Arbitrary constants in the general solution to the Schrödinger equation

Question

Are there some general cases for which the costants in the solution of Schrödinger equation for a one dimensional problem are real?

I know that in general: $$\psi(x)=Ae^{ikx}+Be^{-ikx} \,\,\,\,\,\,\,\,\,\, A,B \in \mathbb{C}$$ But in many problems I notice that in the solution at the end $A,B \in \mathbb{R}$.

Obviously, knowing it at the beginning makes it easier mostly to normalize the wave function.

The two kind of equation are: $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V(x)\psi=E\psi$$

in the case $E>V$ with $k=\sqrt{\frac{2m(E-V)}{\hbar^2}}$

and

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}=E\psi$$

with $k=\sqrt{\frac{2mE}{\hbar^2}}$

Answer 1

Allow me to assume you care more about the physics than the algebra.

Given a potential (for example, a combination of wells and barriers), there are going to be regions (bands) depending on "total energy vs. potential", where the particle behaves differently in different regions.

When $E > V$, there is kinetic energy $T = E - V >0$ for the particle to travel around.

That is, the wave function here is in the form of waves (helical in complex space, sinusoidal when projected to the real) , either standing waves or traveling wave.

In this case, the coefficients of $\psi(x)=Ae^{ikx}+Be^{-ikx}$ are just real $\{A,B\}\in\mathbb{R}^2$ to setup the combination of left- and right- traveling waves to be nailed down by the boundary conditions.

When $E < V$, there is no kinetic energy and the particle is "trapped".

That is, the wave function here is in the form of exponential decay. Classically, this can be either a particle trapped in a well or a particle not having enough energy to go through the barrier. Quantum mechanically, we always at least have a "dying wave", which is exponential decay (in the real space) made up of $\cosh$ and $\sinh$.

In this case, the coefficients of $\psi(x)=Ae^{ikx}+Be^{-ikx}$ are complex $\{A,B\}\in\mathbb{C}^2$ to setup the combo of hyperbolic sine and hyperbolic cosine, which again shall to be nailed down by the boundary conditions.

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I shall emphasize again that there can be different levels of critical $V_i$ (or $V_i(x)$ if you prefer) depending on how the potential is set. You will have different regions (energy bands) accordingly.

Also note that in most case, like the standard harmonic oscillator (which is infinitely wide), your total energy is never going to exceed the potential so the coefficients are complex. However, if you have a parabolic well of finite width (and flat outside, or lower back down outside), then there exists the band where $E-V>0$, where the particle "escapes the well" and can travel (and the coefficients are real).