söndag 25 januari 2015

Physical Quantum Mechanics 2: Schrödinger's Original Wave Equation

Schrödinger first posed a second order Schrödinger equation as the basic wave equation of quantum, in the fourth of his ground-breaking articles in 1926.

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:
  • $H\psi_j =E_j\psi_j$.       (2)
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:
  • $E_j = h\nu_j$                    (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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