From what I understand the Dirac equation is supposed to be an improvement on the Schrödinger equation in that it is consistent with relativity theory. Yet all methods I have encountered for doing actual ab initio quantum mechanical calculations uses the Schrödinger equation. If relativistic effects are important one adds a relativistic correction. If the Dirac equation is a more correct description of reality, shouldn't it give rise to easier calculations? If it doesn't, is it really a more correct description?
Quantum Mechanics – Why Is the Dirac Equation Not Commonly Used for Calculations?
dirac-equationquantum mechanicsspecial-relativity
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This is the subject of an underrated classic paper from the early days of quantum mechanics:
- Mott, 1929: The wave mechanics of alpha-ray tracks.
Mott's introduction is better than my attempt to paraphrase:
In the theory of radioactive disintegration, as presented by Gamow, the $\alpha$-particle is represented by a spherical wave which slowly leaks out of the nucleus. On the other hand, the $\alpha$-particle, once emerged, has particle-like properties, the most striking being the ray tracks that it forms in a Wilson cloud chamber. It is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track; we think intuitively that it should ionise atoms at random throughout space. We could consider that Gamow’s outgoing spherical wave should give the probability of disintegration, but that, when the particle is outside the nucleus, it should be represented by a wave packet moving in a definite direction, so as to produce a straight track. But it ought not to be necessary to do this. The wave mechanics unaided ought to be able to predict the possible results of any observation that we could make on a system, without invoking, until the moment at which the observation is made, the classical particle-like properties of the electrons or $\alpha$-particles forming that system.
Mott's solution is to consider the alpha particle and the first two atoms which it ionizes as a single quantum-mechanical system with three parts, with the result
We shall then show that the atoms cannot both be ionised unless they lie in a straight line with the radioactive nucleus.
That is to say, your question gets the situation backwards. The issue isn't that "free electrons have classical trajectories," and that these electrons are "not able to move classically anymore" when they are bound. Mott's paper shows that the wave mechanics, which successfully predicts the behavior of bound electrons, also predicts the emergence of straight-line ionization trajectories.
With modern buzzwords, we might say that the "classical trajectory" is an "emergent phenomenon" due to the "entanglement" of the alpha particle with the quantum-mechanical constituents of the detector. But this classic paper predates all of those buzzwords and is better without them. The observation is that the probabilities of successive ionization events are correlated, and that the correlation works out to depend on the geometry of the "track" in a way which satisfies our classical intuition.
Best Answer
Think it with an example, Einstein's field equations are much more precise than Newton's law of gravity, but it's much more complicated to solve a Classical Mechanics problem with General Relativity.
More fundamental and precise doesn't mean that it will give easier calculations. If it did, then then chemistry, medicine, etc... wouldn't exist because they can be described almost completely using Dirac's equation.