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Why need *complex* probability amplitude?

  1. Oct 15, 2008 #1
    Does anyone know a deeper reason why the quantum mechanical probability amplitude has to be complex?

    Is it to incorperate time dependence?

    Or maybe the operator/eigenvector formulation is special and since it includes the scalar product, having complex variables is more general and neccessary?

    Or maybe the fact that there is some spin with its transformation means that amplitudes should be complex?

    Hope someone understands these vague ideas :rolleyes:
    No Schrödinger wave argument please, because that doesn't account for the real need of complex numbers.
  2. jcsd
  3. Oct 15, 2008 #2


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    I don't you need to use complex numbers but they are a very convenient way of taking into account the fact that quantum mechanical "objects" have both amplitude and phase.
    E.g. interference effects are much easier to handle mathematically if complex numbers are used.
    It think it is a bit like asking why we use complex number in EM.
  4. Oct 15, 2008 #3


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    It's essentially to deal with the wave aspect. Waves can interfere constructively and destructively. Comple quantities are an easy way to incorporate this in the calculations. This is also the reason people introduce complex numbers when doing classical electromagnetism. The phase is incorporated as the angle in the complex plane.

    EDIT: f95toli posted while I was typing and beat me to it! ;-)
  5. Oct 15, 2008 #4
    Hmm, but then you have to understand what "phase" means for particles.
    Why do particles with phase 0° and particles with phase 180° cancel? In EM it is "simply" the direction of the field and that's why it cancels.

    So the question about complex numbers is equivalent to asking why particles have phase.
    And an explanation is still missing :confused:
  6. Oct 15, 2008 #5


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    This is just a different way of asking why things behave as waves. Nobody knows that.
  7. Oct 15, 2008 #6
    Indeed. The question is why free particles show a periodic phase and why they can interfer linearly just by their presence and no interaction.
    I believe some out there has an idea about it. For example someone could have put my suggestion from above into more concrete mathematical form. Anybody here? :smile:
  8. Oct 15, 2008 #7
    For a free non-relativistic particle the Schrödinger's equation is usually written in form

    i\partial_t\Psi = -\frac{1}{2m}\nabla^2\Psi

    where [tex]\Psi[/tex] is a complex function. If one wants to avoid the use of complex numbers, use two real functions [tex]\Psi_1[/tex], [tex]\Psi_2[/tex], and use the Schrödinger's equation

    \partial_t\Psi_1 \\ \partial_t\Psi_2 \\
    = \frac{1}{2m}\left(\begin{array}{cc}
    0 & -\nabla^2 \\
    \nabla^2 & 0 \\
    \Psi_1 \\ \Psi_2 \\

    It's the same thing.
  9. Oct 15, 2008 #8


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    Consider a free particle with a definite momentum. (OK, that's an idealization because of the uncertainty principle.) You need a complex probability amplitude [itex]\Psi = A e^{i(kx - \omega t)}[/itex], and the usual definition of probability density [itex]P = \Psi^* \Psi[/itex] in order to get a uniform probability density, i.e. equal probability of finding the particle everywhere.

    Or as jostpuur notes, you can define a two-component real field that has the same effect. Either way, a simple real field doesn't work.

    Going beyond that, you're faced with questions similar to "why does F = ma?" in classical mechanics.
  10. Oct 16, 2008 #9
    From my understanding, which admittedly isn't a whole lot, it's just for your convenience.

    The same technique is used in other areas, such as electrical engineering. Complex numbers are used to represent measures of a circuit, such as voltage or current. The *physical* voltage is, of course, a real, not a complex number, and in fact, is equal to the real part of the complex representation. So if you have a voltage notated as re^(ix), the measured result is going to be r*cos(x). The reason for using the complex number, though, is that the min/max of the voltage is simply equal to r and differentiating voltage is a simple matter of multiplication. As long as you restrict yourself to linear transformations of voltage, you can always manipulate it as a complex number.
  11. Oct 16, 2008 #10


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    No, this is NOT correct in the case of QM. While you do not need to used complex numbers you DO need to keep track of both amplitude and phase somehow. However, as jostpuur showed above you are free to do that with a pair of real numbers if you prefer(although it makes things more complicated).
    The important point here is that phase is something "physical" in quantum mechanics, it is a property that is many sitations is as "real" as e.g. charge.
    (charge and phase play the role of generalized momenta and position in electronic systems, in some situations phase is a good quantum numbers while charge is not).

    Moreover, ALL objects have a phase; not only single particles. One obvious example is SQUIDs which are macroscopic objects but the phase of the superconducting wavefunction is quantized.
  12. Oct 16, 2008 #11
    I'm not saying it's the same, but it is similar. In either case, you don't need to represent the mathematics using complex numbers. For example, Feynman's formulation of QM using negative probabilities instead of complex amplitudes.
  13. Oct 16, 2008 #12
    As Schroedinger noted ((Nature, v.169, p.538 (1952)), you don't necessarily need complex wavefunctions (or two real functions) even for a charged particle, as you can make, say, a complex scalar wavefunction real (at least locally) by an appropriate gauge transform.
  14. Oct 9, 2010 #13
    You should check page 15 of the Quantum Mechanics book from Cohen-Tannoudji. It says:

    "Furthermore, we should see that the fact that [TEX] \psi(\vec{r},t) [/TEX], is complex is essential in quantum mechanics, while the complex notation [TEX] E(\vec{r},t) [/TEX] is used in optics purely for convenience (only its real part has a physical meaning). The precise definition of the (complex) quantum state of radiation can only be given in the framework of quantum electrodynamics, a theory which is simultaneously quantum mechanical and relativistic. We shall not consider these problems here (we shall touch on them in complement [TEX] K_V [/TEX]."

    I can't help you anymore since I'm just in chapter 4 :(, but I'm sure there's a good explanation there :)
  15. Oct 9, 2010 #14
    I think f95toli explained it: you can always adjust a single wave function so that it is real.

    Interference was mentioned as a reason for using complex functions, but you also need them if you have more than one orthonormal spanning basis function as part of your wave function.

    jostpuur's notation was cool, but you might recognize it as complex numbers in a different forum: the real and imaginary parts expressed as vectors.

    Can someone please point the way to the documentation that shows how do do equations in forums! I would really appreciate the opportunity to learn that. I'm guessing from andresordonez's post that it is some form of LaTeX coding.
  16. Oct 9, 2010 #15
  17. Oct 10, 2010 #16
    Another way to say the same thing: imaginary numbers enter into quantum mechanics though the uncertainty principle: [X,P] = ih.

    In the wider context of particle physics we can say something like this:
    The geometrical settings for relativistic particle physics is 4 dimensional spacetime with Lorentz metric. Classically this leads to physical objects which are representations of the rotation group SO(3,1) ie. 4-vectors , tensors etc.
    In the context of relativistic quantum physics, we are lead to a larger symmetry Spin(3,1) which is the universal cover of SO(3,1). This leads to additional representations called spinors e.g. electrons. There is in fact a very special role complex numbers play in 4-d Lorentz geometry which can be summarised by the isomorphism Spin(3,1) = SL(2,C).
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