The experience behind a second-order time-dependent Schrödinger equation of the form
- \ddot\psi +H^2\psi =0, (1)
with H a Hamiltonian and the dot signifying differentiation with respect to time, is as follows: Observed frequencies \nu of atomic light emission show to be proportional to differences E_j-E_k > 0 of energy levels E_1\lt E_2\lt E_3... as eigenvalues with corresponding eigen-states \psi_j=\psi_j(x) depending on a space coordinate x, satisfying the time-independent equation:
We can here assume that E_1=0 by replacing H by H-E_1 and what is observed is thus a linear relation between observed frequencies \nu_j and eigenvalues E_j of a mathematical atomic model:
- H\psi_j =E_j\psi_j. (2)
- E_j = h\nu_j (3)
where h is a constant named Planck's constant (which we here normalize to 1).
We now observe that (1) is a natural extension of the real-valued eigenvalue problem (2) to dynamic time-dependency in accordance with (3), as a real-valued wave equation which can be given a physical interpretation in classical continuum mechanical terms with solutions as real-valued linear combinations of \exp(i\nu_jt)\psi_j(x).
We compare with Schrödinger's equation in standard first-order complex form
- i\dot\psi + H\psi =0 (4)
which also respects (3), but does not lend itself to physical interpretation.
We thus have two possible Schrödinger equations with solutions \psi named as wave functions, the second order real-valued form (1) and the standard first order complex-valued form (4), both which fit with the experience of (3).
Here (1) has a physical interpretation in classical continuum mechanical terms, while (4) lacks such a physical interpretation and has only been given a statistical unphysical interpretation.
It seems pretty clear that (1) is to prefer before (4) as a the basic mathematical model of quantum mechanics, and this is the possibility I want to explore. In particular the whole mess of interpreting the wave function in statistical terms can probably be avoided this way, and much now wasted effort saved.
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