måndag 8 december 2014

The Radiating Atom 6: Schrödinger's Equation in Real-Valued System Form

Schrödinger's equation, to start with for the electron of the Hydrogen atom, is usually written in the form
  • $ih\dot\Psi = H\Psi$,
with $\Psi (x,t)$ a complex-valued function of a space-time $(x,t)$,  $\dot\Psi =\frac{\partial\psi}{\partial t}$, $H=-\frac{h^2}{2m}\Delta + V(x)$ the Hamiltonian with $\Delta$ the Laplacian with respect to $x$, $V(x)=-\frac{1}{\vert x\vert}$ the kernel potential, $m$ the electron mass and $h$ Planck's constant.  This equation can equivalently be expressed as follows in real-valued system form, with $\Psi =\phi + i\psi$ and $\phi =\phi (x,t)$ and $\psi =\psi (x,t)$ real-valued functions: 
  • $\dot\psi + H\phi =0$, 
  • $-\dot\phi + H\psi= 0$. 
This system can be viewed as a generalized harmonic oscillator or wave equation, which can naturally be extended to
  • $\dot\psi + H\phi -\gamma\dddot\phi = f$       (1)
  • $-\dot\phi + H\psi -\gamma\dddot\psi = g$     (2)
where $f(x,t)$ and $g(x,t)$ represent external electro-magnetic forcing, and $\gamma\dddot \psi$ and 
$\gamma\dddot \phi$ represents the Abraham-Lorentz recoil force from emission of radiation with $\gamma$ having a dependence on $\Phi \equiv (\psi ,\phi )$ to be specified. A system of this form as a wave equation with small damping subject to near-resonant forcing is analyzed in Mathematical Physics of Black Body Radiation.

The basic energy balance is obtained by multiplying (1) by $\dot\phi$ and (2) by $\dot\psi$, then adding and integrating in space and time, to get for $f=g=0$:
  • $E(\Phi ,T)+R(\Phi ,T)= 0$ for $T>0$,
  • $E(\Phi ,T)=\int (\psi (x,T)H\psi (x,T)+\phi (x,T)H\phi (x,T ))dx$
  • $R(\Phi ,T)=\int_0^T\int(\gamma\ddot\psi^2(x,t)+\gamma\ddot\phi^2(x,t))dxdt$,
which expresses a balance between internal atomic energy $E(\Phi ,T)$ at time $T$ as the sum of "kinetic energy" related to the Laplacian $\Delta$ and potential energy related to V as terms in the Hamiltonian $H$, and total radiated energy until time $T$ in accordance with Larmor's formula stating that radiation scales with $\ddot q^2$, where $\ddot q=\ddot q(t)$ is the "acceleration" of a charge $q(t)$ varying in space over  time. 

Let now $\psi_1=\psi_1(x)$ and $\psi_2=\psi_2(x)$ be two eigenfunction of the Hamiltonian $H$ with corresponding eigenvalues $E_1 < E_2$ and pure eigen-states
  • $\Phi_j(x,t)\equiv (\cos(E_jt/h)\psi_j(x),\sin(E_jt/h)\psi_j(x))$ for $j=1,2$, 
and corresponding charge densities
  • $q_j(t)\equiv\vert \Phi_j(x,t)\vert^2\equiv(\cos^2(Et/h)+\sin^2(Et/h))\psi_j^2(x)=\psi_j^2(x)$. 
We thus find that pure eigen-states have charge densities which are constant in time and thus do not radiate.

On the other hand, the charge density $q(x,t)=\vert\Phi (x,t)\vert^2$ of a superposition $\Phi =c_1\Phi_1+c_2\Phi_2$ with $c_1$ and $c_2$ positive coefficients of the two pure eigenstates $\Phi_1$ and $\Phi_2$,  has a time dependence of the form
  • $q(x,t) = a(x) + b(x)\cos((E_2-E_1)t/h)$  
with $a$ and $b$ coeffcients depending on $x$, and thus is radiating. We are thus led to a dependence of $\gamma$ on $\Phi$  of the form
  • $\gamma \sim\ddot q^2$.
We conclude that (1)-(2) offers a continuum mechanical model of a radiating Hydrogen atom which can be analyzed by eigenfunction expansion as in Mathematical Physics of Black Body Radiation and thus offers an answer to the basic questions of atomic mechanics:
  • Why does a pure-eigen-state not radiate and thus can persist over time as a stable atomic state?
  • Why can an atom radiate under external forcing? 
  • How much is an atom radiating under external forcing? 
Note that the system (1)-(2) in case with $f=g=\gamma =0$ has the equivalent form of a second order wave equation:
  • $\ddot\psi + H^2\psi =0$,
a form which Schrödinger dismissed on the ground that a time dependent potential would cause complications, and probably also because the presence of the term $\ddot\psi$ appears to be asking for a physical interpretation of $\dot\psi^2$ as kinetic energy, which however was already assigned to $\vert\nabla\psi\vert^2$ connected to the Laplacian. 

On the other hand, in the real-valued system form (1)-(2), these complications do no arise, and the extension to forcing and radiation is more natural than in the standard complex form, which is commonly viewed as a complete mystery beyond human comprehension.

What remains to understand is the physical meaning of the system equations (1)-(2), which may well be possible after some imagination, which I hope to report on.  

In short (1)-(2) may be the form of Schrödinger's equation to use for extensions to multi-electron configurations. At least this is the route I am now seeking to explore.

Note that letting $h$ tend to zero, we obtain the dynamical second order system
  • $\ddot\psi (t) = -V^2\psi = -\frac{\psi}{\vert x\vert^2}$
which can be interpreted as Newton's equations for a moving "particle" localized in space. Schrödinger's equation (1)-(2) can thus be viewed as regularized form of Newton's equations with regularization from the Laplacian. In this perspective there is nothing holy about the Laplacian; it is thinkable that the effective regularization in an atom is non-isotropic,  thus with different action in radial and angular variables in spherical coordinates centered at the kernel.  

An equation $\dot\psi +H\phi=\dot\psi + V(x)\phi=0$ with $h=0$ may formally be viewed as some form of force balance expressing a form of "square root of Newton's 2nd law" $\ddot\psi+V^2\psi$.

Note that in (1)-(2) $-H\phi$ connects to $\dot\psi$ and $H\psi$ to $\dot\phi$ and so the dynamics of a pure eigen-state with wave function $\Phi_j$ can be described as a "revolution/oscillation in time" of a space-dependent eigen-function of the Hamiltonian for which the charge density is constant in time without radiation,  while the charge density of a superposition of pure eigen-states varies in time and thus radiates.  With this perspective, an electron is not "moving in space" like some form of planet around the kernel, but instead has a variation in time, which gives rise to a charge density with variation in time and thus radiation, except for a pure eigen-state which does not radiate.  

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