Solution of Schrodinger equation for free electron

[u_know_who]

Hi, i am beginning elementary Quantum Mechanics as my course. While studying one question arise in my mind :

In the solution of Schrodinger wave equation there are two parts.

ψ=A*exp(jKx) + B*exp(-jkx). (for confined electron)

But when dealing with free electron the solution is of the following pattern :

ψ = A*exp(jkx) or ψ = B*exp(-jkx).

Can anyone tell me the reason behind that?

 

[raymo39]

the power of the exponential describes the direction of the evolution.
here what you have is a solution that either describes a particle traveling in the positive or negative x direction. This is fine, because there is no confining potential, so the solution only travels one way through space. In the previous solution the particle is in some potential, so its motion must be considered in both dircetions.

 

[dextercioby]

There are ALWAYS 2 linearly independent solutions to the Schrödinger equation, because it's a second order ODE. Which one "survives" and describes quantum states is a matter of rightfully implementing boundary/limit conditions which are necessary to make the hamiltonian or the momentum operator (essentially) self-adjoint.

 

[u_know_who]

Thank you raymo and dextercioby.