Position or probability

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Position or probability

Post by Jishnu Cv on Wed Aug 24, 2016 5:33 pm

What this equation really gives(physicaly) y=Asin(kx-wt) in a given time?
Position of the particle at that time or the probability of finding an electron

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Re: Position or probability

Post by Sreejith A Nair on Wed Aug 24, 2016 6:00 pm

Physically it makes no sense as the amplitude of the wave is imaginary. And imaginary numbers makes no physical sense. But the square of the modules of y gives the probability of finding the particle at point x at time t.

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Re: Position or probability

Post by Sabiya on Fri Aug 26, 2016 12:58 am

Actually we don't use y=Asin(kx-wt) to describe an electron. This equation is used in SHM or EMW to describe the vertical displacement from mean position and the strength of the electric field respectively.

To describe an electron travelling in one dimension, we use Ψ(x,t) = [Ψ(x)][e^-iwt] where i=√−1, which is a solution of the Schrödinger wave equation in one dimension.
Here,
-->Ψ(x,t) is the variable that is oscillating in a wave like manner and contains information about our electron, just like the displacement in SHM and electric field strength in EMW. It changes with position and time.
-->Ψ(x) is part of Ψ(x,t) which is time independent, i.e. it only depends on position. This may or may not be wavy.
-->e^-iwt is a complex exponential function, which changes only with time. This function actually causes the Ψ(x,t) to rotate with angular velocity w anticlockwise about the x-axis, but you don't need to know that.

So, coming to your question, what does Ψ represent? Believe it or not, nobody has the answer yet. Of course, there was Max Born who said that |Ψ|^2 at a particular x represents the probability density of finding our electron at x, but he was talking about |Ψ|^2, not Ψ. Usually people find it hard to interpret Ψ, because it is a complex number, and people (like Sreejith A Nair) believe that complex numbers don't have any physical significance. To them, I'd like to say that just because you can't make sense of complex numbers in the same way that you make sense of positive real numbers does not mean that they don't make sense at all. Maybe you need to look at your dear positive real numbers in a new way, and then you'll see that negative real numbers, as well as imaginary and other complex numbers also fit into this new world view and suddenly they too make sense to you. Don't close your mind away to new experiences, otherwise you'll find it hard to discover new things.

Go ahead, think about it freely. Complex numbers can be meaningful too, and so can Ψ. Why don't you try to interpret Ψ in your own way? Maybe you'll be the one to come up with a new interpretation of Ψ  Smile .
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Re: Position or probability

Post by Dinesh PR on Fri Aug 26, 2016 4:02 pm

Y (that's psy) doesn't actually have a physical interpretation or meaning to it. Suppose you refer to a distance from zero level or electric field, they can be perceived by humans as a length or something that affects charges based on their previous experience. But Y, is some purely mathematical object that gives you odds of finding position, momentum, energy or any other quantity that describes the state of an electron by solving differential equations obtained from the usage of certain operators. This is Copenhagen's interpretation for you, the most accepted one in quantum mechanics. In a classical wave, if you make distance twice, the wave is different in terms of energy, momentum,etc. In Y, 2 times Y is still same. See, you CANNOT interpret these complex numbers in Y because I can just make it real (at a particular point) by multiplying with the conjugate.
I'll be grateful to know what physical sense you make from a bunch of complex numbers in Y? I'd recommend you to look up the dictionary meaning of 'physical' before replying.

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Re: Position or probability

Post by Dinesh PR on Fri Aug 26, 2016 4:07 pm

One more thing... Remember what the great Richard Feynman said when asked about his own understanding of it. He said, "I can never understand quantum mechanics!". Arguably, there's never been a person who 'imagined' it better!!!

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Re: Position or probability

Post by Sreejith A Nair on Fri Aug 26, 2016 5:05 pm

First of all I NEVER said physical significance but said physical sense.

Sabiya with at most respect I would recommend you to first go look for the meaning of 'physical' in a good dictionary just as Dinesh PR said. After understanding the meaning. Consider the fact that we are just humans. I do not who you are. But still for us we cannot sense i (iota) with our senses (the sense that we can make is just mathematical or sense through representation as Dinesh PR pointed out using the example of electric fields.).So it is nonsense that you believe that just thinking about the physical sense would make sense for something which you cannot sense. (Due to limitations of the human senses )

Also you said that positive real numbers make sense to humans like us. But for your information the sense that we assign to it is just mathematical or representative. They also make no physical sense. (Like we can interpret one tree or car and NOT one itself. ).

And if you believe that you can then start simple and explain the physical sense of number one on the forum.


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Re: Position or probability

Post by Sabiya on Mon Aug 29, 2016 8:20 pm

I have the checked the dictionary, and I admit that I was wrong in using the word 'physical' in my previous reply.

Yes, numbers do not make physical sense to us by themselves. Like Sreejith A Nair pointed out,
Sreejith A Nair wrote:we can interpret one tree or car and NOT one itself.
But we can (and do) associate +ve real numbers (especially natural no.s) with properties of physical objects (like length, mass, temperaure, etc.) in our daily lives. Now this association we make no doubt makes our lives easier, but it's not useful if we want to gain an understanding of other no.s (like -ve reals, imaginary no.s and other complex no.s), as it is biased towards the use of +ve reals (especially naturals). If we base our understanding of numbers on our daily experiences (which are skewed to favour the use of certain no.s over others), then we simply cannot fit other complex no.s into this picture. This is why I said
Sabiya wrote:you can't make sense of complex numbers in the same way that you make sense of positive real numbers

To give an equal footing to the use of all complex no.s, we have to see them in a way that is not skewed towards the use of certain number sets. Seeing numbers as their representations in the Argand plane is a pretty good way to do that, I think. Here each complex no. has a representative significance, hence just about all numbers make equal sense to us.


Dinesh PR, I would like to point out that Ψ is actually different from 2Ψ. You can see this is you use the momentum operator or kinetic energy operator on Ψ. The results for Ψ and 2Ψ will be different. Also, could you please explain what you meant when you said
Dinesh PR wrote: you CANNOT interpret these complex numbers in Y because I can just make it real (at a particular point) by multiplying with the conjugate.
I don't understand it.

Coming to Feynman,
Richard Feynman wrote: I think I can safely say that nobody understands quantum mechanics.
I'm sure this was true at his time and still remains true. But that doesn't mean that we have to keep it true.
Sure, he was one of the most imaginative physicists of all time, and it's right to think that if he didn't understand QM, we're gonna have a hard time too. But that shouldn't mean that we can never understand quantum mechanics. Can't we atleast try to give more meaning to complex no.s and try to interpret Ψ?
Of course, being more imaginative than the most imaginative people of all time is gonna be an uphill task, but that shouldn't stop us from trying. albino
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Re: Position or probability

Post by Dinesh PR on Mon Aug 29, 2016 8:53 pm

No offenses please! You wrote about interpreting the complex number in the function. I meant 'a particular real or imaginary value(s) of Y does not exist'. You can still multiply the Y by any number and still represent the same physical situation! So the argument was, 'How can a physical interpretation exist for each value in Y(x)?'. The statement by Feynman is absolutely true as far as 'Quantum mechanics' is concerned. It may not be as true if things lead to a different, better theory!!!

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Re: Position or probability

Post by Sreejith A Nair on Mon Aug 29, 2016 9:16 pm

I do support Sabiya's observation's after all scientific method revolves around the concept of a better explanation (Which we never attains) . If Einstein had considered Newton to be ultimate then Einstein's existence would have been equal to his non existence.

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Re: Position or probability

Post by Dinesh PR on Mon Aug 29, 2016 10:16 pm

Einstein did not modify Newton's laws but proposed a great theory himself. Remember that the reputation of Newton's laws were not really too much shunned by the new theory. It still works fine in its domain!

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Re: Position or probability

Post by Sreejith A Nair on Mon Aug 29, 2016 10:37 pm

What I ment was not that Einstein proved Newton wrong but was that Einstein did not consider himself to be inferior to Newton. And worked on something that Newton was unable to explain. (Namely the cause of gravitational force)

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Re: Position or probability

Post by Admin on Mon Sep 05, 2016 10:01 pm

As Sabiya mentioned, Ψ(x,t) = [Ψ(x)][e^-iwt] . y=Asin(kx-wt) is not the standing wave equation. Probability is the square of Ψ and it is independent of time. Probability correponds to the probability of finding an electron at a given position x. Ψ does not have any direct physical interpretation in terms of observables unlike Ψ^2 which is probability density -- a physically realizable property. Ψ, of course it has physical significance. Without Ψ, electrons would not interfere.

Position of the electron cannot be known in the present formulation of quantum mechanics. The equation does not govern the motion of the electron like. It does in classical mechanics.

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Re: Position or probability

Post by Holmes on Sun Sep 11, 2016 5:30 pm

it gives the probability of finding a particle at a particular space and time!!!! Laughing

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Re: Position or probability

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