Schrödinger with his cat.
The Schrödinger Equation
The Schrödinger equation
developed by the 39 year old Austrian physicist Erwin Schrödinger
in a sequence of 4 articles in 1926, expresses a balance of kinetic and potential energies
of a set of interacting negatively charged electrons
and positively charged kernels
in the form of a partial differential equation
. The potential energy has contribution from attractive
and repulsive forces
. A solution of Schrödinger’s equation is called a wavefunction
. Schrödinger solved the equation for the one-electron problem of the Hydrogen atom.
Formally the Schrödinger equation is a differential equation in 3N space dimensions plus time, where N is the
number of electrons and kernels, which makes it impossible to solve exactly even for N rather small, and Nobel Laureate Walter Kohn
insists that for N > 100 – 1000
the wavefunction is not a legitimate scientific concept
. In other words, it does not exist! Schrödinger writes in the 4th article:
- The wavefunction itself cannot be given a direct interpretation in three-dimensional space, as in the one-electron problem, because it is a function in configuration space, not in real space.
The wave equation apparently exists as a string of symbols on piece of paper or computer screen,
but a wavefunction as a well defined mathematical object satisfying the equation, does not seem to exist.
What are the consequences of Kohn´s insight? Is it important, or of marginal interest? Let´s seek an answer.
Let´s start with making clear that there is a difference between saying “golden mountain” and a physical mountain made out of gold. Just by pronouncing the properties of something, it does not magically bring what you describe into physical existence, unless you are magician.
The Copenhagen Interpretation
As soon as Schrödinger had formulated his formally multi-dimensional wave equation acting in a 3N
-dimensional “configuration space”
, heated debates started with Bohr
on how to interprete the wavefunction in physical terms. Born came up with a probabilistic interpretation
- the square of the modulus of the multi-dimensional wavefunction is a probability distribution of the configurations of N interacting particles.
Schrödinger objected to any talk about “particles” since his equation was a wave equation and not a particle
But Schrödinger had no better interpretation of the multi-dimensional wavefunction and thus was overpowered in particular during a visit to Bohr’s institute in Copenhagen in September 1926 as described by Heisenberg and in 
- The discussion between Bohr and Schrödinger began at the railway station in Copenhagen and was crried on every day from early morning to late night….It will scarcely be possible to reproduce how passionate the discussion was carried from both sides….After some days Erwin became ill with a feverish cold. Bohr sat on the bed and continued the argument: “But surely Schrödinger, you must see”. But Erwin did not see, and indeed never did see, why it was necessary to destroy the space-time description of atomic processes.
The final shoot-out took place at the famous 5th Solvay Conference in 1927:
Many-Minds wavefunction of the 5th Solvay Conference in 1927 not agreeing on anything.
Schrödinger tried to meet the attacks by his famous thought experiment about Schrödinger’s cat but it did not
A poll taken at the 1997 UMBC quantum mechanics workshop gave the once all-dominant Copenhagen interpretation less than half of the votes according (Interpretation of Quantum Mechanics: Many Worlds or Many Words?). But what interpretation(s) do then most physicists favor, today?
But if the multi-dimensional wavefunction in its configuration space does not exist, then you no longer have to give it a physical interpretation and meaning. For example if you acknowledge that a “round square” does not exist, then you don´t have to worry about what it would look like if you could fabricate one. But if you believe that a “round square” exists, then you face a severe problem if you seek to understand its physics.
If the multi-dimensional wavefunction like a golden a mountain does not exist as a solution of the Schrödinger equation, it is necessary to seek to understand what solutions do exist and their physical interpretation. If we stick to the Schrödinger equation as somehow expressing a correct balance we are then led to the basic question of quantum mechanics:
- What approximate solutions to the Schrödinger equation do exist, what are their properties and how can they be computed and studied?
We address this problem in the knol Many-Minds Quantum Mechanics based on the idea of viewing the Schrödinger’s wave equation as a system of N wave equations, one for each particle with an averaged or blurred presence of the other particles. This natural idea was suggested already in 1920s by Hartree but
was quickly superseeded by the Hartree-Fock method because it seemed as if Hartree alone would lead to trivial electronic structures without any periodic table of elements. But taking stability into account it may be
possible that Hartree´s original idea is fruitful, as discussed on Many-Minds QM.