måndag 26 januari 2015

Physical Quantum Mechanics 4: Interpretation of the Wave Function

We are exploring a second order alternative formulation of Schrödinger's wave equation as the basic model of quantum mechanics, which for a Hydrogen atom takes the form
  • $h^2\ddot\psi +H^2\psi =0$,        (1) 
for all $(x,t)$, where $\psi =\psi (x,t)$ is a real-valued function of a 3d space coordinate $x$ and time coordinate $t$, $\dot\psi =\frac{\partial\psi}{\partial t}$, and $H=-\frac{h^2}{2m}\Delta +V$ is the standard Hamiltonian with $\Delta$ the Laplacian differential operator, $h$ Planck's constant, $m$ the mass of the electron, and $V=V(x)=-\frac{e^2}{4\pi\epsilon_0\vert x\vert}$ is the kernel potential with $\epsilon_0$ the dielectric constant of vacuum and $e$ the charge of the electron. We view (1) as a generalized harmonic oscillator.

We introduce the eigenfunctions $\psi_j$ and corresponding real eigenvalues $E_j$ of the Hamiltonian $H$ satisfying
  • $H\psi_j = E_j\psi$ for $j=1,2,3...$ with $E_1\leq E_2\leq E_3,...$
We then redefine $H$ into $H-E_1$ and $E_j$ into $E_j-E_1$, so that $H\psi_1 =0$ and $E_1=0$ and we consider $\psi_1=\psi_1(x)$ to be the ground state. 

The solution $\psi (x,t)$ of (1) can then be expressed as a real-valued linear combination of eigen-modes
  • $\exp(i\frac{E_j}{h}t)\psi_j(x)$ for $j=1,2,3,...$
What may here be the physical meaning of the wave function $\psi (x,t)$ as a scalar real-valued function?

We seek guidance comparing with the equation for a vibrating thin 2d elastic plate with plane stress-free ground configuration, which may take the form
  • $\ddot\phi +\Delta^2\phi =0$,      (2)
where $\phi (x,t)$ is the transversal displacement of the plate at a position $x$ in the 2d plane of the pale at time $t$. 

We are thus led to interpret the wave function $\psi (x,t)$ of (1) as a "transversal displacement" of a 3d "elastic body" at a position $x$ in 3d and time $t$, with the "transversal displacement" acting so to speak into a "virtual 4th dimension" as a measure of change away from a "stress-free ground configuration".  We may then view (1) to express force balance according to Newton's 2nd law with $h^2\ddot\psi$ rate of change of momentum $h^2\dot\psi$ and $H^2\psi$ a corresponding force.

We seek further guidance in the following conserved quantities of (1) as different forms of energy: 
  • $OE= \frac{1}{2}\int (H\psi )^2 + h^2\dot\psi^2)dx$,                                               (3)
  • $AE= \frac{1}{2}\int (\psi H\psi+h^2\dot\psi H^{-1}\dot\psi )dx$                           (4)
  • $Q=\frac{1}{2}\int (\psi^2+h^2(H^{-1}\dot\psi )^2dx$                                            (5)
as results of multiplication (modulo the ground state) of (1) by  $\dot\psi$,  $H^{-1}\dot\psi$ and $H^{-2}\dot\psi$,  respectively, and integrating in space.

It is here natural to view $OE$ as total oscillator energy, $AE$ as total atomic energy, and it may also be natural to view $Q$ as total charge, viewing thus charge as a form of energy.  We are thus led to define 
  • $\frac{1}{2}(H\psi )^2 + h^2\dot\psi^2) = $ local oscillator energy                                               
  • $\frac{1}{2}(\psi H\psi +h^2\dot\psi H^{-1}\dot\psi ) =$ local atomic energy                    
  • $\frac{1}{2}(\psi^2+h^2(H^{-1}\dot\psi )^2) =$  local charge.                                           
We thus view the wave function $\psi (x,t)$ to represent (scalar) displacement away from a static ground state $\psi_1$ satisfying $H\psi_1=0$, and we view $\psi^2$ to describe charge distribution.

Note that
  • $\int \psi H\psi dx =\frac{h^2}{2m}\int\nabla\psi\vert^2dx +\int V(x)\psi^2(x)dx$,   (6)
with the appearance of $\psi^2$ in the kernel potential directly connection to an interpretation as charge. In particular, the ground state $\psi_1(x)$ emerges as the minimizer of (6) with minimal
potential energy under Laplacian space regularization.

This is radically different from the standard interpretation with $\vert\psi\vert^2$ as a probability distribution of particle position with $\psi$ complex-valued. 

Let us thus compare the two interpretations of (i) $\psi^2$ as charge distribution and (ii) $\vert\psi\vert^2$ as probability distribution of particle position:

1. A function value $\vert\psi\vert^2(x,t)$ is a non-negative number and as such cannot represent a 3d position coordinate, while $\psi^2 (x,t)$ may naturally directly represent a scalar quantity like charge (or mass).

2. The scalar function $\psi^2(x,t)$ generates a 3d charge distribution through the dependence on $x$ and thus has a 3d quality, which gets expressed in radiation from oscillating charges.

3. The only way to connect the scalar $\vert\psi\vert^2$ to 3d space is to interpret
 it as a probability distribution in space, but a probability is not a direct physical quantity.

The conclusion is that connecting the scalar $\vert\psi\vert^2$ to 3d particle position, as in standard quantum mechanics, is both irrational and unneccessary.

We will next extend to radiation summarizing previous posts on the radiating atom.

Then we will extend to atoms/ions with more than one electron. With the wave function $\psi (x,t)$
connecting to charge distribution $\psi^2(x,t)$, we will not be misl(led) to introduce a multi-dimensional wave function $\psi (x1,x2,...,xN,t)$ depending on $N$ 3d space coordinates $x1,x2,...xN$ as in the standard formulation (which makes the Schrödinger equation uncomputable for several electrons) because such a function does not connect to a physical many-electron charge distribution, only to a probability distribution of many-particle positions without direct physical interpretation. Instead we will be led to a system of one-electron Schrödinger equations with direct physical meaning.

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