I Complex representation of wave function

[Runei]

When solving problems, particularly in optics, it is often that we represent the wave-function as a complex number, and then take the real part of it to be the final solution, after we do our analysis.
u(\vec{r},t)=Re\{U(\vec{r},t)\}=\frac{1}{2}\left(U+U^*\right)
Here U is the complex form of the wave function.

What my question is, is whether there exists some analyses regarding the validity of this approach. In general, can we prove that any of the operations we perform in the "complex domain" do not add "extras" to the real function, after we convert back.

Thanks in advance!

 

[S.G. Janssens]

What my question is, is whether there exists some analyses regarding the validity of this approach. In general, can we prove that any of the operations we perform in the "complex domain" do not add "extras" to the real function, after we convert back.

Just a short answer for now: I think that, as long as the evolution equation in question is linear, this approach is valid and not difficult to verify. When the quantity evolves in a nonlinear way, more care and case-by-case assessment are needed.

 

[FactChecker]

When solving problems, particularly in optics, it is often that we represent the wave-function as a complex number, and then take the real part of it to be the final solution, after we do our analysis.
u(\vec{r},t)=Re\{U(\vec{r},t)\}=\frac{1}{2}\left(U+U^*\right)
Here U is the complex form of the wave function.

What my question is, is whether there exists some analyses regarding the validity of this approach. In general, can we prove that any of the operations we perform in the "complex domain" do not add "extras" to the real function, after we convert back.

Just the opposite. If the operations add "extras" to the real function, then they belong there and would have been more difficult to keep track of without the complex analysis approach. The complex plane is a very good way of dealing with cyclic behavior. Suppose a problem has cyclical trade-offs between two things. For instance, the trade-off between the potential and kinetic energy of a pendulum. The real and complex parts are a good way of representing the two types of energy and keeping them coordinated in one entity. The operations in complex analysis are designed to allow the study of such dynamic interactions.