fredag 23 januari 2015

Physical Quantum Mechanics (Based on Second Order Schrödinger Equation) 1

                                              String vibration as deterministic physics.

The founding pillars of modern physics are (i) quantum mechanics of small scale atomistic physics and (ii) relativity theory of large scale physics. Unfortunately (i) and (ii) have shown to be incompatible, which gives modern physics a shaky foundation loaded with mysteries. In particular, quantum mechanics is viewed to be fundamentally different from classical continuum mechanics, and so beyond human comprehension.

Quantum mechanics describes the atomistic world in terms of wave functions $\psi$ satisfying Schrödinger's equation, which for the basic case of the Hydrogen atom takes the (normalized) form
  • $i\dot\psi \pm H\psi =0$,        (1) 
where $\psi =\psi (x,t)$ is a complex-valued function of a 3d space coordinate $x$ and time coordinate $t$, $\dot\psi =\frac{\partial\psi}{\partial t}$, and $H=-\frac{1}{2}\Delta +V$ is a Hamiltonian with $\Delta$ the Laplacian differential operator and $V=V(x)=-\frac{1}{\vert x\vert}$ is the kernel potential. The mystery of the wave function $\psi (x,t)$ and the equation (1), is that $\psi$ has no direct physical meaning, only an indirect unphysical meaning with $\vert\psi (x,t)\vert^2$ viewed as a probability distribution of particle position.  

Schrödinger obtained (1) in 1926 starting from a second-order wave equation    
  • $\ddot\phi +  H^2\phi =0$,      (2)
in terms of a real-valued wave function $\phi (x,t)$, by a formal decomposition  
  • $\frac{\partial^2}{\partial t^2} +  H^2 = -(i\frac{\partial}{\partial t}+H)(i\frac{\partial}{\partial t}-H)$,
thus viewing formally the complex-valued first order equation (1) as the "square root" of the real-valued second order equation (2). 

This decomposition is analogous to the decomposition of the second order wave equation 
  • $\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial x^2}=0$,   (3) 
in a 1d space coordinate $x$, into the first order equations
  • $\frac{\partial\psi}{\partial t}\pm\frac{\partial\psi}{\partial x}=0$.           (4)
We want to compare the physics expressed by (1) and (2), and then start comparing the physics of (3) and (4). 

We know that the function $\phi (x,t)$ in (3) can be interpreted as the transversal displacement of a vibrating elastic string at $(x,t)$ with (3) expressing a balance of inertial and elastic forces according to Newton's law.  

We know that (4) expresses constancy along characteristics $x\pm t=constant$ describing convection or translation of a quantity with speed 1 in the positive or negative $x$-direction. 

We know that the second order wave equation (3) admits waves traveling in both positive and negative $x$-direction, while each of the two equations (4) admits waves traveling in only one direction. We conclude that the physics described by (3) and (4) is different: the elastic string of (3) is not present in (4) and the physics of the translation in (4) is unknown or unspecified. 

We now understand that also (1) and (2) may describe different physics, or no physics. 

Our conclusion is that the second order real-valued form (2), which is close to (3), may describe physics similar to that of a vibrating string as a form of vibrating electron with again (2) expressing force balance, while the physics of the first order complex-valued conventional form (1) has remained a mystery since 1926.

The meaning of the wave function $\phi$ of (2) is "displacement in space" with $\phi^2+(H^{-1}\dot\phi )^2$ representing charge carried as a concrete physical phenomenon. We compare with the accepted meaning of the wave function $\psi$ in (1) with $\vert\psi\vert^2$ a probability distribution of particle position, which is not carried as a physical phenomenon, only as a phantasm in the mind of a physicist.

The difference between (1) and (2) thus appears to be most essential, if external physical reality is maintained to be what makes physics different from mathematics and philosophy, which do not require an external world to exist. 

The unfortunate result is that insisting to take (1) as the basic equation of quantum mechanics while lacking direct physical meaning, has led generations of physicists following Max Born to attribute a non-physical meaning to the wave function $\psi (x,t)$  as a probability distribution $\vert\psi (x,t)\vert^2$ of particle position. 

The result is a collapse of determinism and casuality and thus scientific rationality, which could have been avoided if instead, along with Schrödinger's original thoughts, (2) had been chosen as the basic equation of quantum mechanics. This is the line of thought I would like to explore further with the hope of finding a deterministic rational physical quantum mechanics as a form of classical continuum mechanics, which can replace probabilistic irrational unphysical quantum mechanics as atomistic physics. It is then encouraging to note that the present highest form of modern physics of string theory, connects to (2) rather than (1).

                                        Atomic vibration as deterministic physics.

  • There is nothing more deterministic and with less free will than the ground state of a Hydrogen atom.  (Nietzsche)
  • The assumption of an absolute determinism is the essential foundation of every scientific enquiry. (Planck)
PS Compare with Physicists debate whether quantum math is as real as atoms discussing the difference between ontic (what is) and epistemic (what we know) aspects of quantum mechanics. 
The most clever among us like Motl Lubos insists that the wave function is neither ontic nor epistemic:
  • However, Nature around us doesn't work in either way. Just like the electron in Nature is neither a classical particle nor a classical wave, the wave function is neither "ontic" nor "epistemic". The world is simply described neither by classical physics evolving a point in the phase space; nor by classical statistical physics.
This is so clever: By removing what physics is not, what remains must be what physics is! Right?

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