P.A.M. Dirac attempted to answer this question in his well-know textbook "Principles of Quantum mechanics", so I will not repeat his discussion of the problem. Note that the Schroedinger equation itself can be recast as a hydrodynamic equation for density and velocity (two equations for mass and momentum conservation) where the pressure is given by the "quantum pressure". So if we do not insists on the use of complex numbers and the related concept of wavefunction we can possibly find a more intuitive description of quantum mechanics. However the Schredinger equation is linear something which is not apparent in its equivalent hydrodynamic formulation. So a better question would be why quantum mechanics is linear, or why the superposition principle holds? In the end the universal answer to these questions is that these mathematical constructs are simply a good description of the experimental data and my policy is to avoid philosophical questions such as these since they usually turn out to be sterile from the point of view of physics.

Quantum theory needs existence of an x such that x2 = -1. The reason for this is that orthogonal function spaces, of dimension greater than 2, cannot exist otherwise. In fact the only place where  i  is needed is in the wave packet prior to measurement.  Even the Canonical Commutation Relation doesn't need it.  And nor do the eigenvalue equations.  In those, any general scalar will do.  But in the wave packet, you need an  i .

To include the propagation of a wave packet, along with that wave packet spreading, ,Schrodinger turned a basic diffusion equation into one that propagates and diffuses (or spreads) by introducing  "i=sqrt[-1}"  The spread plus propagation incorporates the major features of QM; namely, that of motion along a trajectory but with increasing uncertainty.

I'd add that there is nothing imaginary about 'imaginary' numbers, despite their name.

The name was first used and popularized by René Descartes in the 17th century (in the 1600s) , who did not understand them and peremptorily derided their use. The proper name is 'complex numbers'.

I'm not entirely sure what you mean by 'funny' ? It's not the only set of numbers that were put down by uncomprehending and overly reactionary know-it-alls : negative numbers and the cipher zero were long scoffed at.... We can only be grateful that there have always been proper scientists ready to challenge the taunts of their more ossified colleagues.

I am sorry, if it felt like a  taunt. 

I do understand and appreciate  the beauty of mathematical structures. The question  is   rather , 'why nature prefers   ONE specific structure over  the other.'    Complex number multiplication  is beautiful, but for sure it   is more than 'six boxes, each having  5  coins'.    The  question is rather, why simple  real  number  multiplication   is insufficient !

I really liked the answer by  Frank (Tabakin) :   to invoke  motion along a trajectory  with increasing uncertainty.    This reduces the question to  uncertainty principle,  which has has long/repetitive answers.

The reason why a given structure is preferred might have to do with levels of complexity, the more complex a physical phenomenon, the more complex the numbers attached to its description become ....

we need imaginary numbers to make operators hermitian so as to make

real observations

Dear P.K.

prefers not nature, but we. It is the question of our comfort.

Regards,

Eugene.

Dear Eugene,

If it is a question of our comfort (which I think it is) then  there  could be alternatives ? The question  on uniqueness remains ....

I disagree.   My position is math is invented rather than discovered, therefore there is nothing fundamental about complex numbers.  They are completely artificial. See my article here:

http://www.eleceng.adelaide.edu.au/personal/dabbott/publications/PIE_abbott2013.pdf

Note that you can do all of science in a real space without complex numbers. Our article demonstrates this here:

http://www.eleceng.adelaide.edu.au/personal/dabbott/publications/PIE_chappell2014.pdf

Then we show that we can do quantum mechanics without complex numbers here:

http://www.eleceng.adelaide.edu.au/personal/dabbott/publications/QIP_chappell2013.pdf

In my opinion, complex numbers appear in quantum mechanics out of a mathematical necessity. See my paper at:

http://hal.archives-ouvertes.fr/docs/01/05/49/53/PDF/quantum_explanation.pdf

The question on uniqueness always remains, P.K. See Derek Abbott.

I saw Derek  and also [doi:10.1063/1.3680558]. As a opinion, and  some case studies  (on alternatives) are fine.  But   it is not clear  QM, QFT,  and all can  in fact be re-said  with out     invoking  (a,b)*(c,d) = (ac-bd, ad+bd).  And of course,   this   structure  of  imaginary numbers makes the description much more simpler, why ?    I don't have  problem with quantum formulation -beautiful, as it is.   Just   asking,  if  there is a deeper understanding,   why  the whole  thing looks so beautiful  through THIS glass ?

Dear P.K.

QM and QFT are not complete. Time is absent (or introduced formally) in these theories.

Regards,

Eugene.

Actually, this is something I have solid insight into. The imaginary number arises due to a fundamental error in physics. 

In the cgs system of units, the dimension of charge is always distributed (squared) relative to the dimension of mass. When the MKS system was developed, and incorrect assumption was made that charge could be a linear dimension relative to linear mass. This resulted in the absurd units involving square roots.

However, only certain units dimensions were "corrected." The units of resistance, current, and potential were among those that were, and since the simplified version of Ohm's law involves only those units, the formula still holds. It is equations that involve the five uncorrected units (conductance, permeability, permittivity, capacitance, and inductance) and any other electric-based unit that causes the imaginary number to appear.

For example, the impedance equation attempts to add the wire resistance to the LC resistance. It turns out the LC resistance should actually be magnetic flux, and when resistance and magnetic flux are added, well you can't add unlike units. The imaginary number is the result of trying to do so. 

The imaginary number represents the correction for the missing dimension of charge, which is required to balance the units. If the units were corrected, the imaginary number would disappear. However, that requires the simultaneous correction of all the errors in physics that arose from using the wrong understanding of charge.

P.A.M. Dirac attempted to answer this question in his well-know textbook "Principles of Quantum mechanics", so I will not repeat his discussion of the problem. Note that the Schroedinger equation itself can be recast as a hydrodynamic equation for density and velocity (two equations for mass and momentum conservation) where the pressure is given by the "quantum pressure". So if we do not insists on the use of complex numbers and the related concept of wavefunction we can possibly find a more intuitive description of quantum mechanics. However the Schredinger equation is linear something which is not apparent in its equivalent hydrodynamic formulation. So a better question would be why quantum mechanics is linear, or why the superposition principle holds? In the end the universal answer to these questions is that these mathematical constructs are simply a good description of the experimental data and my policy is to avoid philosophical questions such as these since they usually turn out to be sterile from the point of view of physics.

Dear colleagues!

All our knowledge (probabilistic) about future is extrapolation. If reality is continuous, then first approximation is always linear and principle of superposition takes place.  This states QM. It is not about existence, but about possibility to exist. If measurement is done, then all this world of possibilities changes. Keep in mind, not real world, but world of possibilities! Reality may be in symmetries (forms of operators). They are external factors for QM. We can only guess them. C'est la vie.

Regards,

Eugene.

Dear Eugen,

What do you mean by "reality  is  continuous"  ? You mean  "time" ? 

And  why "first approximation is always linear"  ?   Rather,

if self-organization is an answer to   'how nature works,'  [Per Bak]  then 

criticality ( and thus  non-analyticity) is   dominant - then linearity could not

be assumed. Linearity is an approximation, however  good it may be.

PK.

Yes, I mean changes in time. You are right, concerning self-organisation. In linear approximation nothing happens. It is only coordinates, as cartesian. Linearity (QM) is only the language to describe reality.

Regards,

Eugene.

How non-linear problems are solved, P.K.? See Poincare. First approximation is the solution of linearized problem.

Regards,

Eugene.

True, there too linearity is an approximation (good though).   Even, such  approximation would  not work at  a singular (critical)  points :  f(x)= x^(2/5) cannot be  linearized.  [for self-organized criticality   see 'how nature works' http://jasss.soc.surrey.ac.uk/4/4/reviews/bak.html]

PK.

Exact elementary functions are idealizations. Every real (continuous) function can be extended in Tailor series.

Eugene.

Sorry, expanded.

That is not true Eugen. That,  magnetization (M)  of  Ising  model in two dimension    goes as (T_c-T) ^(1/8)  is not an idealization !    In fact M can not be expanded  in Tailor series    about T=T_c.

Hi Pradeep,

Interesting question this one. I think it is not just quantum mechanics. The same goes with Classical electrodynamics as well. As long as we are getting the real values straight, we can use anything including complex numbers for our convenience.  However, if you are thinking about physical significance of complex/imaginary numbers then it is a different matter altogether.

As you have proposed in the explanation itself, the funny composition rule is in terms of fully real quantities. Imaginary numbers are for convenience. Everything that is done using complex quantities in physics can very well be expressed in terms of purely real quantities. Or are tehre exceptions?

Regards,

Rajat

And in two value code (01), Rajat.

Ising model is itself idealization, P.K.

That is not right  at all Eugen. Please see  this 1/8  observed in 1976 (PRL, 37, pg 29, table I, 4th line from below)  or even earlier,  and  many times   in later years.    This discussion is going too far from  the initial Question and  I will not attend  to  answers  in these directions.

I don't reject critical points, Pardeep. I only say, that they are beyond QM and linearity.

Regards,

Eugene.

Dear All,

Is there anything in entire physics where use of imaginary or complex quantities is absolutely essential and we cannot get the same results using real quantities alone? This is  a pertinent question. If the answer is no, then Pradeep's question is answered automatically: complex numbers only add to the convenience of description. However, if the answer is no, then we have to consider seriously the original question of Pradeep.. One may think of complex dielectric functions in electrodynamics or Schrodinger prescription the QM for momentum and hamiltonian and so on.

The real question is: is it absolutely essential to use complex numbers, without which we cannot get the required results?

Regards,

Rajat

Rajat,

in physics all mearsurable quantities are real. That is the answer.

Regards,

Eugene.

Rajat, the question is   not whether  complex numbers can be   written without using "i= sqrt(-1)". Of course they can be, if one uses a   different kind of multiplication  and addition of number pairs. Even Schrodinger's  can be expressed as two non-linear equations (with nice   interpretations, as Sebastiano pointed out);  just that the  linear  form   is  simple (even though   it needs   so called 'complex'    numbers) !

With 'natural numbers' the description is  complex and with 'complex numbers' the description is simple.    It seems, to explain/understand, one need to   built complexity in language  to keep the description simple or  vice versa.    Just a choice,  or   there is something deeper there ? 

Quantum theory needs existence of an x such that x2 = -1. The reason for this is that orthogonal vector spaces, of dimension greater than 2, cannot exist otherwise.  Such orthogonal spaces imply existence of an x such that x2 = -1.  Work through it and see.  Or refer to:

W E Baylis, J Hushilt, and Jiansu Wei, Why i?, American Journal of Physics 60 (1992), no. 9, 788–797.

Dear Pradeep,

your question can be reformulated as follows: Why does symmetry exist, that makes usage of complex numbers in QM convenient? You can ask also, why does any symmetry exist, particularly, periodicity, as a kind of symmetry and Time, as it's (periodicity) manifestation? Moreover, You can ask also, why does anything exist? The answer is: Because Nothing can not exist (by definition). Rajat knows this very well.

Question Why? has it's limltations, Pradeep.

Regards,

Eugene.

Orthogonal vector space is already discussed linearity, Steve, but, of course, You are right.

Regards,

Eugene.

Thank you Steve for the reference. It is like putting the cart before the horse. They give an interpretation of the imaginary unit in terms of vector spaces, which Pradeep has mentioned and its uses are well-known. To say that otherwise orthogonal vector spaces of more than two dimensions cannot exist is wrong. The various quantities can be expressed or rephrased using complex numbers does not mean that i is absolutely necessary. The i is purely imaginary in the truest sense of the term and hence exists only in the mental/psychological realm. It cannot have a direct and absolute physical significance, which significance cannot be obtained entirely in terms of real quantities.

Regards,

Rajat

Thank you Rajat.

Adapting from the  Baylis, Hushilt, and Jiansu Wei paper I mentioned above, I show in Section 11 of my paper below, that orthogonal spaces, dimension greater than 2, imply existence of an x such that x2 = -1.  I'd be pleased to have pointed out any flaw in my workings.

Article How Arithmetic Generates the Logic of Quantum Experiments

Dear Steve,

I do not think you need to go to orthogonal spaces for demanding the existence of i. Very simply, we can say that just because we admit the existence of an integer called -1 ( or any other negative integer for that matter) and because there is a process called taking a sqrt of a number, it follows that there must exist a quantity i= sqrt(-1).

Regards,

Rajat

NOTE added 4 hrs later: This was intended as a contribution to a different, but related, thread (link below), 

I find it instructive to see how many different directions people are taking to approach this question of complex numbers and their use in QM.

Another direction, not explicitly mentioned so far (I think), relates to the description of waves such that they incorporate position-momentum uncertainly (Dicke & Wittke, Intro to QM, Addison Wesley, 1960 and ff., p. 24 in the 1969 Third Printing – and showing my age):

“…as will be discussed in detail later, there are reasons for believing that a particle with its momentum exactly known is in a state such that its position is completely uncertain. In such a case, the probability distribution measured by the wave intensity | ψ |^2 should be independent of position. This suggests that the wave function to be associated with a particle of definite momentum should be given by Eqn 2-2: ψ = A exp[ i (kx - wt)], rather than by Eqn 2-5: ψ = A sin (kx – wt + α)].”

As already mentioned, all observations record real numbers. So Is it, in the end, simply a matter of taste as to whether one uses one complex variable to encode a pair of variables in one equation, or uses two sets of equations?

Personally, I like to view the complex i (and its various hypercomplex manifestations) as something allowing an easy encoding of direction, phase, rotation etc., as distinct from magnitude.

https://www.researchgate.net/post/How_did_Schrodinger_concluded_that_adding_a_complex_number_to_wave_function_is_important_Whats_the_physics_behind_that?_tpcectx=qa_overview_following&_trid=5415e7b9d2fd64a7218b4567_

Dear P.K. Mohanty,

  To my knowledge, it can be traced back to the axiom of quantum mechanics, [x,p]=ih/2π.  Of course you can redefine the imaginary number i in another way, but that could raise another question, why do you want to define it that way. Or if you can find another equation to replace the quantized condition, that would be good too.

Rajat,

all numbers exist only in mental/psychlogical realm.

Concerning i, there are situations in physics (exept QM), where it is inevitable. For example, the absolute! value of magnetic dipole moment (observerble!) may have two signs, plus and minus. How can we describe this situation without i or something analogous?

The truth of the matter is that the i in the Schrodinger-Pauli equation can have quite a different interpretation than the i in the Dirac equation. In my paper "Vector Analysis of Spinors", I show that the i is conveniently associated with the pseudoscalar element, or directed volume element, of the (Clifford) geometric algebra G(3) of space. In the Dirac equation, the i becomes something quite different, as I show in my newest paper, "Spacetime Algebra of Dirac Spinors", which I am just now completing. My paper "Vector Analysis of Spinors" can be found on my Research Gate page, or on my website.

http://www.garretstar.com/

I am sorry that I found this question so later after its formulation. The answer is simple: QM needs complex number from the same reason as electromagnetic fields need them. Because certain quantities have a PHASE, or saying otherwise, are not characterized by a single real number, but by a pair of real numbers.

Let's go directly to the QM. The wave-function has an intensity, and a phase - i.e. two numbers are needed. Recall interference experiments in which we need to consider not only the intensity of different contributing waves, but also their phases. To manipulate this pair of numbers in a compact form, i.e. for writing a single expression instead of two, we use the complex amplitudes. For a certain wave, the amplitude is equal to the square root of the wave intensity, times the phase-factor. So, we get a complex amplitude.

Of course, there may be additional advantages of the complex numbers in QM, (e.g. the complex plane is convenient in calculating the poles of the scattering matrix). However, the first typical thing that QM teaches us to do for a quantum system, is to solve the Schrodinger equation and find the wave-function.

Notwithstanding Garret Sobczyk’s analyses/distinctions regarding i  in QM (2 posts above), I think the following is also an interesting addition to the discussion of the need for complex numbers  in QM, filched from the first link below, in which Radha Pyari Sandhir attributes it to a lecture by Scott Aaronson at MIT (2nd link below)

"As for why we're dealing with complex numbers in QM in the first place, check out the Real vs. Complex Numbers section in a lecture by Scott Aaronson from MIT.

An extract:

Why did God go with the complex numbers and not the real numbers?

Years ago, at Berkeley, I was hanging out with some math grad students -- I fell in with the wrong crowd -- and I asked them that exact question. The mathematicians just snickered. "Give us a break -- the complex numbers are algebraically closed!" To them it wasn't a mystery at all.

But to me it is sort of strange. I mean, complex numbers were seen for centuries as fictitious entities that human beings made up, in order that every quadratic equation should have a root. (That's why we talk about their "imaginary" parts.) So why should Nature, at its most fundamental level, run on something that we invented for our convenience?

Answer: Well, if you want every unitary operation to have a square root, then you have to go to the complex numbers... Scott: Dammit, you're getting ahead of me!

Alright, yeah: suppose we require that, for every linear transformation U that we can apply to a state, there must be another transformation V such that V2=U. This is basically a continuity assumption: we're saying that, if it makes sense to apply an operation for one second, then it ought to make sense to apply that same operation for only half a second.

Can we get that with only real amplitudes? Well, consider the following linear transformation:[1 0; 0 -1]

This transformation is just a mirror reversal of the plane. That is, it takes a two-dimensional Flatland creature and flips it over like a pancake, sending its heart to the other side of its two-dimensional body. But how do you apply half of a mirror reversal without leaving the plane? You can't! If you want to flip a pancake by a continuous motion, then you need to go into ... dum dum dum ... THE THIRD DIMENSION.

More generally, if you want to flip over an N-dimensional object by a continuous motion, then you need to go into the (N+1)st dimension.

But what if you want every linear transformation to have a square root in the same number of dimensions? Well, in that case, you have to allow complex numbers. So that's one reason God might have made the choice She did."

Incidentally, I came across this as a result of looking up stuff on phase in QM, after checking out Sofia Wechsler’s latest paper, posted today (3rd link).

http://functionspace.com/topic/169/What-is-meant-by-phase-in-quantum-mechanics

http://www.scottaaronson.com/democritus/lec9.html

Nobody invented complex numbers. They simply exist.

Well, Chang, that is a  challenge!

I think that at least part of the plain-language answer you seek, from a physical perspective, might be that the Complex Number system (Imaginaries together with the Reals) provides a neat (and at times virtually essential) way to encode / model, and manipulate, directional (angular) as well as extended (length) information.

A simple explanation from the maths perspective necessarily involves some maths, but at its simplest, as you probably know already, the Imaginaries allow one to solve equations requiring the square root of a negative number.

If you look at my longer post further up this question-thread, you'll find more on the mathematical perspective.

(Remembering the Gauss-inspired phrase recommended by mathematical educator, Arnold E. Ross: "think deeply of simple things": I'd also be interested in better or additional such plain language explanations from those more expert than me.)

If you ever wanted to pursue it in depth, mathematically, you'd need to study Clifford Algebras (based on the work of a C19th German schoolteacher called Hermann Grassmann) that generalise many different "hypercomplex" number systems, such as the quaternions, as well as Reals and the usual Imaginaries that together form the traditional Complex Numbers, relating to your question. 

Incidentally, as to answers such as Eugene's, that "they simply exist", that assertion can be seen as one (indeed, major) school of thought in the philosophy of mathematics, as discussed in the Wikipedia entry linked to this post.

I hope this helps - Paul

https://en.wikipedia.org/wiki/Philosophy_of_mathematics

Interestingly, in special relativity we also need complex numbers. Time is imaginary.

If the Dirac matrices are both orthogonal and involutory (the square of a matrix being equal to the unit matrix), then the imaginary unit is implied in their algebra. Any orthogonal space of 3 or more dimensions implies existence of this scalar.  This is because orthogonality implies existence of scalar products.  In a space where the scalar products between 3 distinct pairs are zero, the imaginary unit is implied.  Therefore, Implicit in any physics using products between unit vectors or basis vectors,  i  is implied.  That said, this number may not enter equations of dynamics.  It does show up due to orthogonality in the Fourier transform, in the Pauli algebra. If relativity is expressible using matrices that have rational or real enties, the square root of minus is in the background.  Somewhere the maths will assert the existence of a number such that x^2+1=0.  This is because certain arrangements of rationals in rational matrices do imply  i.  This means the rational or real numbers in the mathematics rest in a field with closure that extends over the complex numbers, with rational and real numbers systems being inadequate.  The reason we don't generally see this is because we have invented unit vectors i j k and we play with these.  The orthogonality of these unit vectors is in our minds.  The do not truly communicate orthogonality with the scalars in the  mathematics.

http://labs.adsabs.harvard.edu/adsabsadsabs/abs/1992AmJPh..60..788B/

http://geocalc.clas.asu.edu/pdf-preAdobe8/VeSp&ComUpdated.pdf

By imaginary time I mean simply Minkowski metric, Christian, but there can be huge philosophy in this, begining with Hamilton. It was one of his intentions in introducing quaternions.

Much food for thought...

I have just now posted a short note showing why complex numbers appear in physics and where in physics they come from.

Article A short note on why the imaginary unit is inherent in physics

[EDIT; I only noticed Eugene's last reply after posting - but it may still interest others]

In reply to Eugene and Christian (with a nod to Steve Faulkner) re "imaginary time" (which also caused some confusion among the general public via Stephen Hawking's "A Brief History of Time"), isn't it simply that the term "ict" appears in some formulations of space-time vectors, leading some people to use the potentially misleading phrase "imaginary time"?

Those of Hawking's readers without knowledge of complex numbers will presumably have interpreted "imaginary time" as suggesting something like "time doesn't really exist, but is an illusion of the (mental) imagination". [I'm deliberately labouring the point.]

Appreciating Steve Faulkner's succinct summary above, I'll assume (until, if need be, corrected) the following is consistent with it:

In SR the complex i can appear, simply as one needs to take the square root of a difference (that between the squared 3-space distance components and the squared duration ('1-space' temporal-distance component), in order to represent the observed constancy of light-speed within a consistent space-time - the well-known Minkowski space).

So the fact that the time coordinate is, in a sense, 'inverted' in direction relative to the spatial coordinates ((+++-) or (---+) signature) gives rise to a negation in the distance metric, which in turn involves, visibly (in at least some formulations) a 4-vector capable of squaring with a negative component - hence the i in "ict".

I think this satisfies Steve's summary paper - Minkowski space is a vector space of more than two dimensions, though I'm curious as to whether his description admits more than one "type" of complex i, as asserted in the works of David Hestenes.

For example, the i described in this post would presumably have entered Dirac's equation by way of his building a relativistic wave equation, yet the i in Schroedinger's equation should have nothing in common with Special Relativity.

Hi Paul. My immediiate gut reaction to Hestenes's other i is that it is the same i  we all know and love.  I am currently working on exactly this area.  My interest is logical independence of i in spacetime.

I have only browsed it, but it seems there is expert coverage on spacetime as a geometric theory.  I attach their paper:

Christian,

As to your first part: Of course I agree completely that we need structures that square to -1.  Sorry if I introduced any confusion - I was just trying to step around your coverage of the alternative representations.

Many thanks too for the Dirac ref.. If Dirac was saying his equation is fundamentally non-complex, then I'm quite surprised. Do the complexities cancel out somehow (in spite of Steve's demonstration above and 4-component spinor solutions)?

I'll have to consider it more carefully when I'm back from holiday and freer of family...

Season's Greetings - Paul

Hi Steve,

I'd been wondering whether it was a matter of the structure of the space involved making a distinction between Hestenes' different "types" of i, but I've not got very far with that, and you may well be right that they're ultimately all the same.

Many thanks for Lasenby et al.. I've been watching their Cambridge MRAO group, but not seen that particular paper before.

I trust you'll post your own results on RG.

Season's greetings - Paul

Christian,

Thanks for the clarification.

Christian,

"i" is shorter, than matricies in that cases, when it is enough.

The word "imaginary" has only historical meaning. "Imaginary" numbers are no less real, than "real".

Merry Christmas!

P.S. Measuring complex quantity means receiving two somehow connected real numbers. There are many cases, when one is not enough. Their connection (as well as for the other mathematical structures) is governed by logic.

Of course. But it must be explored more carefully. I had no time to do this till now.

Christian,

Having recently read through your paper (Minkowski Spacetime and QED…), posted above on Dec 23rd, when you write:

“The unit imaginary is nothing mysterious that distinguishes quantum from classical mechanics. It is just a compact form of writing Hamilton’s equations of motion (EQOM) of a harmonic oscillator.”

…can I therefore take it, for the purposes of intuition (and with a nod also to Eugene’s PS to his post on Dec 24th, above), that the unit imaginary in QM, via Hamilton’s equations, is in effect simply a way to encode together pairs of conjugate (real) variables, such as coordinate and momentum?

I’ve struggled long and hard with the idea of the unit imaginary in QM, and to find such a clear resolution, if correct, is an epiphany.

Thank you - Paul

Yes, Paul, unit imaginary in QM expresses the physical fact of conjugacy of some of the variables.

Regards,

Eugene.

to me, the complex  #s are used, so as to allow the ability to capture phase, so evident in easiest diffraction and inteference effects.. sorry abt typo.

should have added that the del squared op serves  the function of

be having diffusively, so informatuin (ie prob operator) diffuses with widely spreading

info.

In most general terms, Hamiltonian mechanics is formulated on a manifold of even dimension, endowed with a symplectic form, that is, a skew symmetric and non degenerate bilinear form. In matrix representation, this form can be written where is an inner product for tangent vectors, and J is a matrix satisfying J^2 = -1. The same matrix can be used to introduce a complex structure in the tangent linear space. Less abstractly, that means we can use the complex variables q+ip and write a single complex Hamilton equation instead of two different ones:

@_t(p+iq) = -i(@_q+i@_p)H

The wave equation is a Hamitonian system, it is a classical field equation. It describes a quantum particle only with Born's statistical interpretation and the projection postulate.  The probability is proportional to q^2+p^2, which in classical mechanics is merely the energy for a harmonic oscillator.

Can we not "decide this" via the simplest answer, that provides the best insight to this question, a la Ocham's razor? 

To me, it is that there are two "orthogonal" quantities, that need to divide their

energies, so as to couple together with orthogonal relationships we see:

e.g. x, px, etc. The imaginary/.complex numbers are just the simplest way,

to express their relative phase relationship.

I would like to make a comment on the “Magical Complex Numbers” as Roger Penrose entitled chapter 4 of his book "The Road to Reality" (Jonathan Cape, 2004). I read this book in its French translation (“A la découverte des lois de l’univers”, Ed. Odile Jacob, 2007). It makes 1063 pages in small typography!

On view of this book, one can understand that Penrose uses the word “magic”. He is first of all a mathematician and it seems that he “fell in love” with the number “i” from his youth. In his book, he shows a.o. how using complex numbers brings marvels in a lot of fields: Fourier analysis, holomorphicity, quaternions algebra, QM and several others subjects already quoted in the above posts to the present question. At the end, his “twistor theory” (which could be used in a theory of quantum gravity) rests on complex numbers.

Penrose appears as the follower of a long cohort of admirers of “i” since centuries.

Just to mention the equation of Euler (1707-1783), called by generations of mathematicians the “most beautiful equation”:

eiπ = -1

Feynman, who was a physicist, called it “our jewel” or “the most remarkable formula in mathematics”.

As “-1 = i2”, one even finds:

(lni)/i = π/2

One has “i”, which is imaginary, being duly related to 2 universal constants: “e” and “π"

In chapter 5 of his book, Penrose plays showing that:

ii = e-π/2 = 0.2078795....

I'd like to further comment that complex numbers can be found in a lot of everyday situations.

Let's just consider systems (see, e.g. J.W. Forrester "Principles of Systems", Wright-Allen Press, 1968). There are lots of systems around us. Humans can be seen as systems made of sub-systems (immune system, organs, cells, ..).

In systems, embedding a first order negative feedback loop (with time constant "-T1") in a first order positive feedback loop (with time constant "T2") will result in a second order negative feedback loop with time constant "T3" such that: T3=(-T1*T2)1/2=i*(T1*T2)1/2, the system showing a wavy behavior. This happens in economy, biology, ....

For instance, looking at the dynamic working of the immune system (biology). One has an intruder (antigen, virus, …) penetrating the human body. The immune system reacts by producing antibodies, T lymphocytes, etc. in great number (first order positive feedback loop) up to annihilation of the intruders. After a while, as the concentration of intruders strongly diminishes, the need of antibodies, lymphocytes, etc. progressively disappears and they are less and less produced (first order negative feedback loop). Then if they have not completely disappeared, the intruders take vigor again and multiply as the reaction has weakened. But soon, there is a new reaction of the immune system followed by a new decrease of the intruders and so on. The system behaves as a wave (with maxima, minima, amplitudes, frequencies in the reaction). Same in economy when hiring additional staff for seasonal work ….

Here also “i” appears as a result of the mathematical formalism of the dynamical behavior. An “exp(k.i.t)" appears in the equations.

So, it might be that complex numbers are used in QM to “manipulate a pair of numbers in a compact form”, i.e. to write a single equation instead of two.

But, complex numbers might also reflect something more basic in nature because of their general appearance in a lot of fields and situations.

Then, the word “magic” would not have to be taken as meaning a tricky illusion, but rather as reflecting the fundamental beauty of nature.

There are only three division algebras: the reals, the complexes, and the quaternions.  The quaternionic field is non commutative, then the richest algebra with the most good properties is the set of complex numbers. That's why they are used in so many mathematical disciplines, even for proving theorems with real numbers. There is nothing magical nor mystical in that, and even less in the appropriateness for describing Nature. The complex algebra is an efficient tool in mathematical physics, and that's all folk.

One essential reason  is that the general solution to the Schrodinger equation must  be complex  due to the fundamental theorem of algebra.  see equation 6.72 for a wave hitting a barrier in this link. http://physics.mq.edu.au/~jcresser/Phys201/LectureNotes/SchrodingerEqn.pdf  .

Regarding complex numbers, the Schrödinger equation and every other quantum wave equation have nothing special with respect to classical field equations. In no way, complex numbers can explain the peculiarities (or problems) of quantum mechanics. That's a wide spread wrong idea that should be fought.