How Computation Changes Theoretical Physics of TD and QM

The physical theories of Thermo Dynamics TD and Quantum Mechanics QM were both formed before the computer, and so do not include the aspect of computation, computability and computational work. Both theories focus on separate equilibrium states rather than dynamical evolution between different states. Rather statics than dynamics, because dynamics is more demanding by including evolution in time.  

The computer changes the game by offering computational power allowing computational simulation of evolution in time of dynamical systems and so opens to a better understanding of the World as it evolves from one instant of time to a next in a process of time-stepping. 

The change is fundamental and opens entirely new possibilities and also resolution of fundamental unresolved problems in TD and QM connected to the static nature of these theories in standard form. 

So can the 2nd Law of TD be given an explanation based on finite precision computation confronting instability as explained in Computational Thermodynamics. 

So can the basic unresolved foundational problems troubling QM since 100 years, be circumvented by a reformulation into Real Quantum Mechanics RealQM which can be explored in dynamical form by computation. 

Let us here focus on the dynamical aspects of RealQM, which come in two forms (i) time-periodic and (ii) dynamical evolution between equilibrium states. 

RealQM takes the following form for an atomic system consisting of $N$ electrons as non-overlapping unit charge densities and a set of atomic nuclei for simplicity as particles at fixed positions, interacting by Coulomb forces, which can be described by a complex-valued wave function $\psi (x,t)$ depending on 3d spatial coordinate $x$ and time $t$ satisfying a Schrödinger Equation SE of the form 

• $i\dot\psi (t)+ H(\psi (t) ) = f(t)$       (SE)

where the dot denotes differentiation with respect to time, $H(\psi )$ is a Hamiltonian depending on $\psi$ and $f(t)$ is an exterior driving force. Here $\vert\psi (x,t)\vert^2$ has a direct physical meaning in 3d space as charge density. 

We note that (SE) has the form of a classical dynamical system in a function $\psi (x,t)$ with direct physical meaning depending on a 3d spatial coordinate $x$ and $t$.  Given an initial value $\psi (x,0)$ the value of $\psi (x,t)$ at a later time $t>0$ can be determined by resolving (SE) by time-stepping with computational work scaling linearly with $N$.

A time-periodic solution (with $f(t)=0$) can take the form 

•  $\psi (x,t)=\exp(iEt)\Psi (x)$. 
• $H(\Psi )=E\Psi$. 
• $\Psi$ eigenstate and $E$ (real) eigenvalue as energy.

Here the eigenstates appear as static states. The eigenstate with smallest energy is the groundstate of the system. It is possible to compute the groundstate by parabolic relaxation in the form of time-stepping of 

• $\dot\Psi + H(\Psi )=0$ with renormalisation to unit charge. 

which can be seen as a gradient method towards minimum energy as an actual physical process when an atom or molecule finds its minimum energy equilibrium state.

But (SE) opens to computational simulation of genuine dynamical evolution between physical states described by charge density under exterior forcing. 

RealQM thus offers a new capacity of computational  simulation of complex atomic systems in terms of charge density as real physics with clear meaning.

RealQM should be compared with StdQM based on a multi-dimensional Schrödinger Equation StdSE with only probabilistic physical meaning with exponential computational complexity requiring drastic reduction into physics with unclear meaning.

Notice that RealQM stays within the classical world of continuum physics and so does not meet the unresolvable problems of StdQM of (i) meaning of wave function, (ii) role of measurement and (iii) computational complexity. There is no need of any special Philosophy of RealQM as for StdQM. 

Summary: Computation offers a new tool to simulation and understanding of atomic systems when applied to RealQM as a computable model with clear physics. Thus 100 years after conception QM may take a leap into a new era of Computational QM, leaving the unresolved foundational problems of StdQM behind as irrelevant. 

PS The typical reaction to RealQM is not a welcome as possibly something offering new capabilities and relief from old troubles, but rather the opposite as an unwanted disturbance to a status quo in full agreement that "nobody understands QM" as a "soft pillow" in the words of Einstein.