Relation between potential and bound states in Schrodinger equation.

[arroy_0205]

Suppose I have Schroedinger equation in the form:
$$-u''(x)+V(x)u(x)=Eu(x)$$
The potential is such that as |x| -> Infinity, V(x) reaches a constant positive value. In this case can we have bound state/plane wave solutions for u(x) with E > 0 ?

 

[Meir Achuz]

You can have bound state solutions with E>0.
Just take any V that-->0 and add any positive constant to it.
You can't have plane wave solutions if V depends on x.

 

[denian]

The relationship between potential and bound states in the Schrodinger equation is crucial for understanding the behavior of quantum systems. In this case, the potential is a key factor in determining whether or not bound state solutions with positive energy can exist.

The Schrodinger equation describes the behavior of a quantum particle in a potential field. The potential, represented by V(x), determines the energy levels and the allowed states of the particle. In a system where the potential approaches a constant positive value as |x| -> Infinity, it is possible to have bound state solutions with positive energy.

Bound states are characterized by the confinement of a particle within a finite region of space. In the Schrodinger equation, this confinement is represented by the potential well created by V(x). When the potential approaches a constant positive value, it creates a finite well that can trap the particle and allow for bound state solutions.

On the other hand, plane wave solutions are characterized by a constant energy and a wave function that extends to infinity. In a system with a potential that approaches a constant positive value, it is still possible to have plane wave solutions for u(x) with E > 0. However, these solutions would not be bound states as the particle is not confined within a finite region.

In summary, the potential plays a crucial role in determining the existence and behavior of bound states and plane wave solutions in the Schrodinger equation. In a system with a potential that approaches a constant positive value, it is possible to have both bound states and plane wave solutions with positive energy, but they would have different characteristics.