Why energy level of one dimension problem is descrete?

[sadegh4137]

in the book of quantum mechanics concept and application by Zettili, chapter 4 write a theorem, that is:

in one dimensional problem the energy level of a bound state system are discrete and not degenerate.

i can not prove this theorem.

can you help me to do this!

 

[DrDu]

First there are exceptions, e.g. when you have a symmetric potential with an infinitely high barrier.
In that case you can find two degenerate bound states.
Generally, you can refrain to the theory of differential equations.
For any energy the Schroedinger equation, which is a differential equation of second order, has two solutions (which may not be normalizable), so there can be at most two degenerate bound states.
In the case of regular (not infinite) potentials, fiddling around with the Wronskian, you can show that at most one of the two solutions can be a bound state, the other solution will diverge and not be in the Hilbert space.

 

[sadegh4137]

i can't understand your answer!

we always find two solution for Schrödinger equation.
for example well potential has two answer and the others!

please explain it for me!