This recent discussion on LinkedIn asks for applications of complex numbers which can be explained to High-School students.
The case of AC electric circuits is one familiar application which was mentioned by several respondents and one of them pointed to http://www.picomonster.com/ where an attempt is made to motivate their use for describing relationships between other cyclical phenomena.
Another commenter mentioned the use of complex numbers in Quantum Mechanics but I find this hard to explain in elementary terms.
With regard to the Alternating Current (and mechanical vibration) applications, the picomonster site seems to be well reviewed by those who have already actually used complex methods in exercises without really understanding why they worked, but gets less favourable comments from those who have not already been motivated by seeing the ideas used for something.
In order to provide real motivation, we need to see how the use of complex amplitudes helps to simplify the solution of a real problem such as finding the current in a complicated electric circuit corresponding to a given applied AC voltage.
It might help to emphasize the fact that these applications are dealing with a fixed frequency and that the “imaginary amplitudes” are representing a real quantity (oar position or electric current for example) whose time variation is periodic and just 90degrees out of phase from the case of real amplitude. Then the complex amplitude A+iB corresponds to a current of Acos(wt)+Bsin(wt) (in Amperes for example), and the corresponding voltage across a resistance R ohms is R(Acos(wt)+Bsin(wt))=RAcos(wt)+RBsin(wt) which corresponds to the complex amplitude RA+iRB=R(A+iB).
That’s not very exciting of course, because the result is just as easily obtained without the complex amplitude, but we can now quickly go on to more interesting cases of circuit elements such as inductances and capacitors which affect the relative phase of current and voltage.
For an inductance of L henrys the voltage (which is proportional to rate of change of current) is 90degrees out of phase from the current and proportional to both the magnitude of the current and the frequency of its oscillation. The Acos(wt) of current then gives wLA(-sin(wt)) of voltage and the Bsin(wt) gives wLBcos(wt), so the resulting voltage is wLBcos(wt)-wLAsin(wt) which corresponds to a complex amplitude of wLB-iwLA = -iwL(A+iB).
When two components are connected in series, the current is the same for both (because it is a flow of electrons which don’t “pile up”) but to get the voltage from one end to the other the individual voltages are added. So the total voltage is given by RAcos(wt)+RBsin(wt)+wLBcos(wt)-wLAsin(wt)
= (RA+wLB)cos(wt)+(RB-wLA)sin(wt) which corresponds to the complex amplitude (RA+wLB)+i(RB-wLA)=(R-iwL)(A+iB)
Since the process of going from current to voltage corresponds to multiplying the complex amplitude by the complex “impedance” R-iwL, the reverse process of figuring out the current from a given applied voltage V=Ucos(wt)+Wsin(wt)then just corresponds to dividing the voltage’s complex amplitude U+iW by the “impedance” to get I=Acos(wt)+Bsin(wt) with A+iB=(U+iW)/(R-iwL).
To solve the same problem without using complex numbers would involve solving for A and B in the equation RAcos(wt)+RBsin(wt)+wLBcos(wt)-wLAsin(wt)=Ucos(wt)+Wsin(wt) This would be possible (by matching the sin and cos multipliers on each side to get 2 equations in 2 unknowns for A and B), but it is both more involved to work out and not so easy to generalize to other kinds of circuit (whereas the rules for working with combined series and parallel circuits with complex impedances turn out to be exactly the same algebraic form as those for dealing with just pure resistors)
I think all of the above could be understood by high school students who have started working with trig functions to describe periodic phenomena – without any need for calculus or even for using Euler’s formula to write Acoswt+Bsinwt as Re((A+iB)e^-iwt). The presentation I have given might not be gentle enough for all students to absorb directly but it certainly should be good enough for anyone who has any business teaching the subject.
With regard to Quantum Mechanics however, I know of no way to motivate the use of complex amplitudes for normal, or even quite high performing, high school students. And to motivate a topic by saying “you will find this useful later” does not fit with my ideal of avoiding appeals to authority rather than the students’ own experience and understanding.
This doesn’t mean that there aren’t good explanations for why complex amplitudes are used. The occurrence of i in the Schroedinger equation can be motivated to a student who has studied differential equations on the basis of wanting a first order equation related to the wave equation (even though this may not be strictly necessary), and various other special properties which make the complex case more appealing seem to depend on even higher levels of abstraction:
- An appeal to the value of matching elements of the Jordan algebra of OBSERVABLES with those of the Lie algebra of SYMMETRY GENERATORS (“Only in this case can we turn any
self-adjoint complex matrix into a skew-adjoint one,
and vice versa, by multiplying by i.”) is
attributed by Tony Smith to John Baez, but I can’t find the quote and in fact John actually
says“Why does nature prefer the Jordan algebras hn(C) over all the rest? Or does it? Could the other Jordan algebras – even the exceptional one – have some role to play in quantum physics? Despite much research, these questions remain unanswered to this day.” -
The property of multiplying degrees of freedom (ie “The dimensions of hn(C) behave quite nicely:
dim(hnm(C)) = dim(hn(C)) dim(hm(C))
But, for the real numbers we usually have
dim(hnm(R)) > dim(hn(R)) dim(hm(R))
and for the quaternions we usually have
dim(hnm(H)) < dim(hn(H)) dim(hm(H))")is mentioned by John Baez with reference to this paper by Lucien Hardy
See also this lecture by Scott Aaronson
and
this paper on the non=existence of a quantum De Finetti representation in the case of real Hilbert spaces