Physics as Oscillating Systems

To get an overview to prepare entering into realQM it may be illuminating to recall that the mathematical model of a physical system which has permanence over time in the form of oscillations between two different states, typically takes the form:

• $i\dot\varphi + A\varphi = 0$      (1)

where $\varphi =\varphi_1 +i\varphi_2$ is a complex-valued function with $\varphi_1(x,t)$ and $\varphi_2(x,t)$ representing the different states being real-valued functions of a 3d space coordinate $x$ and a time coordinate $t$, with $i$ the imaginary unit and the dot signifying differentiation with respect to time, and $A\varphi \equiv A\varphi_1+iA\varphi_2$ with $A$ a symmetric operator acting on real-valued functions. We understand that (1) is a condensed complex-valued form of the following system on real-valued form:

• $\dot\varphi_1 =-A\varphi_2$, $\dot\varphi_2 = A\varphi_1$   (2)

where the oscillatory nature is exhibited: $\varphi_1$ changes with input from $A\varphi_2$ and $\varphi_2$ changes with input from $A\varphi_1$. We understand that there is nothing magical in the complex-valued form (1) as it is only shorthand for the real-valued system (2) simply expressing oscillation between two states.

With different operators $A$ the equation (1) (or system (2)) covers:

1. Physics (Harmonic oscillator): $A = identity$, $\varphi_2$ position, $\varphi_1$ velocity,
2. Electro-magnetics (Maxwell’s Equations): $A =\nabla\times$, $\varphi_2=E$ electrical field, $\varphi_1=B$ magnetic field,
3. Mechanics (Vibrating Elastic Plate): $A =\Delta$, $\varphi_2$ displacement velocity, $\varphi_1$ moment,
4. Chemistry (Foxes and Rabbits): $A=identity$, $\varphi_1$ and $\varphi_2$ species densities,
5. Quantum Mechanics (Schrödinger’s Equation): $A=H$ with $H$ Hamiltonian, $\varphi$ wave function.

We see that (1) encompasses the basic models of physics with quantum mechanics on the same footing as classical mechanics and electro-magnetics. We see oscillations between kinetic and potential/elastic energy, between species densities (foxes and rabbits), and in particular between electric and magnetic fields giving perspective on the oscillation between the real and imaginary parts of the wave function of quantum mechanics.

Hopefully, this can help to reduce the mystery of the complex form of Schrödinger's equation and give incentive to check out realQM.

The basic feature of (2) obtained by multiplication of the first equation by $\varphi_1$ (or $-\dot\varphi_2$) and the second by $\varphi_2$ (or $\dot\varphi_1$) and addition followed by integration in space, is conservation in time of

• $\int (\varphi_1^2+\varphi_2^2)dx$,
• $\int (\varphi_1A\varphi_1+\varphi_2A\varphi_2)dx$,

capturing oscillation between two states; when $\varphi_1$ is big $\varphi_2$ is small, and vice versa.

Inviscid fluid mechanics can also be formulated as (a generalisation of ) (2):

• $\dot v + \nabla p= 0$, $\dot p + \nabla\cdot v=0$ with $v$ velocity and $p$ pressure.