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I am taking undergraduate quantum mechanics currently, and the concept of an imaginary number had always troubled me. I always feel that complex numbers are more of a mathematical convenience, but apparently this is not true, it has occurred in way too many of my classes, Circuits, Control Theory and now Quantum Mechanics, and it seems that I always understand the math, but fail to grasp the concept in terms of its physical mapping. Hence my question, what does imaginary number maps to physically?

Any help would be much appreciated

Qmechanic
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user2654176
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    More on complex numbers and physics: https://physics.stackexchange.com/q/11396/2451 , https://physics.stackexchange.com/q/32422/2451 , https://physics.stackexchange.com/questions/tagged/complex-numbers?sort=votes – Qmechanic Sep 07 '13 at 22:18
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    It seems to me that negative numbers are also a 'mathematical convenience'. – Greg Sep 08 '13 at 03:13
  • Eigenvalues distinguish a quantum state. As we can't measure two quantities simultaneously, eigenvalues values are real numbers. And to produce real eigenvalue we need hermitian matrix with complex element. – Self-Made Man Sep 08 '13 at 17:19
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    I concur with Greg. In history there was a period where algebra was used without explicit negative numbers. Every problem can be stated in such a way that explicit negative numbers are evaded. Once mathematicians started using negative numbers explicitly many problem could be stated much more efficiently, enabling more efficient mathematical thinking. It's analogous for complex numbers. It's possible to create a set of rules for manipulating pairs of numbers that never mentions the square root of a negative number explicitly, yet mathematically equivalent to manipulating complex numbers. – Cleonis Sep 08 '13 at 22:24

7 Answers7

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As far as describing nature is concerned mathematics is another language that can describe it.

In ancient times there was no mathematics other than arithmetic, and that was the way nature was described: cycles of the moon, trajectories of planets etc.

Then came geometry and it used mathematical tools to describe nature further, based on arithmetic but using geometrical axioms and theorems.

Algebra came late, from the arabs, during the middle ages where the dogmatism of the church had stopped progress in science. With enlightment, mathematics took off and became a beautiful tool describing/modeling nature by the time of Newton, the co-inventor of calculus. Negative numbers and complex numbers became necessary by using algebra and are now incorporated in all mathematics useful for describing nature, as you have found out.

We can plot negative numbers in an x plot, if that is what you mean by "maps to physically". Complex numbers, one axis real one imaginary are another plot where the solutions for physical systems can be displayed economically. They simplify notations and calculations. A relevance to physics problems comes in that they describe/incorporate trigonometric functions which continually appear in solutions of physics boundary problems due to the harmonic nature of many set ups. They are economical, in the same way that algebra became economical, not needing the convoluted sentences of the classical (BC) times mathematics.

Complex numbers for physics are, as all mathematical tools used, a convenient representation fitting the data.

anna v
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  • Thanks for the explanation, although I am still very confused, for example I always thought that everything in this world can be map with real numbers, energy, distance, time, temperature....etc, because they give meaning such as order and reference to the physical concept, but what does imaginary number give, what is a quantity that has an imaginary value and what does it even mean? How does an imaginary object interact with real object? Those are the questions in my mind that has been unanswered – user2654176 Sep 08 '13 at 04:28
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    the use of the word "imaginary" is confusing. It is just another type of coordinate. It does map one to one to the real numbers with an i infront to indicate it is in a different "space". There are even more complicated extensions of this, called "quaternion" http://en.wikipedia.org/wiki/Quaternion . Complex numbers are useful for calculation in usual physics problems so we know about them:). If you think of mathematics as a tool it is not necessary to angst about a physical identity other than the convenience of calculations. – anna v Sep 08 '13 at 05:32
  • Sorry for using the word "imaginary", it is confusing when asking questions, I do realize imaginary numbers are just another mathematical space, just like real number is an extension of rational numbers, complex numbers are the extension of real numbers, but what is intriguing is the fact that we can directly measure integer, rational, and to some degree of precision irrational and real, but we can not directly measure complex, and this really troubles me as it seems that you cannot map to anything physically using complex numbers? – user2654176 Sep 08 '13 at 05:39
  • In truth, every type of coordinate is mapped one to one with real numbers, with just a designation to separate it from 0 to infinity. For example negative numbers map on real numbers with just a - in front meaning subtraction. the i in fron of the imaginary axis of complex numbers is infront of a one to one map to real numbers. If we take complicated vector spaces the same is true. we just have to keep track of which direction we are talking about by a symbol that differentiates this axis from that axis. There is nothing more esoteric about it. – anna v Sep 08 '13 at 05:55
  • I apologize for my failure to understand the concept, but yes I do know that the cardinality of complex and reals are the same, but it just seems that complex numbers lacks a fundamental mapping that gives meaning when we describe the law of universe using it, perhaps as Manishearth described below, it actually is merely a mathematical convenience, to understand the meaning of it is by nature meaningless. However, I have seen other mathematical convenience and in my best knowledge, all of them are measurable, complex number is the only system I know that is directly unmeasurable – user2654176 Sep 08 '13 at 06:07
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    @user2654176 Do you have some examples of mathematical conveniences that are directly measurable? In other words, what do you mean by "directly measurable"? If you mean "measurable without calculation", then almost nothing is directly measurable as it involves calculation in the measuring process. If you mean "all information can be extracted via measurements", well, the complex numbers used in physics can be measures in their real and imaginary parts, and the information can be joined together to get the complex number. – Manishearth Sep 08 '13 at 12:16
  • to expand a bit on anna's beautiful answer. A "trajectory" is represented as a differentiable function of a parameter (e.g $f(t)$). This represnetation among others, is meant to state that a trajectory includes information about the "velocity" (i.e the derivative) or other things. This is a modelling example which under some cases might hold or not. The same way a complex number encodes information about both magnitude and orientation and allows these to be combined (by using compelx arithmetic).Physicaly this allows (f example) rotations to be represented – Nikos M. Jul 05 '14 at 09:05
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Phase and amplitude! While complex numbers are used in the analysis of Classical Harmonic Oscillator or the LCR circuit as a part of Mathematical technique, wherein once the solution is obtained the real part is extracted. However in Quantum mechanics they play a central role, and QM cannot be formulated without them.

We necessarily need both the concept of Phase and Amplitude in the description of quantum mechanics. When 2 event are separated by multiple paths, then the information about later event occurring given first happens is encoded in the Probability Amplitude. Amplitude of a Single path is essentially the quantity $e^{iS}$, where S is the action. To calculate the Probability amplitude sum over the amplitude for each and every possible path in which the final event can be realized if the Initial event had happened. The Probability that the final event happens is Modulus square of the probability amplitude.

The use of Complex numbers(And Probability Amplitudes) in Quantum Mechanics follows very naturally as a generalization of the concept of Probability, And Feynman very nicely Illustrates this in his paper, "Space-Time Approach to Non-Relativistic Quantum Mechanics" or "The Concept of Probability in Quantum Mechanics"

While one can always rewrite all the formulas in Quantum mechanics as 2 real numbers, However then the beauty, and its simplicity is lost.

Glorfindel
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Prathyush
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    +1 This is the correct answer. QM mechanics can not be formulated without complex numbers or something isomorphic to that field, which again, will be complex numbers regardless of how we call them. – Bubble Sep 08 '13 at 15:39
  • This answer seems the best so far. The theory of epistemology put forth by Objectivism requires that concepts are grounded to reality via perceptual evidence. Im not satisfied i can explain explain complex numbers to teenager yet but this helps me get started. – Aaronium112 Oct 01 '16 at 14:48
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It's just a mathematical tool1, which is especially useful to portray some kinds of vectors and sine waves. It doesn't have a real "meaning" here, and we can do fine whthout it (calculations just become more tedious)

For example, in circuit analysis, we convert the coordinate pair $(V,\phi)$ (which isn't linear; adding two voltages need not add their phases) into the coordinate pair $(V\cos\phi,V\sin\phi)$, which is linear. Now, instead of writing it as a coordinate pair, we write it as a complex number (which is essentially just a pair of numbers) and then notice that we can extend this to the concept of impedance and even be able to divide by these numbers and get the correct answer. Writing voltage as $(V\cos\phi)\hat i +(V\sin\phi)\hat j$ would still give a linear system, but we can't divide by vectors so it's limited in application.

In quantum mechanics, the use is similar -- we are expressing a linear quantity with a phase. Again, this is a nonlinear coordinate pair, but rewriting it as a complex number gives us a linear coordinate pair that plays nice with multiplication and division. When we bring linear algebra into it, one sees that the representation is even more apt.

Again, it's just a tool or a representation. It has no real meaning.

1. As @MBN mentioned below, real numbers are a mathematical tool, too. We can't "directly" measure real numbers, either.

Manishearth
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  • So does the complex part of the impedance exist? – user2654176 Sep 08 '13 at 05:41
  • @user2654176 Impedance is a human constructed quantity. This is like asking "does resistance exist?" -- resistance is just a quantity that we defined. Yes, impedance is measurable. All you have to do is measure the current waveform as you pass a voltage $(V,0)$ through it, and record $(I,\phi)$. Now, $\frac{I}{V}e^{i\phi}$ is your complex impedance. If you're asking "can we directly get the complex part without calculations" -- I find the question meaningless as we can't directly measure much anyway. – Manishearth Sep 08 '13 at 05:46
  • "It's just a mathematical tool" can be said about any part of maths, but it is missleading. – MBN Sep 08 '13 at 12:03
  • @MBN What do you mean? Basically, there's no real physical significance to it in the sense that you can't "measure" a complex number. Also, we're talking about physics, not maths. Mathematical tools in mathematics have no physical meaning. Representations in physics can have a physical meaning. – Manishearth Sep 08 '13 at 12:06
  • Once upon a time complex numbers were treated as a pair of numbers. You can treat it as a pair or a number in the number plane or a vector in a 2D plane but all have the same property. – Self-Made Man Sep 08 '13 at 17:29
  • @Manishearth: You cannot measure real numbers either, but you single out complex numbers as "just a mathematical tool" as if they are different from other maths objects. – MBN Sep 09 '13 at 12:13
  • @MBN Ah, I see. I wasn't singling them out, no, I'll edit that in. – Manishearth Sep 09 '13 at 12:15
  • @Manishearth: Ok, I like it better now. I was complaining because there is a psychological difficulty for some people when they see complex numbers for the first time. – MBN Sep 09 '13 at 13:06
  • @MBN Yeah. I tried to make it clearer in the comments; there's no such thing as being "directly" measurable (except for a couple discrete quantities) – Manishearth Sep 09 '13 at 13:08
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If you really wanted to, you could formulate the laws of physics without using imaginary numbers - see, for example, Can one do the maths of physics without using $\sqrt{-1}$?

Let's say you need to do a fourier decomposition of a function $f$ in order to find how some responds to being driven at $f$.

If you think back, fourier decomposition is normally introduced with only real numbers, and a function $f$ was decomposed into odd and even parts, which can be represented as infinite series of cosines and sines respectively, and you could certainly do that. However, if we view $f$ as the real part of some complex-valued function, the equations become much simpler.

If this makes you wonder whether the complex-valued function is "more real" than the real part, and what is the physical significance, you could see $f$ as $\frac{1}{2}(z + \bar{z})$ where $z$ is the complex function; then, $z$ and $\bar{z}$ both have no physical significance, but they formally satisfy the equations governing the system, hence their sum must also.

Community
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xuanji
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@Prathyush's answer is completely correct and provides nice reference, but I just wanted to add that behind his discussion about phases and path integrals is the concept of unitary and conservation of probability, which is a physical concept, and the OP asked about what imaginary numbers "map to physically". Thinking in the Heisenberg picture one can write every operator as a real diagonal matrix and a set of mappings (rotations) between these operators, but in order for everything to conserve probability it is absolutely essential that these rotations contain imaginary numbers in general as any unitary operator may be written as $\exp{i \hat{H}}$, where $\hat{H}$ is a Hermitian operator (with real eigenvalues). It is similar for another very special rotation in Hilbert space, which is time evolution, without which there would be no physics as such. This has a deep connection to the relation between Lie algebras and Lie groups.

Without complex numbers we would have no unitarity and no conservation of probability or would be led to a trivial quantum theory which is not dynamical!

Bubble
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Both sides of a complex number are each real numbers. Either the real part or the imaginary part can be used for computing the value of a measurable quantity. And the results are always the same. Then we are left with only 'i', to determine it's Mathematical and then physical meaning. The allegory of 'i' harbors a real Mathematical meaning which when unveiled, discloses the 'reality' of a 'complex' number. In other words, 'complex numbers' are 'real numbers' in disguise.
One can say the above in another way by saying that (-1)^1/2 is completely a real number.

Ugo
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A rotation by a right angle. So complex numbers come up whenever you have periodicity/oscillations/phases etc.

For a simple example, consider the physical intuition for Euler's formula -- if you push something with force $F=kx$, you get exponential motion, but $F=-kx$ is periodic motion. The latter, where you have periodic, trigonometric functions, is where there are complex numbers/imaginary exponents.

Similarly when describing light, the polarisation of a light wave behaves as a phase. The same idea carries over to quantum mechanics.

Abhimanyu Pallavi Sudhir
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