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If someone asks you what freedom is, what do you say?

Many people think that freedom is real freedom: freedom to go where you want and do what you want at any time.

And in turn, what exactly are your unfreedoms?

After careful consideration, it will be found that there are two aspects to non-freedom:

One is that your behavior is limited by time or space.

For example, you can't do something for a certain period of time, or you can only do something for a certain period of time.

The other is that some pressure or push keeps you from stopping.

For example, as a chicken baby, your parents encourage you every day, in order to test a good university in the future, in addition to sleeping and eating, you are studying hard.

If we turn to physics, there are two corresponding cases of non-freedom.

In the first case, the motion of an object is restricted, or constrained. It is achieved through a class of actions called binding forces (also known as binding reactions).

The variables describing the motion of a body under constraints satisfy equations, and the more equations there are, the more unknowns there are to be solved, and the fewer independent variables there are left.

If there is a particle, and theoretically, the position of each particle needs three independent coordinates (,) to determine, then each particle needs a total of variables to describe it.

Assuming that they are subject to a total of constraints, since each constraint corresponds to an equation, an equation can theoretically determine a variable, so that a total of variables are determined, then the number of independent variables remaining is We call the number of independent variables the degree of freedom of the system.

For example, a system of particles, if the particles are vibrating but the whole is stationary, then the degrees of freedom are. Why? Let the reader think for himself.

In the second case, the body is driven by an external force, or acceptor force. For example, the gravity of free fall and the horizontal tension of trailer are all active forces.

Binding force and active force together is external force. Therefore, the two cases of non-freedom are ultimately caused by a mechanism, which is the action of external forces.

Generally speaking, freedom means that there is no external force, and an object that is not subject to external force is free.

So what is the picture of this freedom without force?

Newton's first law answered this question: an unstressed body will remain at rest or in uniform motion in a straight line.

In other words, a free body will remain stationary or in uniform linear motion.

But you realize how unfree this freedom really is!

When we say,"A bird flies freely in the air," do we mean that it flies uniformly in a straight line? Obviously not! We mean that it flies anywhere at will, and it is equally likely to fly anywhere.

Therefore, if it were truly free, objects should be able to appear anywhere in space! This is an intuitive understanding of freedom by a normal logical mind unaffected by Newton's first law.

But Newton's first law gives it the freedom to either stay there or move uniformly along a straight line.

Of course, you may also say that this is normal ah, freedom is not absolute, always subject to the underlying laws of physics constraints. For example, the law of conservation of energy, the law of conservation of momentum and the law of conservation of angular momentum, etc.

Since an object is not acted on by external forces, then according to the law of conservation of momentum, its momentum is constant. Since momentum is a vector, conservation of momentum means that the direction of motion is constant! So there are only two possible choices for an object: stationary or moving in a straight line at a constant speed.

In fact, in the macroscopic world, there is no and cannot be a violation of Newton's first law, otherwise Newtonian mechanics would have been abandoned long ago!

It seems that freedom in the macro world can only be so!

So is this true of freedom in the microcosm?

Newton's laws apply only to the macroscopic world. For microscopic particles, they obey the laws of quantum mechanics, and quantum mechanics has no statement consistent with Newton's first law.

So, in the microscopic world, free particles can really appear everywhere?

The answer is: really! This is exactly what quantum mechanics says.

Although quantum mechanics is incredibly counterintuitive, it also fits perfectly with our most general intuitions on many issues! The law of motion of free particles is one example.

So, specifically, what exactly is a free particle in quantum mechanics?

In quantum mechanics, the state of a particle is described by a wave function. According to the statistical interpretation of the wave function, the square of the modulus of the wave function represents the probability of a particle appearing at a certain point in space. If you want to predict where it will most likely occur, you can simply calculate as the square of the modulus of the wave function.

You only get one chance to find it, and wherever you find it doesn't mean it came from somewhere else, because it was there. Although it's actually somewhere else, your observations cause its wave function to collapse where it was found.

It is worth emphasizing that the particles are not dispersed in space, but are distributed throughout space with probabilities determined by the modulus of the wave function as a whole. So when it is discovered somewhere, it will be acquired as a complete individual, with its original properties, including charge, mass, spin, etc.

Since it is a free particle, then naturally, it has the same chance of appearing at any point in space. So it doesn't stop somewhere, it doesn't move uniformly along a straight line, but it appears at any point in space with equal probability.

See--same probability at any point in space! Is there a more perfect freedom in the world than this?

Some people may have some unrealistic "fantastic ideas," for example, he thinks "anytime, anywhere, and immediately" is the real freedom.

Come on, this isn't physical freedom! On the contrary, it is not freedom at all, because this "freedom to come and go" depends on the speed (because it needs to be fast) and acceleration (because it needs to change direction at any time) of the particles, and the particles must be acted on by strong external forces at all times. How can it be physical freedom?

In quantum mechanics, physical freedom means that particles can appear everywhere, regardless of spacetime. And add another condition-the same probability!

You may think this condition is superfluous, because you may want to stay more in one place and less in another, or you may even want to change the probability of being at a certain point at will, but now it is the same probability, which does not seem free enough!

This idea, like the "whimsy" above, is not "free" in physics, but not free.

Having said this, you should understand that the freedom given to free particles in quantum mechanics is the most real and complete freedom in physics. By contrast, the freedom described in Newton's first law is not real freedom.

Now you may have a question: since it's a free particle, its momentum is conserved. So, its momentum should always be in a certain direction, so doesn't it always move in a straight line? If so, how could it possibly run around in space?

Yes, free particles conserve momentum, which means that the momentum of a particle does go in one direction. However, this does not mean that particles always move in one direction!

According to the statistical interpretation of quantum mechanics, a particle just appears unnoticed where you detect it, and you don't need to know where it came from, because the probability of it appearing in different places is simultaneous. Its entirety can appear at any point in space at any time.

Therefore, the free particle is not confined to a point or line by conservation of momentum, it appears equally everywhere in space, and its state is a completely free state.

Finally, what is the description of the state of a free particle satisfying this uniform probability distribution? And how did you get it?

As mentioned earlier, quantum mechanics uses wave functions to describe the state of particles, and wave functions are solutions to the basic equations of quantum mechanics, the Schrödinger equation. So, to describe free particles, we have to solve the Schrödinger equation.

The basic form of the Schrodinger equation is

It represents the action from the outside. Since it is now a free particle, then the Schrodinger equation satisfied by the free particle is its solution

This is the wave function of a free particle. According to the statistical interpretation of the wave function, the probability density of finding a particle at any point in space is 0 This means that the probability of finding a particle at any point is the same.

Explanation: Since it is a free particle, its space is infinite, and the particle has the same probability of discovery everywhere, then this probability can only be zero. Therefore, free particles cannot be stationary. But this does not affect the conclusion that free particles are equally likely everywhere in space.

According to de Broglie relation

and the equiphase plane of this wave at time can be obtained as the projection value of the potential vector of the point on the wave front on the wave vector. For the same time, is a fixed value, indicating that the equiphase surface is plane, so the wave function of free particles has the form of plane waves.

It should be noted that, although the wave function of a classical plane wave can also be written in the form of a real number-or function-the wave function of a free particle must be complex to be a solution to the Schrödinger equation with imaginary numbers. In fact, only this can guarantee the uniform distribution of particle probability, because the modulus of sine or cosine function is still periodic fluctuation in space.

One interesting thing about wave functions for free particles is that many people like to use wave functions for free particles to "reverse" the Schrödinger equation.

Because people intuitively believe that the wave function of a free particle should have the form of a plane wave. By calculating the time and space partial derivatives of this wave function, we can obtain the squares of energy and momentum satisfying the following conditions:

Combining the relationship between momentum and energy in classical mechanics yields an equation that is exactly the Schrödinger equation for free particles. On this basis, considering that the energy should include potential energy, we obtain the Schrödinger equation satisfied by any particle.

The reason for this quote is that the Schrödinger equation is now regarded as a fundamental assumption of quantum mechanics that cannot be deduced from any other conclusion. In other words, any attempt to derive Schrödinger's equation is meaningless in itself-although you can indeed retrieve Schrödinger's equation in your mind by this "backward extrapolation."

In conclusion, quantum mechanics is a much more advanced philosophical view of freedom than Newtonian mechanics!

In fact, quantum mechanics is more advanced and profound than Newtonian mechanics in almost every respect, not just freedom. It is considered the crowning achievement of man's insight into the natural world. So far, its theories have described the world perfectly, although humans have not yet been able to understand its true nature.

references

Griffiths, David J.; Schroeter, Darrell F. (2018). Introduction to Quantum Mechanics(Third ed.). Cambridge University Press.

This article comes from Weixin Official Accounts: University Physics (ID: wuliboke), by Xue Debao

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