Complex numbers in physics

Question

Can someone please explain the origin of complex numbers in physical values. For instance, denoting a plane wave with Euler's identity and also the complex relative permittivity?

Thank you.

Answer 1

Exponentials differentiate and integrate better than trig functions, and in general are “easier to combine” than working with trig functions, v.g. complex impedance in a circuit. Taking the real part at the end “brings you back” to the physical fields, voltages, currents etc.

Answer 2

By "physical values" I assume you mean observables, i.e. quantities that one can measure in real life. Observables are always real number ($\mathbb{R}$) -- at least so far. If you manage to measure a $3\mathrm{i}$ long slab of wood, let me know.

Complex numbers enter physical problems in two ways:

1:

They are integral part of a theory (e.g. quantum mechanics)

For instance, the Schroedinger equation looks like: $$ \mathrm{i}\hbar\frac{\mathrm{d}|\psi\rangle}{\mathrm{d}t} = H|\psi\rangle.$$

The imaginary unit appears in the equation of motion itself, and is hence an integral part of the dynamics and of the physics of the system. As it well known, however, you can only access $\langle \psi|\psi \rangle = |\psi(x)|^2$ which is real. This results in the wavefunction $\psi(x)$ always being defined up to an arbitrary (absolute) phase factor that you cannot experimentally measure (see here).

It is important to note, however, that phase differences can still be measured. In fact, $|\psi(x)(e^{i\phi_1}+e^{i\phi_2})|^2 \propto \sin(\Delta\phi)$.

2:

They are an artificial extension that allows the maths to become easier.

Mainly, complex numbers have a nice phasor representation, and complex exponentials are easy to differentiate and to integrate.

The latter reason is the one that applies to electromagnetism: while in quantum mechanics $e^{ikr}$ is the actual solution for a travelling wave, in electromagnetism it would $\cos(kr)$. Because there are no imaginary units in Maxwell's equations.

The derivative of $\cos$ is $-\sin$, whose derivative is $-\cos$ etc. which tired physicists to the point that they decided to artificially expand our physical (real) space to complex numbers: $$ \cos(kr) \rightarrow \mathrm{Re}[e^{ikr}], $$

which makes the maths easier.

What does this mean?

• Say we have a complex valued wavevector $k = k_0 + i\kappa$:

$$\mathrm{Re}[e^{ikr}] = \mathrm{Re}[e^{ik_0r}e^{-\kappa r}] = e^{-\kappa r}\cos(k_0 r), $$ i.e. that just gave us a wave with a decaying envelope.

• Say we have a complex permittivity $\epsilon = \epsilon_r + i\epsilon_i = |\epsilon|(\cos\delta + i\sin\delta) = |\epsilon|e^{i\delta} $:

$$D = \epsilon E = (\epsilon_r + i\epsilon_i) E = |\epsilon|e^{i\delta}E, $$

where $D$ is the displacement field and $E$ a wave of the form $E_0 e^{ikz}$.

Then, $$ D = E_0 |\epsilon|e^{i(kr+\delta)}, $$ $$ \mathrm{Re}(D) = E_0 |\epsilon|\cos(kr+\delta),$$ i.e. a wave with a modified amplitude and now a phase delay $\delta$.

The imaginary part, as per the phasor formalism, usually gives you information about the phase delay introduced by components or media. This is on top of any modification to the amplitude.

You can get the same results without resorting to complex numbers, but the maths you would harder -- for instance, $e^a\cdot e^b = e^{a+b}$ whereas $\cos(a)\cos(b)$ is less straightforward...