måndag 23 maj 2011

Monstrosity of Quantum Mechanics 2

The simplicity (linearity) of the Schrödinger equation is seductive and has misled many minds.

Quantum mechanics as a description of the microscopic world of atoms and molecules is based on Schrödinger's wave equation, which as a mathematical object is (see above)
  • scalar
  • linear
  • multidimensional in 3N coordinates for N electrons/kernels
with solutions called wave-functions commonly denoted as Psi:
  • Psi (x1, x2, ..., xN, t)
with xj representing the three position coordinates of particle j with j=1,...,N, and t denoting time. The wave function Psi thus depends on 3N independent real variables plus time.

The simplicity of the Schrödinger wave equation (scalar and linear) as a description of a complex reality, is thus balanced by an extreme richness of the wave function depending on
3N + 1 independent variables.

The richness of the wave-function thus makes it impossible to give it a physical meaning as
representing a configuration or distribution of electrons and kernels, which threatened to kill quantum mechanics at birth but was rescued by Max Born declaring that
  • | Psi (x1,....,xN,t) |^2
represents the probability of the configuration given by the coordinates (x1,...,xN,t), and by Niels Bohr declaring that the wave function as a probability distribution, upon observation could collapse into a definite physical state, as when opening the box containing the Schrödinger cat.

Born and Bohr thus developed the Copenhagen Interpretation (of quantum mechanics), which is today the officially accepted truth although contested by alternatives as hidden variables and many-worlds interpretations, without any winner.

Schrödinger himself left quantum mechanics as soon as the Copenhagen Interpretation captured the minds of most physicists.

The richness of the wave function is in fact a monstrosity already for small systems with N = 100 say, not to speak of real systems of 10^23 particles in a mole of gas, as pointed out by Walter Kohn, Nobel Prize in Physics in 1998:
  • The wave function does not exist for N larger than 100.
  • Why? Because it cannot be computed, because of the many dimensions.
Kohn got the Nobel Prize for computing electron densities instead of probabilities as solutions of a non-linear version of the Schrödinger equation in 3 space dimension, referred to as density functional theory.

If now the wave-function as solution to the Schrödinger equation does not exist, there must something fishy about the Schrödinger equation. What? We saw that the equation is scalar and linear and thus has a simple structure, which is not problematic in itself, but if it necessitates a monstrous richness in dimensions, it seems that one should question the very formulation of the Schrödinger equation as a scalar linear multidimensional equation.

From where did Schrödinger get his equation? Did he derive it from basic principles? Not really. It is more of an ad hoc invention expressing particle interaction by
electrostatic Coulomb potentials combined with a new mysterious form of kinetic energy.

How can we know that the equation is a good model of physics if it cannot be solved? How can we check that its solutions give correct predictions if they cannot be computed and thus determined?

Nevertheless it is mantra of modern physics that the Schrödinger equation is a good model, but it is a mantra without physical meaning about an equations which cannot be solved. It is like claiming that a certain truth is hidden in a riddle which cannot be solved.

Thus, new versions of the Schrödinger equation are needed. I explore one such line of thought in Many-Minds Quantum Mechanics in the spirit of the Hartree method as a non-linear coupled system of one-electron/kernel Schrödinger equations.

The simplicity of linearity (and superposition) of the multi-dimensional Schrödinger equation is here replaced by a non-linear complexity, but the system solutions only depend on three space dimensions which makes a direct physical interpretation possible, without probabilities and wave function collapse. This is a realist approach as compared to the non-realist Copenhagen Interpretation.

Compare with Lars-Göran Johansson: Interpreting Quantum Mechanics: A Realsist's View in Schrödinger's Vein suggesting a form of realist wave-particle duality as continuous waves for propagation and discontinuous particles for exchange of energy.

4 kommentarer:

  1. Can a computer solve the Helium-atom Schrödinger equation? How does it compare with experiment?

  2. Yes it can and the solufion is neither symmetric nor anti-symmetric and thus violates the basic mantra of quantum mechanics, while it should be in accordance with experimental observation.

  3. There is another thing i wonder about MMQM. Suppose you want to calculate the first and second excited states of the Helium atom. In traditional QM this is done by finding the eigenvalues of the Hamilton operator. How do you go about in MMQM. You find the ground state by minimizing the energy functional which is fine, but how do you proceed after that?

  4. This depends on the energy of the initial value if there is no incoming radiation, or of the intensity of incoming radiation.