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Strictly speaking, it is "wrong" because it does not describe spin-1/2 particle like an electrons. Why in every QM textbook is it taught, not as a historical equation, but as a current equation?

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    $\begingroup$ Can you elaborate on why you think this? The Schrodinger equation as presented in a first course in QM $(i\hbar\frac{d}{dt}\psi = \hat H\psi)$ is perfectly capable of describing the dynamics of a non-relativistic spin-1/2 particle, and it is used constantly by working physicists. $\endgroup$
    – J. Murray
    Oct 11, 2020 at 3:59
  • $\begingroup$ Murray, Pauli equation has spin built in, under the non-relativistic condition. It is the non-relativistic limit of the Dirac equation but Schrodinger equation is not. $\endgroup$
    – verdelite
    Oct 11, 2020 at 4:02
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    $\begingroup$ So? That doesn't mean that the Schrodinger equation is inadequate to describe spin-1/2 particles. $\endgroup$
    – J. Murray
    Oct 11, 2020 at 4:04
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    $\begingroup$ The Pauli equation is the Schrodinger equation applied to spin-1/2 particles in an electromagnetic field. It is simply a particular choice of $\hat H$ for the Schrodinger equation. $\endgroup$
    – J. Murray
    Oct 11, 2020 at 4:11
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    $\begingroup$ Probably my question is a duplicate of this one : physics.stackexchange.com/q/129667 $\endgroup$
    – verdelite
    Oct 11, 2020 at 4:23

2 Answers 2

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Why is Schrodinger equation taught while it does not describe an electron?

This question is of the same order as asking:

"Why is Newtonian gravity still taught everywhere since it has been seen that General Relativity is the underlying theory of gravity" and thus should explain everything?

There are even observations in the solar system.

Physics is the observation and of nature and the specific use of measurements to record observations.Mathematical theories are used in order to model observations and , important, predict new observations so that the theory is validated.

In physics mathematical theories, there is a definite process of emergence, classical emerges from quantum, thermodynamics from statistical mechanics etc. Each mathematical format of equations is used in the appropriate range of variables.

Schrodinger's equation modeled the concept of quantum mechanics which , historically was developing, and fitted the hydrogen spectra in a consistent way. Fitted them within errors in measurement at the time. This is a fact and useful. As the observations increased and the errors decreased the necessity of spin for the fine structure of the spectra became necessary. This does not invalidate the previous solutions, it adds details in fine structure finding that the effect of the spin of the electron can be detected in the spectra .

So the Shrodinger equation is still used because its solutions are a first order fit to quantum mechanical observations, spins adding a small measurable effect.

For the same reason the harmonic oscillation model is widely used because it is a first order approximation to quantum mechanical potentials about their minimum. Even though it is a simple model it is still very useful in disciplines that need quantum mechanics to model their data.

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I think you are confusing the treatment of relativistic (spin-$\frac{1}{2}$ particles) electrons as compared to the non-relativistic case. The Schrodinger equation can perfectly describe the properties of the non-relativistic electron. The Dirac equation describes the interactions of relativistic electrons (and other spin-$\frac{1}{2}$ particles).

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