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Why amplitude squared gives probability / Schrodinger Equation

  1. Nov 20, 2009 #1
    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
     
  2. jcsd
  3. Nov 20, 2009 #2
    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 [tex]\int^{1}_{5} x^2 dx [/tex] 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.
     
  4. Nov 20, 2009 #3
    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?
     
  5. Nov 20, 2009 #4
    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.
     
  6. Nov 21, 2009 #5
    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
     
  7. Nov 21, 2009 #6

    Vanadium 50

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    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.
     
  8. Nov 21, 2009 #7
    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.
     
  9. Nov 21, 2009 #8
    (a+i) (a-i) = a^2 + 1
     
  10. Nov 21, 2009 #9
    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.
     
  11. Nov 21, 2009 #10
    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).
     
  12. Nov 21, 2009 #11
    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?
     
  13. Nov 21, 2009 #12
    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.
     
  14. Nov 21, 2009 #13
    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?
     
  15. Nov 21, 2009 #14
    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.
     
    Last edited: Nov 21, 2009
  16. Nov 21, 2009 #15

    RUTA

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    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.
     
  17. Nov 22, 2009 #16

    DrChinese

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    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."
     
  18. Nov 23, 2009 #17
    Oh god, I have a feeling I'm getting myself deep into things that are way, wayy beyond my level.
     
  19. Nov 25, 2009 #18
    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
     
  20. Nov 25, 2009 #19

    RUTA

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    There are those who believe they have it "understood," e.g., Cramer's transactional interpretation http://www.npl.washington.edu/ti/ [Broken] , but I agree with Truecrimson.
     
    Last edited by a moderator: May 4, 2017
  21. Nov 25, 2009 #20
    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.
     
    Last edited: Nov 25, 2009
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