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I am almost sure that the wave function is the most important figures in modern physics book. On the other hand I know that wave function even do not have a physical meaning it self alone!

  • Why is wave function so important?

  • How many types of wave function exist?

  • What means time and space coordinates inside of wave function? (I know how to find wave vector and angular frequency but I don't know how to find time and space coordinates.)

$$\Psi=\exp i({\bf {k}\cdot \bf {x}} -\omega t)$$

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4 Answers

what ever time $t$ and position $\bf r$ coordinates is, relation between them is velocity of particle: $$v=\frac {\bf r}{t}$$ wave function is important because quantum mechanics is a kind of wave mechanics.

  • wave function symmetric antisymmetric of many-particle systems of three or two particle systems
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When doing any form of quantum mechanics calculation we have to choose a mathematical model to use. In this context a mathematical model is just a set of equations that describe some approximation to the real world (whatever "real" means!).

If you learned QM at college you almost certainly started out by learning the Schrödinger equation, and it's in this context that you probably first heard about wavefunctions. This is because the Schrödinger equation equation is analogous to a wave function, and it generally has solutions that look like waves. The equation you give is of a plane wave, and is a solution of the Schrödinger equation for a free particle.

But even right at the beginning of QM Heisenberg came up with a different mathematical model to describe quantum mechanics called matrix mechanics and this doesn't use the idea of a wave function in the same way Schrödinger equation does. Dirac proved that Schrödinger's equation and matrix mechanics describe the same physics, and this is a good example of how different and apparently unrelated mathematical models can describe the same physics. This can be very useful as different models are often complimentary i.e. in some circumstance one model is easier to work with while in others a different model may be easiest.

Since the development of QM in the 1920s we have quantum field theory and indeed string theory, and these are also mathematical models to describe quantum phenomena, though they're more accurate and of wider scope than the early models.

Anyhow, the point of all this is that I wouldn't agree with your statement:

I am almost sure that the wave function is the most important figures in modern physics book.

Wavefunctions are very important because more most applications they are the easiest way to describe physics, but the wavefunction/Schrödinger approach is far from the only way physicists work with QM.

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"This is because the Schrödinger equation equation is analogous to a wave function, and it generally has solutions that look like waves." you mean analogous to a wave equation? Also, just looking at the archetypical schrodinger equation with two space and one time derivatives, it actually looks like the diffusion equation... –  kηives Oct 13 '12 at 23:26
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In short and simply...

Why is wave function so important?

Because it's powerful, and it is powerful because it's encoding all the information that you need to predict the future dynamic properties (energy, impulse...) and state of the particle or system that it describes, thus putting all that information in one function honestly is a very neat thing, you just need to know the right way to "extract" this info from it, that much better of having a set of different numbers that just makes thing more complex.

How many types of wave function exist?

it depends on what you mean by "type" because we can "classify" them in different ways, if we put in mind the physical meaning of what it will describe, I would say that there is only two types, the one for Bosons and second for fermions, if you would like to classify them by it's math, then there are infinitely many of them, thus because there mathematical structure highly depends on the boundary conditions of your system, think about usual sound speaker, if you will but it inside a box or inside a big hall, despite the waves sound and music that goes out of it is the same, you will hear different "quality" of music, the wave functions behaves similarly, it depends on in which room/box you will insert your particle/system.

What means time and space coordinates inside of wave function? (I know how to find wave vector and angular frequency but I don't know how to find time and space coordinates.)

this question actually means that you misunderstand what is the wave function, I will ask you where is my sound wave will be after 5 seconds? your answer should be it's insane unless we are speaking about the "wave front", anyway in QM we can't speak about that because sound wave speed is limited while the "wave" of our wave function has "almost" infinite speed (this don't contradict relativity, to understand how is that please read about wave Phase & Group velocity, and which one currying information), thus you "can" think about your particle that it is "no more in some particular location" at some time, because it is now wave not a "ball" so it's existence is spread out along all our universe, and there is no particular position at particular time, you just can "hope" that you will find it there at some time (here more correctly to say at some period of time) with some particular probability, and nothing more. So the coordinates and time that appears in wave function, are not your particle location, it is the coordinates of the space that your particle in, just like that.

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Why is wave function so important?

Just to clarify, the question should be "Why is the wave function so important to us?"

The reason for the distinction is that we define the wave function and attach certain meaning to its behavior under mathematical manipulation, but ultimately it is a tool that we use to achieve some purpose. The purpose of this tool is to make predictions regarding certain measurable features of the external world.

So what does the wave function represent?

By definition the wave function represents probability amplitudes, and the square of the modulus of the wave function represents a relative probability. We can multiply the wave function with its complex conjugate in order to define a real function that tells us the probability of an event within some interval of spacetime.

$$\Psi(x,t) = Ae^{-i(kx-\omega t)}$$

$$|\Psi|^2 = Ae^{-i(kx-\omega t)}Ae^{i(kx-\omega t)}=A^2$$

$$\int_{-\infty}^{\infty}|\Psi|^2dx = 1$$

The importance of the wave function really comes out when we actually manipulate it. It turns out that probability amplitudes evolve deterministically, which means we can identify the probability of certain future events. The fact that we can represent nature this way mathematically gives the wave function special meaning to us as creatures who are interested in predicting future events. A nice analogy can be made to the power of the wave function and the power early astronomers had in predicting the seasons. Early civilization was extremely concerned with the growing season. The fact that one could watch the stars and make precise enough measurements so as to predict when the next growing season occurred was extremely important. Similarly, the ability to predict certain outcomes of experiments with extreme accuracy allows for the ability to do things like MRI's and other high tech applications.

How many types of wave function exist?

How many do you want? TMS's point about there being difference between bosonic and fermionic wave functions is a good start, however, the better point is that the wave function is situationally dependent. One could try to define a universal wavefunction which in principle describes the entire universe. However, in reality, even if we defined such a wave function, we would still be forced to only consider a portion of it in most practical situations. This I would argue is possible only because some long range effects are small enough to be ignored in certain applications. In any case, in any given situation one can define a single wave function for the evolution of the system, which will be composed of many superposed independent wave functions which represent some sub component of the system. We are principally interested in how these subcomponents interfere with each other so as to effect the outcomes of experiments.

What means time and space coordinates inside of wave function? (I know how to find wave vector and angular frequency but I don't know how to find time and space coordinates.)

The exponent of the wave function $-i(kx-\omega t)$ can be understood in terms of action, which is defined as the difference between kinetic and potential energy (the Lagrangian) over time.

$$S = \int L dt$$

It is possible also to understand this in terms of how this varies over space and time, which is how it is frequently understood in most physical applications.

The variation of the Lagrangian over space and time (x is space and t is time) is of principle concern in physics. When we see $-i(kx-\omega t)$, we understand we are looking at a very simple version of the action. When we start talking about systems with more particles, this equation becomes more complicated.

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