Fourier transform). There is no doubt that it is one of the most powerful and commonly used mathematical tools.

Before we give some of its characteristics, let's talk about some general properties of this Fourier transform:

A Fourier transform is an integral transform (that is, an operator) that takes one function and returns another.

As an operator in the function space, we can regard it as the object of pure mathematics, but we can give it a good physical explanation. We can use it in both physics and mathematics.

Today, we will mainly consider it from the perspective of physics.

In the following discussion, we assume that the integral always converges.

Ling is an integrable function. The Fourier transform of f is given by the following integrals:

If it represents the change of sound waves with time, then the result of Fourier transform represents the frequency that makes up sound waves, so it can also be regarded as a function of frequency.

The dynamic diagram below shows how sound waves (the unit pulse signal in the picture) are composed of many sine waves, and the superposition of sine waves produces a function, that is.

The Fourier transform of a pulse signal is a sinc function. All signals can be constructed using sine waves represented by time or frequency. Source: Lucas Vieira.

It is important to understand that signals can always be expressed in two equivalent ways. As long as one of them is given and the other is only certain, we have a formula to calculate them. How we choose depends only on how we want to express a signal.

The unique inverse Fourier transform is derived from the following formula:

The nature of Fourier transform Fourier transform can not be explained in one or two lessons, we can only scratch the surface in this article. However, some amazing features of the Fourier transform must be mentioned:

The first is the influence of translation. Hypothetically. Through the transformation of variables, we get:

The time shift (signal delay) causes the frequency function to produce a phase shift. What effect does the scaling of the variable have?

Hypothetically. We will discuss it separately from a0.

Substitution u=at is used. Let's see when a

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