måndag 12 januari 2015

The Radiating Atom 10: Restart from Schrödinger IV 1926

Schrödinger in Quantization and Proper Values IV, 1926:
Meantime, there is no doubt a certain crudness in the use of a complex wave function. If it were unavoidable in principle, and not merely a faciliation of the calculation, this would mean that there are in principle two wave functions, which must be used together in oder to obtain information on the state of the system. This somewhat unacceptable inference admits, I believe, of the very much more congenial interpretation that the state of the system is given by a real function and its time derivative. Our inability to give more accurate information about this is intimately connected with the fact that we have before us only the substitute, extraordinarily convenient for the calculation, to be sure, for a real wave equation of probably fourth order, which, however, I have not succeeded in forming in the non-conservative case.

In the present series of posts on the radiating atom, I have restarted from the last of Schrödinger's four legendary 1926 articles formulating Schrödinger's equation, for Hydrogen to start with:
  • $i\dot\Psi + H\Psi =0$         (1) 
where $\Psi (x,t)=\psi +i\phi$ is a complex-valued electronic wave function of a space coordinate $x=(x_1,x_2,x_3)$ and time $t$, with real-valued real and imaginary parts $\psi (x,t)$ and $\phi (x,t)$, $H$ is the Hamiltonian defined by
  • $H =-\frac{h^2}{2m}\Delta + V$,
where $\Delta$ is the Laplacian with respect to $x$, $V(x)=-\frac{1}{\vert x\vert}$ is the kernel potential, $m$ the electron mass, $h$ Planck's constant and the dot signifies differentiation with respect to time $t$.

Before ending up with (1) Schrödinger considered the following second-order equation in terms of a real-valued wave function $\psi (x,t)$, which can be the real or imaginary part of $\Psi (x,t)$:
  • $\ddot\psi +H^2\psi = 0$      (2)
which formally follows by writing (1) as the first order system
  • $\dot\psi -  H\phi =0$         (3)
  • $\dot\phi + H\psi =0$          (4)
and then eliminating $\phi$ by differentating (3) with respect to time and replacing $H\dot\phi$ by
$-H^2\psi$ after application of $H$ to (4).

Schrödinger thus considered both (1) and (2), but then decided to choose the complex first order form (1), while regretting that the real-valued second-order form (2) in principle was to prefer, because of its very much more congenial interpretation

What Schrödinger referred to was that (2) could be given a physical interpretation as force balance pretty much as in classical mechanics, while the physical meaning of (1) was mysterious to Schrödinger and has so remained to all physicists into our time:

The accepted wisdom, presented in all books, is that (1) arises from classical Hamiltonian mechanics by formally representing momentum by the differential operator $-ih\nabla$ acting in space and energy by the differential operator $ih\frac{\partial}{\partial t}$ in time, which is however ad hoc and without physical reason as expressed by Schrödinger himself and acknowledged by all physicists into our time.

The lack of physical interpretation of (1) means that modern physics as the foundation and model of modern science rests on a quantum foam of mystery, which is the opposite of scientific enligthenment.

Schrödinger stated that his choice of (1) before (2) came from a perceived difficulty of generalizing (2) to a non-conservative case including radiation. But maybe Schrödinger dismissed (2) too quickly.

To check this out, let us consider the following natural generalization of (2) to include radiation as a direct generalization of the classical mechanical or electromagnetic wave equation with (small) radiative damping under near-resonant forcing considered in Mathematical Physics of Black Body Radiation (and Computational Black Body Radiation):
  • $\ddot\psi +H^2\psi -\gamma\dddot\psi = f$      (5)
where $f(x,t)$ is external forcing,  and $\gamma =\gamma (\psi )$ is a (small) positive radiation damping coefficient. The equation (5) has the physical meaning of force balance with
  • $\ddot\psi +H^2\psi$ out-of-balance force of an electronic resonator 
  • $-\gamma\dddot\psi$ the Abraham-Lorentz radiation recoil force
  • $f$ component of an electrical field force. 
Let us now subject the model (5) to a basic study:  First we observe that if $f=0$ and  $\gamma =0$,  then conservation of total charge expressed as
  • $\frac{d}{dt}\int\rho (x,t)dx =0$,                   (6)
with $\rho =\psi^2+(H^{-1}\dot\psi )^2$ the charge intensity, is obtained by multiplying (5) with $H^{-2}\dot\psi$ and integrating in space.

Next, letting $\{\Psi_1,\Psi_2,\Psi_3....\}$ be an orthonormal basis of eigenfunctions $\Psi_k=\Psi_k(x)$ of the Hamiltonian $H$ satisfying $H\Psi_k =E_k\Psi_k$ with corresponding sequence of eigenvalues $E_1\le E_2\le E_3 ...$ , we spectrally decompose $\psi (x,t) =\sum\psi_k(t)\Psi_k(x)$ and $f(x,t)=\sum f_k(t)\Psi_k(x)$  and obtain after multiplication of (5) by $\Psi_k$ and integrating in space, for $k=1,2,..$ and for all $t$ :
  • $\ddot\psi_k(t) +E_k^2\psi_k(t) -\gamma\dddot\psi_k(t) = f_k(t)$   (7)
which is a set of harmonic oscillators with damping under forcing, each one which can be analyzed as in Mathematical Physics of Black Body Radiation.

Let us now consider the basic case $\psi (x,t) = \psi_1(t)\Psi_1(x) +\psi_2(t)\Psi_2(x)$ with $\Psi_1$ the ground state eigenfunction with smallest eigenvalue $E_1$ and $\Psi_2$ an eigenfunction of the next eigenvalue $E_2 > E_1$ with non-vanishing $f_2(t)$ (assuming $f_k=0$ for $k>2$). By a shift of the Hamiltonian by $E_1$, we may assume that $E_1=0$ and then also that $f_1=0$. We thus have the system
  • $\ddot\psi_1(t)  -\gamma\dddot\psi_1(t) = 0$,  thus $\psi_1(t)=\psi_1=constant$, 
  • $\ddot\psi_2(t) +E_2^2\psi_2(t) -\gamma\dddot\psi_2(t) = f_2(t)$,   
and conclude under an assumption of near-resonant forcing $f_2(t)\sim \cos(\nu t)$ with $\nu\approx E_2/h$ and small damping as in Mathematical Physics of Black Body Radiation:
  • $\int\gamma\ddot\psi_2^2dt \approx \int f_2^2(t)dt$   (8)
or in terms of the wave function $\psi$ and the forcing $f$ 
  • $\int\gamma\ddot\psi^2dxdt \approx\int f^2dxdt$       (9)
which expresses that in periodic equilibrium state:
  • outgoing radiation = incoming radiation.        (10)
We now recall the basic energy balance of (5) obtained by multiplying (5) by $\dot\psi$ and integrating in space:
  • $\dot A(t) +R(t) = W(t)$, 
  • $A(t)=\frac{1}{2}(\int\dot\psi^2dx+\int (H\psi )^2dx)$ = internal oscillator energy
  • $R(t)=\int\gamma\ddot\psi_2^2dt$ = outgoing radiation (per unit of time)
  • $W(t) = \int f(x,t)\dot\psi dx$ = work by incoming radiation (per unit of time),
with in equilibrium periodic state, $\dot A(t)=0$ and $R(t)=W(t)$ effectively expressing (9) or (10).  

Of particular concern is now the charge conservation in (5). We note that the internal oscillator energy $A(t)$ may increase under forcing with $W(t)>R(t)$, or decrease if $W(t) < R(t)$,
reflecting a change of balance of the spectral weights $\psi_k(t)$. The question is then if such a change of internal oscillator energy may take place under conservation of total charge, and we 
are then led to compare the work $f_k\dot\psi_k$ connected to energy and $f_kE_k^{-2}\dot\psi_k$ connected to charge with the corresponding coefficient $E_k^{-2}$ for $k>1$. 

Now, in typical cases, $E_k\approx 10^{15}$ and thus $E_k^{-2}\approx 10^{-30}$, which signifies that in the model (5) energy may change under almost perfect charge conservation.

Note that (8) can be expressed as
  • $\gamma \nu_2^4\int\psi_2^2dt\approx \int f_2^2dt$,
thus connecting the amplitude of the excited component $\psi_2\Psi_2$ to the forcing $f_2^2$, which itself may be of the form $\gamma\nu_2^4$ with a possibly different $\gamma$. The radiation balance (10) can thus be viewed to express radiative equilbrium of a collection of atoms under mutual radiative absorption/emission.  

We sum up the virtues of (5) as a semi-classical continuum wave model of a radiating atom subject to forcing,
  1. (5) lends itself to physical interpretation as force balance in a classical sense with the Laplacian representing some form of elastic energy, and the value of wave function $\psi (x,t)$ at position $(x,t)$  representing the "displacement" of the electron at $(x,t)$ from a ground state. 
  2. The Abraham-Lorentz recoil force is small compared to forcing and oscillator imbalance,  because $\gamma$ is very small, which means that self-interaction is avoided and the forcing $f$ can be viewed to be independent of the wave function $\psi$.
  3. (5) lends itself to mathematical analysis as energy balance under charge conservation. 
  4. (5) has a natural extension to a model for a many-electron atom as a system of one-electron equations, which is computable and thus potentially useful. 
  5. (5) coincides in the case $f=0$ and $\gamma =0$ with the standard model (1) and thus with experiments.
  6. (5) admits the ground state to be independent of time as a stable solution without radiation and forcing.
  7. (5) fits with observed radiation of frequency $\nu =(E_k-E_1)/h$ under near-resonant forcing.
  8. Outgoing and incoming radiation can be shifted in (10), which allows (5) to model both absorption of radiation and stimulated or spontaneous emission of radiation. 
Our conclusion is that maybe (5) is the basic model of quantum mechanics asking for thorough analysis and waiting for extensive practical use, rather than (1), corresponding to a restart from the original idea of Schrödinger as the true father of quantum mechanics.

It appears that the advantage of (5) allowing natural extension to radiation and forcing, was (paradoxically so) by Schrödinger perceived instead as a disadvantage making him prefer (1). Too bad that Schrödinger is not around anymore, so that he could have clarified the reason for his choice.

The above virtues 1-8 of (5) may be compared to the following acknowledged deficiencies/difficulties of (1):
  1. The physical meaning of (1) as a strange ad hoc "square-root" of (5) is unknown.
  2. Extension to radiation of (1) is typically accomplished through a time-dependent potential representing forcing, which does not include the Abraham-Lorentz recoil force and thus appears to miss essential physics. 
  3. The attribution of kinetic energy to $\vert\nabla\psi\vert^2$, resulting form formally replacing classical momentum by the differential operator $-ih\nabla$, is irrational from physics point of view.
  4. The generalization of (1) to include radiation under forcing is commonly viewed to require extensions to QED which is further away from classical mechanics, and thus loaded with difficulties. 
  5. Extension of (1) to many electrons introduces a multi-dimensional wave function, which makes (1) uncomputable and thus useless.
PS Note that the wave equation $\ddot\psi +H^2\psi -\gamma\dddot\psi = f$ is a scalar equation in a real-valued function $\psi (x,t)$ with scalar forcing $f(x,t)$, which may be any component of the electrical field, with non-zero $f_k(t)$ in near-resonant interaction with an eigenmode $\psi_k(t)\Psi_k (x)$. It is thus the multiplicity of eigenvalues with in particular 3 independent $2p_{x1}$, $2p_{x2}$ and $2p_{x3}$ eigenstates oriented in the coordinate directions $x =(x1,x2,x3), which in the basic case of resonant radiation connects the scalar wave equation to the  vector $E=(E_{x1},E_{x2},E_{x3})$ of the electrical field, see The Radiating Atom 9.

Interpreting the scalar wave function $\psi$ as an (oscillating) virtual "displacement" connects to a corresponding (oscillating) real physical displacement of charge in 3d space, as the connection between the scalar $\psi (x,t)$ and the vectors of charge displacement and related electrical field.   

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