Interpreting Schrödinger's PDE often leads to common mistakes: (1) confusing the wave function ψ(x,t) with a physical wave, when it is a probability amplitude; (2) ignoring its probabilistic nature, as ∣ψ∣2 gives probability density, not deterministic trajectories; (3) misunderstanding the Hamiltonian's role in governing dynamics and energy eigenvalues; (4) neglecting boundary conditions and normalization, which are essential for physical solutions; and (5) overlooking the importance of the wave function's complex nature, as its phase is critical for quantum interference and superposition. Avoiding these errors ensures a correct understanding of Schrödinger's equation in quantum mechanics.

This is a brief response, first to shed light on the question and its answer, and second to thank our Pakistani collaborator friend for his brilliant analysis.

This motivates me to confirm his answer and add two more points:

6- the wave function in QM PSI is continuous but its derivative is not continuous?

In physics textbooks it is written that the Psi wave function and its first derivative must be continuous.

Meanwhile, in these same books you find many situations where the Psi solution is continuous and its derivative is not, the simplest example is the particle in a one-dimensional box.

Psi. Psi* and its derivative must be continuous while the derivative of Psi alone may or may not.

Useless quantum mechanics is actually inherent to Schrödinger's PDE since it lives and operates in an incomplete R^4 space where time is not properly defined as embedded in 3D geometry.

7-We assume that it is time to replace the classic Schrödinger PDE with its square or Laplacian PDE where time is a dimensionless integer embedded in a unitary 4D x-t space.

The question arises:

Did E. Schrödinger deeply understand quantum mechanics?

I believe that E. Schrödinger understood quantum mechanics deeply.

Only, he did not have the power to impose his own ideas against the giants N. Bohr, Heisenberg, Einstein, etc.

Check out his brilliant quotes:

i-Entanglement is not one but rather the characteristic feature of the quantum mechanical.

ii-He wrote about the probabilistic interpretation of quantum mechanics saying:

"I don't like it and I'm sorry I ever had anything to do with it."

Today, after 100 years, we are more reliable about what is true and what is false:

1- The three giants Einstein, Bohr, Schrödinger were all wrong in confining entanglement to quantum physics phenomena when it applies to classical physics in the same way as in QM (Sound waves in audio rooms for example).

2-Most of the great scientists of the time and the iron guards who followed believe that entanglement transfers instantly (spooky action at a distance) while the statistical theory of Cairo techniques predicts that the entanglement signal has a maximum speed equal to the speed of light C.

3-Quantum entanglement comes before Schrödinger PDE and not after.

4-Sentimentally I feel discomfort every time I hear or read the sentence:

Separation of variables

in time-dependent PDE.

To me this looks like separating the head from the body of a living subject (entanglement).

"I think I can safely say that nobody understands quantum mechanics," Richard Feynman.

I'm not sure interoperation of a physical theory belongs on a forum devoted to mathematics - maybe better a forum on philosophy. In reality theoretical physics has come a long way since the 1920's over a century ago. Not only that Schrödinger's equation is not invariant under the Lorentz so not compatible even with special relativity much less general relativity.