Is the Wave Function a Fundamental Hypothesis in Quantum Mechanics?

[kroni]

Hello,

I would like to understand why particle are caraterized by their wave funtion ? Why parameters are probabilisticly defined ? I see no contradiction, physics problem or mathématical reason to this. Is wave function is a fundamental hypothesis of quantum mechanic or there is a proof.

Thanks for your answer or a link to a previous thread speaking about it.

Kroni

 

[Matterwave]

The Born rule, assigning a probabilistic interpretation of the wave function is a postulate of Quantum Mechanics. Schrödinger himself never accepted this probabilistic interpretation of the wave function. He tried unsuccessfully to interpret the wave function as an expression of electron density. There is no proof, other than that this formulation of quantum mechanics agrees with experiment.

Even though there is no proof, that doesn't mean it wasn't motivated by some physical insights. For this you might want to take a look at the history of quantum mechanics.

 

[DrChinese]

Hello,

I would like to understand why particle are caraterized by their wave funtion ? Why parameters are probabilisticly defined ? I see no contradiction, physics problem or mathématical reason to this. Is wave function is a fundamental hypothesis of quantum mechanic or there is a proof.

Thanks for your answer or a link to a previous thread speaking about it.

Kroni

Welcome to PhysicsForums, kroni!

There are several way to interpret the wave function itself. None of these change the actual nuts and bolts of the math used in working with physics.

For example, the so-called Bohmian interpretation has a physical rationale for the wave function. And yet there is nothing about this interpretation that is demonstrably "superior" to any other interpretation. So beauty is in the eye of the beholder.

 

[bhobba]

It lies at the very foundation of QM.

See this for its conceptual core:
http://www.scottaaronson.com/democritus/lec9.html

Basiclly QM is an extension of probability theory that allows the continuous transformation between so called pure states.

At a deeper level probabilities are tied up with something called Gleason's theorem:
http://en.wikipedia.org/wiki/Gleason's_theorem

We have also recently been discussing it on this forum (at an advanced level - don't be too worried if it's gibberish - its just to give a flavour about what going on here beyond the usual often quite poor treatments in the pop-sci literature):
https://www.physicsforums.com/showthread.php?t=758125

Thanks
Bill

 

[kroni]

Thanks a lot for your answer,

Ok, the Born rule assign probability to the wave function , this choice agree the experiments. What i don't understand is why there is a wave function ? I can imagine a world with perfect located particle, with state well defined and interacting together. Is there a mathematical proof or a contradiction which prove the necessity of having a particle-wave duality ?

Thanks

Kroni

Bhobba, your link is very interesting.

 

[DennisN]

What i don't understand is why there is a wave function ?

"Why questions" are very hard to answer, see this clip .

[...]having a particle-wave duality ? (my bolding)

Regarding the term "particle-wave duality", please see the FAQ on it here:
https://www.physicsforums.com/showthread.php?t=511178

Is there a mathematical proof or a contradiction which prove the necessity of having a particle-wave duality ?

How about looking at what really happens in the actual world we live in?

Look at the double slit experiment with electrons:
http://www.hitachi.com/rd/portal/research/em/doubleslit.html
(there is a video clip on the middle of that page)

A single electron is always detected at only one location in the detector, yet electrons that are fired one by one through an "electron biprism" will together build up an interference pattern over time. We can't predict where a single electron will be detected, but we can predict what the interference pattern will look like.

Bonus material: you could also have look at "The Original Double Slit Experiment" (Veritasium)

 

[bhobba]

I can imagine a world with perfect located particle, with state well defined and interacting together. Is there a mathematical proof or a contradiction which prove the necessity of having a particle-wave duality

Well actually the wave particle duality is a crock of the proverbial:
https://www.physicsforums.com/showthread.php?t=511178

Thanks
Bill

 

[DrChinese]

I can imagine a world with perfect located particle, with state well defined and interacting together. Is there a mathematical proof or a contradiction which prove the necessity of having a particle-wave duality ?

Yes, there is. It does not prove "particle-wave duality" but it does disprove your hypothesis of "perfect located particle, with state well defined". It is called Bell's Theorem. There are a number of additional theorems in the same class, they are called "no-go" because they all throw out local realistic (local hidden variables or hidden state) theories as a batch. Bell's Theorem states (from Wiki):

"No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."

Bell tests confirm QM and reject local realism.

 

[kroni]

Thanks a lot for the answer.
DrChinese, i would like to understand the problem by a contructivist approach, not from experiment.
Ok there is no physical theory of local hidden variable that reproduce prediction of quantum mechanics but it's not the core of my problem.
My problem is why quantum mechanics ?

I think i have a tiny way of explanation.
Imagine a perfectly localized particle that desintegrate, the residu of désintegration must go in a direction and every choice of direction is arbitrary. A good solution to avoid this choice is to propagate a field representing probability of presence in all the direction until observation. May be somethink like this justify the existence of wave function, and its probabilistic interpretation.

 

[UltrafastPED]

...understand the problem by a contructivist approach, not from experiment

By "contructivist approach", do you mean:
http://en.wikipedia.org/wiki/Constructivism_(philosophy_of_education )


I doubt you will be anymore satisfied than Einstein was.

 

[DrChinese]

...My problem is why quantum mechanics ?

I think i have a tiny way of explanation.
Imagine a perfectly localized particle that desintegrate, the residu of désintegration must go in a direction and every choice of direction is arbitrary. A good solution to avoid this choice is to propagate a field representing probability of presence in all the direction until observation. May be somethink like this justify the existence of wave function, and its probabilistic interpretation.

QM works. Why would you use something that didn't work?

As to your example: Bell's Theorem limits descriptions such as this. For your example to work, the future must be able to affect the past. I would recommend reading about Bell.

 

[bhobba]

My problem is why quantum mechanics ?

Why the big bang? Why is the electrons mass exactly what it is? The list of whys scientists do not know the answer to is pretty well endless.

I can't tell you why QM. I can however explain its conceptual core as an extension of probability theory - but that's as good as I can do.

Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Standard probability theory is basically a theory about such mixed states. However it has a problem with continuous transformations which in physics you more or less require. To see this consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

Basically QM is the theory that makes sense out of weird pure states that are complex vectors. The way it does that is via the Born rule which says the probability of outcome i is the square of the associated complex number. Why is that? Gleason's theorem sheds some light on it but doesn't entirely prove it, a few other assumptions are required eg non contextuality and the probability must be a function of the complex number.

The bottom line is its just how nature is.

Thanks
Bill

 

[kroni]

Thanks for your clear and helpful answer !
I am sorry for my poor English but i don't live in a English-speaking country so i do the best i can.

UltraFastPED : I speak about constructivism in mathematics. Before using something, before write an équation, on prove a théorem you need to prove that why you will use exist. Simple example : If you want to solve $ax = b$ on $\mathbb{R}$, you need to explain how you construct $\mathbb{R}$.

Bhobba and DrChinese, my thought experiment explained in the last post is not related to Bell's theorem. I know that no local hidden variable theory are valid. But my objective is to prove by a contradiction that wave function is a mathématical nécéssity to avoid arbitrary or random choice and the probability interpretation is implicit.

Now forgot all you know about QM, just imagine a point-like object in 3D-space. This object disintegrate in two piece, all the direction for the two piece are equiprobable. If you chose a particular direction it will be a random or an arbitrary choice. So, you absolutely need to model particle state by a probabilistic field to avoid a random or an arbitrary choice of direction.

Thanks

Clément

 

[bhobba]

But my objective is to prove by a contradiction that wave function is a mathématical nécéssity to avoid arbitrary or random choice and the probability interpretation is implicit.

That's impossible. For example Bohmian mechanics (BM) is fully equivalent to standard QM meaning there is no way to tell them apart and its deterministic. In BM probabilities arise due to lack of knowledge about initial conditions.

It evades Bell by being explicitly non local. It evades Kochen-Specker by being explicitly contextual.

I can see you are into math. Its exactly the same situation with the parallel postulate in geometry. Its neither right or wrong. If its different you end up with a different geometry - that's all. Determinism in QM is neither right or wrong - you just end up with different theories depending on what you accept - with a bit of a twist - unlike the parallel postulate it doesn't necessarily lead to experimental differences.

Thanks
Bill

 

[kroni]

Thanks for your answer.

I understand better the different view of the same problem.
I need to learn more about QM history before use it !

Thanks

Clément