Why Complex Numbers for Real Physics?

The basic model of physics is the harmonic oscillator represented by a body  of mass 1 oscillation back and forth in time along an x-axis subject to a spring force

  • dv/dt = -x(t),   (1a)
  • dx/dt  = v(t),    (1b)

where x(t) is the position of the body at time t with corresponding spring force – x(t) and velocity dx/dt = v(t),  see Harmonic Oscillator. The solution is x(t) = sin(t) with v(t) = cos(t) if started at t = 0 with x(0) = 0 and v(0) = 1.

The solution can also be depicted in a plane (x, y) – coordinate system as points (sin(t), cos(t)) moving around a unit circle with speed 1, see Polar Coordinates.

This connects to expressing (1) in terms of complex numbers in the form

  • i*du/dt + u = 0,   (2)

where u(t) = v(t) + i*x(t) = cos(t) + i*sin(t) with i the imaginary unit, see Complex Numbers.

The mathematical model for the Harmonic Oscillator can thus take equivalent forms, as (1) with the body oscillating back and forth on an axis subject to a spring force, or as (2) with the position-velocity pair (x(t), v(t)) moving around a unit circle.

Is there then any difference between (1) and (2) apparently expressing the same thing? Yes. The model (1) has a clear physical meaning as a body-spring system, while the physics of (2) is unclear in the sense that the physics of moving around  unit circle is, if not unthinkable, definitely more complicated in terms of forces.

This discussion connects to quantum mechanics with the Schrödinger equation usually expressed in complex form as the following generalisation of (2):

  • i*dΨ/dt + HΨ = 0,        (3)

where H is a real-valued Hamiltonian and Ψ = ψ + i*φ is a complex-valued was function with real-valued ψ and φ (real and imaginary part of Ψ, or in system form as an analog of (1):

  • dψ/dt = -Hφ,    (4a)
  • dφ/dt = Hψ.     (4b)

The appearance of the imaginary unit i (3) is viewed to be part of the mystery of quantum mechanics as the mystery of the physical meaning of (3). Writing the Schrödinger equations in system form (4) removes the mystery of the appearance of using complex numbers to express real physics, and the challenge then is to understand the physics of (4). To this end we rewrite differentiate (4a) with respect to t to get

  • d2ψ/dt2 = – Hdφ/dt = – H^2ψ

which has the form of a wave equation

  • d2ψ/dt2 + H^2ψ = 0     (5)

with positive definite H^2 just as (1) can be written as a wave equation d2x/dt2 + x = 0 with a plus sign for x.

We thus see that Schrödinger’s equation has the form of a wave equation in a real-valued function ψ with the Hamiltonian H expressing real physic connecting to different forms of physical energy (potential and “kinetic”), just as the for the body-spring system with energy stored in the spring.

Writing wave equations in complex form is convenient from a symbolic analysis point of view, since multiplication of exponentials with complex exponent is direct, but expressing the wave equation in real form connects better to real physics.

Real physics does not ask for complex numbers. The Schrödinger equation in complex form (3) is picked out of the air, while the system form (4) has a direct connection to real physics suggesting that quantum mechanics is not so much different from classical mechanics. Puh!