# Orbital wave functions and probability density - interpretation issue

We're always converting between "abstract mathematical constructs" and "physically real entities" when we do physics. That's what having a mathematical model that predicts what happens in (an idealization of) the world *means*. This is not at all unique to quantum mechanics, it is inherent in the idea that mathematics can tell us *anything* about the real world.

For a purely classical instance of this, see e.g. the question *"
How can energy be useful when it is 'abstract'?"*.

For a famous essay pondering the larger philosophy behind the use of mathematics in physics, see Wigner's *"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"*.

Furthermore, the "most accurate way" is not always the most *useful* way to think about a problem. Yes, sure, our most "accurate" understanding of the quantum world is not non-relativistic quantum mechanics as wave mechanics, but relativistic quantum field theory, just like our most "accurate" understanding of gravity is not Newtonian gravity, but general relativity.

But we don't use the most "accurate" understanding for predicting everything. No one does general relativistic calculations to figure out how long a thrown rock will take to hit the ground, and likewise, no one uses full quantum field theory to get a picture of what orbitals are.

Finally, you should be careful to distinguish between the formal predictions of quantum mechanics - "The square of the wavefunction is the probability density to detect the electron at a particular place and time" - and its interpretations - e.g. "The electron really is smeared out and has no position", "The electron has a definite position and is guided by the pilot wave", "There is a world for every possible position of the electron", etc. It is the former which are objective and can be experimentally tested, while the latter cannot.

You are right. One has to learn stuff and then unlearn it again.

The picture of a de Broglie electron wave neatly fitting a whole number of wavelengths into the circumference of a Bohr orbit - which lies behind your assumptions 2 and 3 - and is nicely explained, for example, in https://physics.stackexchange.com/a/318638/194034 is, at a higher level, just plain wrong.

Electrons do not orbit the nucleus like planets going round the sun. They have a wave-function which has to be a solution of the Schrodinger equation (or the Dirac equation - relativity is not the issue here).

But we will go on teaching the Bohr model, as it is a useful step up the ladder of fuller understanding.