Does QM really need to be complex?
Why can't we just use real wavefunctions?
🧵 Some math, non-relativistic QM:
First of all, not all wavefunctions are complex. The Schrödinger equation tells us that time-independent (stationary) solutions can be real. These are energy eigenstates, e.g. a bound state in an hydrogen atom. But even if you start with a real wavefunction, time-evolution introduces an imaginary part.
Time-evolution for a real-valued wavefunction correspond to a global phase. But global phases don't have observable consequences.
Another way of seeing this is that for real wavefunctions, as well as for wavefunctions with global phases the probability current is zero. They might as well be real and one can consider Psi as an equivalence class of all wavefunctions differing only by a global phase (a ray).
But not all states can be equivalent to real states. Consider non-stationary states. Since energy eigenstates form a complete basis, we can write any non-stationary state as a linear superposition, e.g.
In this case the probability current doesn't vanish unless E_1 =E_2. This means that the probability density isn't constant either and the state 'moves', whereas stationary wavefunctions are just rotating in the complex plane.
You can tell the difference between a stationary state and a non-stationary state just by looking at their wavefucntions evolve with time. Give it a try: Which of these isn't stationary.
Clearly eigenstates of operators that don't commute with the Hamiltonian aren't in an energy eigenstate and are therefore not equivalent to real wavefunctions.
There is a second class of wavefunctions that aren't equivalent to real wavefunctions and that are those that differ by a local phase.
I this case the norm (probability density) is unaffected. probability of finding the particle at position x doesn't change. But the probability current is non-zero:
This is only possible if the current is divergence-less dj=0 (like water flow in a pipe: the local density is constant yet there is flow). An example for such a local phase are plane waves or angular momentum eigenstates.
Fields with arbitrary local phases are particularly interesting. In contrast to plane waves, a wavefunction with an arbitrary local phase doesn't solve the Schrödinger equation- the derivative acts on the phase (unless theta(x)~ x as for the plane wave). It does solve a modified Schrödinger equation with an extra term in the derivative with an additional field A(x). So a field with an arbitrary local phase is a charged field and A(x) is the gauge field.
This modified current is gauge invariant, but now can be non-zero even if the phase gradient vanishes (the Aharanov Bohm effect. So you can do quantum mechanics with real wavefunctions, but only if you're interested in stationary energy eigenstates. It can't describe any non-energy eigenstates or charged fields.
