Why amplitude squared gives probability / Schrodinger Equation

[phina]

We started Quantum Physics in class, and I tried working out the Schrodinger Equation (not mathematically, of course - that's far beyond my level. Just the vague concepts)

I understand it's basically a function that shows how something changes in time, and a snapshot of it at one particular moment describes possible positions of something.

What I really don't understand is why you square the amplitude to get the probability. I understand that the amplitude is a complex number, and squaring it would solve that (I believe... I've never formally learned about complex numbers), but I'm really confused as to why you would square it.
I assumed that it's not so much that you take amplitude and square it to find probability, but that you take probability and root it to find amplitude... but still, why? What does amplitude by itself represent?
The amplitude allows for interference, whereas the probability doesn't... I also figured that much... but still, I'm at a bit of a loss.

To confirm, matter-waves are probability waves for each particle? They are used for the Schrodinger Equation?

I'm having so much trouble tying this all together, haha.
Thanks in advance for any explanation

 

[RedX]

I think you hit it on the head when you said that you need to square the amplitude to get a non-negative number. The probability or probability measure should always be positive or zero (I once saw in a calculus book the integral $$\int^{1}_{5} x^2 dx$$ and I was puzzled if that was really defined, as the measure would be negative: does anyone know if that integral makes sense?).

If you imagine the complex plane, the only thing that is really physical is the length of a complex number. You could perform rotations in complex space which would change the complex number, but not the length or square of the complex number. So squaring a complex number still gives you the important information.

 

[phina]

Could you by any chance expand on the second paragraph? How are you not changing important factors?
If you have a+i, and you square it, would you not end up with a+2i-1?
Am I completely misunderstanding complex numbers?

 

[Einstein Mcfly]

You don't "square" it, you multiply it by its complex conjugate. THAT gives you the absolute magnitude of the vector. This isn't only true for wave functions by the way, it's true for any probability amplitude used in any statistical calculation. The question is then, "why do we need probability amplitudes to describe microphysics", and at this point (and perhaps for your whole life) you're just going to have to accept "because" or "because it works", as the best minds of the last eighty years haven't been able to hash that one out.

 

[phina]

Is it just a semantics thing? Does "squaring" a complex number actually mean multiplying by its complex conjugate?

And if you multiply (a+i) by (a-i), wouldn't you end up with (a^2 - 1) which is still not what you started out with?

Sorry, sorry, please bear with me

 

[Vanadium 50]

I would look at it another way - the Schroedinger equation is wave equation. (This is an important concept, not to be missed) With all waves, the intensity is the amplitude squared. In QM, the probability density is the intensity.

 

[kanato]

Is it just a semantics thing? Does "squaring" a complex number actually mean multiplying by its complex conjugate?

And if you multiply (a+i) by (a-i), wouldn't you end up with (a^2 - 1) which is still not what you started out with?

Sorry, sorry, please bear with me

Squaring a complex number is different than multiplying by its complex conjugate. You can square a complex number just like any other number: (x + iy)^2 = x^2 - y^2 + 2xyi, which is generally complex.

Also, if you multiply (a+i) by (a-i) you get (a^2 + 1), not minus one. That is not what you started with, but if you square anything except 1 or zero you get something other than what you started with, so I'm not sure what your objection is.

 

[meopemuk]

(a+i) (a-i) = a^2 + 1

 

[phina]

But why is probability density equal to the amplitude of the equation?

My objection comes from RedX saying that squaring a complex number still gives you the important information.

I just don't understand where amplitude comes from, since the wave equation is not "real" and is just a mathematical tool. Is amplitude just derived from probability to make the equation work?

If squaring a complex number gives you a complex number, you've got i in your probability, so how does that solve anything?
If amplitude means nothing by itself, and if the wave equation is a mathematical tool, why isn't it made so amplitude IS the probability?

Also, sorry, algebra mistake, haha.

 

[RedX]

My objection comes from RedX saying that squaring a complex number still gives you the important information.

I worded it poorly. You do lose a lot of mathematical information by multiplying by the complex conjugate. Just squaring it however, you only lose if it was plus or minus (unless you specify a Riemann sheet in your mapping).

 

[phina]

You only lose the sign? So x+iy = |x^2 - y^2 + 2xyi| ?

But... how? Are these just properties of complex numbers I'm completely unaware of?

 

[transience]

You only lose the sign? So x+iy = |x^2 - y^2 + 2xyi| ?

But... how? Are these just properties of complex numbers I'm completely unaware of?

If you have a number that you know has been squared then by finding the square root of this number you will find all of the information about the original number EXCEPT for its sign. For instance if you know that x^2=9 then you know that |x|=3 but you don't know whether x=+3 or -3 unless you can find that from something else, for instance if you find a negative number while computing a number that by definition must be positive you can safely discard that part of the solution.

Multiplying a number by its complex conjugate gives you the magnitude of the number in the complex plane. If you want to go backwards with this operation you will only be able to say what distance this point is from the origin, which will give you a circle that the number must lie on, you cannot know at what point on the circle the number is unless you have other information. So by taking the complex conjugate you have lost a lot of information.

The wavefunction contains all of the information that it is possible to know quantum mechanically about the system. The wavefunction will tell you what states are mutually exclusive and what happens when you do certain things to the system. When you take the amplitude you are mapping the wavefunction onto real space so you can know where particle you are looking at is. This is an operation that you perform to extract a certain piece of information from the wavefunction.

 

[phina]

Oh, okay. Though I still don't see why it's necessary to square it, rather than just... make it positive.

Another question:
The amplitudes of waves can interfere with each other... but probability can't, of course. That wouldn't make sense. Why?

 

[Bob_for_short]

As Vanadium 50 said, the answer follows from the equation. It can be shown that there is a "continuity" equation for the wave function "squared". For one particle it describes the probability flow, for N >>1 particles it describes the particle flux.

 

[RUTA]

See section 7, "Origin of the Born Rule and Quantum Dynamics," in "Causality, Symmetries and Quantum Mechanics," Jeeva Anandan, Foundations of Physics Letters v15, #5, October 2002, pp 415-438.

 

[DrChinese]

See section 7, "Origin of the Born Rule and Quantum Dynamics," in "Causality, Symmetries and Quantum Mechanics," Jeeva Anandan, Foundations of Physics Letters v15, #5, October 2002, pp 415-438.

Great reference, RUTA, I found this at:

http://arxiv.org/abs/physics/0112020

"...In particular, an argument is made for why there are probability amplitudes that are complex numbers, which obey the Born rule for quantum probabilities. This argument generalizes the Feynman path integral formulation of quantum mechanics to include all possible terms in the action that are allowed by the symmetries, but only the lowest order terms are observable at the presently accessible energy scales, which is consistent with observation."

 

[phina]

Oh god, I have a feeling I'm getting myself deep into things that are way, wayy beyond my level.

 

[Truecrimson]

Oh god, I have a feeling I'm getting myself deep into things that are way, wayy beyond my level.

If you want "the" derivation, there's no such thing yet, as far as I know. People've been trying to come up with a more convincing argument than, say, the heuristic analogy with the intensity of light. But we can think about it the other way around, for example, by asking "what if it's not the square of the amplitude?" to get the better idea of the situation we're in. It's like asking "why complex number?" No one knows an absolute reason. So don't worry about it if you see people arguing about these things not on the level of introductory QM.

I recommend that you take a look at this lecture http://www.scottaaronson.com/democritus/lec9.html

 

[RUTA]

If you want "the" derivation, there's no such thing yet, as far as I know. People've been trying to come up with a more convincing argument than, say, the heuristic analogy with the intensity of light. But we can think about it the other way around, for example, by asking "what if it's not the square of the amplitude?" to get the better idea of the situation we're in. It's like asking "why complex number?" No one knows an absolute reason. So don't worry about it if you see people arguing about these things not on the level of introductory QM.

There are those who believe they have it "understood," e.g., Cramer's transactional interpretation http://www.npl.washington.edu/ti/ , but I agree with Truecrimson.

 

[Truecrimson]

Oh, and there's Gleason theorem.
http://en.wikipedia.org/wiki/Gleason's_theorem

I've heard that the theorem "motivates" Born's rule (the squaring rule) so I'm not sure if it can be taken as a final answer or not. Anyone wants to clear me up on this?

To the OP, you can think of the statement of the theorem in Wikipedia saying that the only possible measure of the probability in this Hilbert space framework is the square of the wave function, with the caveat that the dimension has to be greater than 2. With wave functions, we're in an infinite dimensional space (i.e. you can write a wave function as an infinitely long column vector of complex numbers), so there's no peoblem here.

On a side note, I think the theorem has been proved in 2 dimensions using POVM.

 

[ArjenDijksman]

What I really don't understand is why you square the amplitude to get the probability.

I like to think about quantum particles as represented by vectors, which you may view as little spinning arrows or needles in 3D. A quantum measurement involves at least two particles: 1. the detected particle and 2. the detecting particle. So the probability of detection is proportional to the cross section of the detected particle times the cross section of the detecting particle, both projected on a fixed axis. If both particles rotate in phase (due to a pilot wave), the detection probability is the square of the complex amplitude of the slowest spinning particle.

There are some visuals at the http://en.wikiversity.org/wiki/Making_sense_of_quantum_mechanics/Principles_of_Quantum_Mechanics" .

 

[phina]

But.. argh. It's so frustrating. This seems like such a simple question, but there doesn't seem to be any simple answer.

What is amplitude, then?

Anyone know of any books that'll talk about this with some depth, but for a beginner? All books I've read have just sort of... mentioned it briefly and moved on. I want further explanations and discussions, but wouldn't be able to handle advanced jargon and stuff.

 

[Sillyboy]

I am just a greenguy, so I have no deep comprehension. I just conprehend the wave function compared with the description of optical wave.
I think if you describe it in terms of exp(complex) form, you can have a better understanding~
But, at least I am studying the mathematical methods for the QM, so I do not have a good comprehension. just ignore what I said~

 

[RUTA]

But.. argh. It's so frustrating. This seems like such a simple question, but there doesn't seem to be any simple answer.

Welcome to the world of intellectuals If you want simple answers, you can, for example, ask a "priest" (any religion, any demonination) for their worldview. Most people want simple answers, regardless of their intellectual merit. Thus, there are far more "religious zealots" in the world than scholars. Oh, about the nature of "simple questions," you'll find they are often the most profound.

Enjoy!

 

[Truecrimson]

Anyone know of any books that'll talk about this with some depth, but for a beginner? All books I've read have just sort of... mentioned it briefly and moved on. I want further explanations and discussions, but wouldn't be able to handle advanced jargon and stuff.

Isham's https://www.amazon.com/dp/1860940013/?tag=pfamazon01-20 may help.

I really think that you'll be less frustrated when you learn more about the mathematical formalism, at least up to the notion of state vector or postulates of QM.

 

[ArjenDijksman]

But.. argh. It's so frustrating. This seems like such a simple question, but there doesn't seem to be any simple answer.

What is amplitude, then?

OK, I didn't get your primary question first. So, what is amplitude? Sillyboy's answer gives a good hint:

I think if you describe it in terms of exp(complex) form, you can have a better understanding~

Remember that a particle can be seen as having the shape of an arrow. The amplitude of the vector representing the arrow-particle is then the complex number modulus(A).exp(i.phase) that represents its orientation with respect to an initial orientation of the vector. For example, if the vector |psi> rotates at angular velocity w, its amplitude is exp(i.w.t).

I like Feynman's explanation. For example, at the http://vega.org.uk/video/subseries/8" , watch at about min 57:00 - 1:00:00.

 

[jambaugh]

You can derive the Born Probability formula from certain assumptions.

Start with the representation of system modes projectively as elements of a Hilbert space.

Assert that a zero transition amplitude indicates a forbidden (zero probability) transition.

Assert that for normalized mode vectors the unit magnitude transition amplitudes correspond to assured transitions.

Now the key to it being a square of the probabilities is the assertion that multiple independent instances of a physical system is represented as a single large system by taking the tensor product of many copies.

You consider the limit as N goes to infinity where you replicate N copies of the system and expand the initial mode into parallel and orthogonal parts. You group terms by components which have a definite number of assured transitions and pay attention to what percentage has the peak coefficient. In the limit all coefficients go to zero except the sum of modes where there are exactly |a|^2 * N assured transitions. With normalization this sum of modes goes to unit norm in the limit and thus it is assured that |a|^2 is the percentage of transitions which will be observed.

If you like I can type up a formal derivation and post it.

The mathematical key is the fact that to get a norm for a big space you take the sum of the squares of the norms in the subspaces. Since when combining independent systems you both add probabilities and add square norms the probabilities must be squares of the norms.

Now you can make it be any power you want of the magnitude of the transition amplitude by using a different kind of normed space than a Hilbert space. Look at e.g. Soblov spaces and Taxicab norms. But you then find they are not isotropic in the same way as are Hilbert spaces. There will be preferred directions and bases.

Hilbert spaces with their square summing norms are the most isotropic with their unitary group of symmetries.

 

[Fra]

What does amplitude by itself represent?

This is interpretational questions but in one view it is supposed to represent the observers state of information about the system here and now.

ie. all the information knows about the system is "encoded" in this state vector (wave function). And all the possible information states maps out a hilbert space.

The schrödinger equation determines the time evolution of this information, in between measurements.

The various ways of computing expectation values or things, from this "information state" is not really yet understood in a deeper sense that everyone agrees upon.

You can even ask what exactly is the physical and informational basis od probability, and it's not trivial. So before we can answer why the square of the wavefunction is probability, we might want to understand what probability means.

The usual meaning, operationally defined in terms of distributions of repetitive samplings etc, really is only a mathematical abstraction. Infinite sequences and infinite ensembles seriously do NOT have a real physical representation, that begs the question what exactly do we mean by probability of future events? To talk about the probability of past events in the historical frequency sense is one thing, but what exactly does probability of the future mean? And what does a relative representation of probability look like?

Both the notion of probability and the notion of information state of a system are thus in the interpretation I describe here, relative notions. This generates a second set of "issues", such as what the implications of the fact that two observers in general might have totally different descriptions of the same thing are.

So I think that those who question the meaning of the wavefunction, but claim to understand perfectly the physical basis of probability of the future, is missing both a deeper sense of the problem, and a possible key to resolution.

There is also a problem with that idea that there is an observer independent view of things, simply because it's a non-physical and non-verifiable statement. A group or famility of observers can negotiate a consensus, but what is the meaning of this to a larger group?

Of course, one can also ask what is "information", and where is this information stored? Are there any infinite memory sinks to encode/store information or is the information capacity generally bounded?

Most of these questions are good, but their final resolution is still open for suggestions.

But this doesn't prevent progress, and the development of most of quantum physics doesn't need the answer these questions. The questions are of more relevants for philosophers, and those who want to develop the theory further into yet unknown territory, or for those what ponder of some of the large number of unsolve or open questions in physics.

/Fredrik

 

[bcrowell]

In my opinion, the use of complex numbers is mostly irrelevant to phina's basic question here.

First off, there's a general wave idea that has nothing to do with quantum mechanics. This idea is that the energy of a wave is proportional to the square of its amplitude. (For many types of waves, this is only a good approximation for small amplitudes.) So you automatically need to consider two things: A and A^2. A is always the thing that adds when waves come together. If you're representing a sound wave as a scalar that measures the pressure, then A is a scalar, and A^2 is a scalar. In the case of an electromagnetic wave, A is a vector (the electric and/or magnetic field), and to get a scalar that will give you an energy, the only mathematically sensible thing to do is to take the square of its magnitude.

When it comes to quantum mechanics, I would start from the double-slit experiment: http://www.lightandmatter.com/html_books/6mr/ch03/ch03.html#Section3.3 We observe an interference pattern, so it must be a wave phenomenon. We also observe discrete "hits," like bullet-holes, so it's simultaneously a particle phenomenon. We conclude that light exhibits both particle and wave characteristics, in the same experiment. You don't need any complex numbers here. The wavefunction, which is in the case of an electron is kind of a mystical unobservable creature, is in this case simply the electric field (or the magnetic field, whichever you like). Nothing changes about the argument above concerning EM fields. A still has to be the field vector, and the energy density still has to go like A^2, which has to be interpreted as the square of the field vector's magnitude. The only difference is that we observe that the outcome of the experiment is probabilistic, and the correspondence principle requires that the probability go like A^2, since the probability has to relate to the energy density in the classical limit of large numbers of photons.

Even when it comes to electrons, you can do quite a bit of physics without having to bother with the fact that the wave really needs to be a complex number. You can do all the states of hydrogen, for example, with real-valued wavefunctions. The only reason you really need the complex numbers is to represent traveling waves.

Lots of people are answering Phina's question with big words and references to hard books. His question was a beginner's question, and he needs a beginner's answer and pointers to good beginner's books. A good beginner's book on the conceptual stuff is Feynman's QED: The Strange Theory of Light and Matter.

 

[ArjenDijksman]

A good beginner's book on the conceptual stuff is Feynman's QED: The Strange Theory of Light and Matter.

I agree. That book is a second version of Feynman's Douglas Robb Memorial Lectures mentioned earlier. Key message of Feynman in these lectures: "All we do is draw little arrows on a piece of paper - that's all!" Complex numbers only serve to get track of the changing angle of the arrow. Real numbers only serve to give lengths of projections of the arrow and hence probabilities of interaction between two arrows. But the physics of quantum particles is conveyed by "arrow-particles", as opposed to "point particles" in a classical view.

 

[phina]

I actually read QED about a year ago.
I watched that part of that lecture, where it seemed Feynman basically said "Why does it work this way? I dunno! No one does!" so... I guess I'm not finding my answer, haha.

 

[ArjenDijksman]

I actually read QED about a year ago.
I watched that part of that lecture, where it seemed Feynman basically said "Why does it work this way? I dunno! No one does!" so... I guess I'm not finding my answer, haha.

Come on, don't let yourself be impeded by such negative statements, even by Feynman. Quantum mechanics stems from a very simple fundamental principle: a quantum particle is represented by a spinning arrow. The motion of the tip of a spinning arrow is always perpendicular to the arrow itself. Draw this on paper. In complex notation, perpendicular means exp(i.pi/2) or in shorthand i. So the equation that describes the spinning of the arrow |psi> is simply:

|psi(t+dt)> - |psi(t)> = i.|psi(t)>.omega.dt

with omega the angular velocity of the tip of the arrow. This is a generalized form of the time dependent Schrödinger equation for arrow-like objects (needle, rod, twirling baton...).

 

[RUTA]

Come on, don't let yourself be impeded by such negative statements, even by Feynman. Quantum mechanics stems from a very simple fundamental principle: a quantum particle is represented by a spinning arrow.

“All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics. It has survived all tests and there is no reason to believe that there is any flaw in it. We all know how to use it and how to apply it to problems; and so we have learned to live with the fact that nobody can understand it.” Murray Gell-Mann in L. Wolpert, The Unnatural Nature of Science (Harvard University Press, Cambridge, MA, 1993), p. 144.

 

[ArjenDijksman]

“All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics. It has survived all tests and there is no reason to believe that there is any flaw in it. We all know how to use it and how to apply it to problems;..."

Yes.

"...and so we have learned to live with the fact that nobody can understand it.” Murray Gell-Mann in L. Wolpert, The Unnatural Nature of Science (Harvard University Press, Cambridge, MA, 1993), p. 144.

Too sad. I can understand that senior physicists express this disillusion but in order to ensure progress for the future, it's important to encourage junior physicists to question it.

"Murray Gell-Mann said that we all know how to calculate, how to use QM, but none of us really understands what is behind the formalism, what it is saying about nature. That has to be answered in some way." Basil Hiley in an http://www.goertzel.org/dynapsyc/1997/interview.html".

 

[ytuab]

“All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics. It has survived all tests and there is no reason to believe that there is any flaw in it.

We often see these phrases which are full of confidence in the QM textbooks.
But I think we need more explanation about QM.

For example, the probability density of the electron of the point at infinity (near the point at infinity) in the hydorogen atom is not zero.
And when an electron is released by an apparatus, if the electron's position at first is expressed by a wave packet,
(or the instant the delta function which means the electron's position becomes a wave packet,)
the probability density of the electron of the point at infinity(near the point at infinity) is not zero.

I want to know how QM textbooks explain this phenomina.
Are there any flaw in QM?