Understanding the Wave Equation

[sciencenoob]

Can anyone explain to me, the significance of the wave equation such that I can understand it.

 

From what I gather (and do correct me), the wave equation is 'something' who's squared (i.e. something^2) is a good enough description of the probability density of finding the particle in a bound state.

 

The founding of the wave equation and the use of the wave like nature of particles to describe particle systems is nothing but luck , hope, and messing around with mathematics that magically seem to work. In that nobody knows the underlying theory or understnading of why particles behave as waves in the first place or why mathematical wave equations and operations can be used to describe particle behavior.

[swansont]

The wave function multiplied by its complex conjugate gives you the probability density.

[timo]

A wave equation is a differential equation describing/modelling waves. What you're talking about is the wave function. Consider the seriousity of your judgement about quantum mechanics in the light that you cannot tell an equation from a function.

[sciencenoob]

yes but what 'is' the wave equation in itself. for the sake of argument I could care less about the mathematical operations that leads to the solutions of, but the physical significance of the wave equation still seems to bear no much meaning to me. Of course there are other constants and values and equations etc in phsyics that simply have little physical signifiance, but QM is so abstract as to use wave mathematical relationships to abstractly and arbitrarily put it into context for QM, such that you get completely abstract terms

 

ok my bad for the wrong use of technical terminology, i am a noob at this,

 

I did mean the wave function

[iNow]

I'm a layman, so I'll offer some layman's terms. Apologies to those well versed should I misspeak. Please correct my interpretation as needed.

 

The wave function describes all possible states of an object. This is really the heart of the whole "Schroedinger's Cat" issue. Let's say you put a cat into a box. In that box is a poison in a glass vile, and the box gets dropped (I've changed the real thing a bit, but this is close).

 

So, cat in box, glass vile of poison in box with cat, box gets dropped. This is where we are.

 

So, the wave function which describes this discusses all possibilities.

 

The vile didn't break, and the cat is alive.

The vile did break, but the cat survived.

The vile didn't break, but the cat died (maybe he had really bad vertigo .

The vile did break, and the cat died.

 

Before you open the box, according to QM, the cat is actually all of these things at the same time. Since you don't know, all you have is a wave function which describes the probabilities of each state.

 

Here's where it gets weird though.

 

The cat is also anything else, just at a lower probability. The cat is a dog. The cat is your grandmother. The cat is elvis. The cat is George Washington. The cat is a pterodactyl pulling a meerkat out of a blackhole... According to probability, it's actually all of those things until it's observed, and the wave function is used to calculate the probability of each potential state... Most are just so very unlikely that they can be disregarded, and although still possible, we tend to look at the "cat is alive/cat is dead" only split.

 

So, once you've opened the box and you look in, you actually learn the state of the cat. You know if it's dead or alive, you know if the vile of poison broke or did not, you know if Ben Franklin wearing a ball gag was in there the whole time... and "the wave function collapses."

 

All of the possibilities that made up the wave function go away (hence the collapse), because you've observed the system and only one possibility was realized.

 

Just wait until you read about double slits. Check common sense at the door and prepare to enter the rabbit's hole.

[swansont]

QM is so abstract as to use wave mathematical relationships to abstractly and arbitrarily put it into context for QM, such that you get completely abstract terms

 

Yes, this is somewhat true, especially from a perspective such as yours. But intuitive, classical physics fails to explain behavior on the atomic scale and smaller. QM works.

[sciencenoob]

I can understand and have looked past the fact that QM is just abstract so we need to drop our intuitions when using QM. But I still want to try to understand the signifiance of the wave function. For example, plank has been obsessed with the signifiance of the planks constant, as have mathematicians with pi, etc. Perhaps that is naive and but though naive I think this quest is still important towards developing better intuitions about the very abstract world we live in.

 

Nobody has explained what the wave functon is by definition though and why its use for QM works the way it does

[fredrik]

but the physical significance of the wave equation still seems to bear no much meaning to me. Of course there are other constants and values and equations etc in phsyics that simply have little physical signifiance, but QM is so abstract as to use wave mathematical relationships to abstractly and arbitrarily put it into context for QM, such that you get completely abstract terms

 

I'm not sure what physical means, but if I assume you are new to QM, I would say that the closest thing to attaching and "significance" to the wave function is in terms of the observers information about the system the wavefunction describes. You can say that the wavefunction encodes the observers information about something.

 

Clearly information about something can come in different forms. Direct, or indirect. In the case of indirect information, there exists a relation between the information, and an equivalent direct information. For example, there is a relation between information about position and momentum. The uncertainty relation follows directly from the relation.

 

It's easy to see how can can add similar types of information, but how to do add dissimilar types of information? I think this is what QM is about, or should be about. But others may disagree, that's just my personal private opinion.

 

Moreover the dynamics equations of QM, should I think be interpreted as evolution of expectations, which i think is best viewed in terms of information and probabilities.

 

When I was first encountered quantum mechanics from the "mechanical" point of view, it sure is hard to understand what is going on. I think to appreciate it, one should think about how we learn things. How does the process of aquiring the knowledge of a fact look like? And does perhaps this process itself have any implications on thinkgs? can we really separate the process of learning facts, from the facts themselves? Analyze that and it makes more sense.

 

In this sense, classical mechanics is much more speculative than is QM. But once I got over that first edge, I noticed more things that was not satisfactory.

 

I think few "experts" would bet their grandmother on the QM won't be revised, although I'm sure some probably would But if I have to guess, this revision is not going to make it any easier to understand from the point of view of classical mechanics, it's probably going to get even worse.

 

/Fredrik

[BenTheMan]

noob---

 

Are you talking about the Schroedinger equation or the wave equation? Because the Schroedinger equation isn't a wave equation, it's a dispersion (i.e. heat) equation.

 

The founding of the wave equation and the use of the wave like nature of particles to describe particle systems is nothing but luck , hope, and messing around with mathematics that magically seem to work. In that nobody knows the underlying theory or understnading of why particles behave as waves in the first place or why mathematical wave equations and operations can be used to describe particle behavior.

 

This is kind of trivializing some pretty brilliant work. I think it was a gradual realization, as opposed to ``fiddling around with numbers untill it worked''.

 

yes but what 'is' the wave equation in itself. for the sake of argument I could care less about the mathematical operations that leads to the solutions of, but the physical significance of the wave equation still seems to bear no much meaning to me.

 

Presumably you mean ``What IS the Schroedinger equation?'' The Schroedinger equation describes the time evolution of a probability density. When you know the initial conditions of the system, you know the probabilities at all future times that the system will be in a given state.