Hello!
"DEGENERATE EIGENVALUES IN QUANTUM MECHANICS"
Remember that we know the Schrodinger equation
We write a time-independent potential
We have eigenfunctions or eigenvectors
We have also eigenvalues
The solutions of such a time-independent Schrodinger equation are eigenvectors and eigenvalues
We  have the same "k" for two different wave functions
In this case, for each eigenvalue we can associate two different wavefunctions which propagate to the right or to the left
One energy and two different wavefunctions, that's we have got
When we have two different  wavefunctions (or more) for a single eigenvalue, we can say such an eigenvalue is degenerate
The eigenvalue is degenerated
The energy is degenerated
We have two different wave functions and an unique energy
What does it happen with momentum?
"P" is the momentum operator
We substitute "P" by its value
We rearrange
We may write this
We substitute the momentum operator by its value and rearrange the formula for both solutions
We have "+k" to the right
Now we apply "P" to the other solution
We substitute again
We rearrange again
Now we have "-k" to the left
The momentum operator is able to distinguish both solutions which have the same energy
The momentum operator breaks the degeneracy
It breaks the degeneration of the energy operator
We can distinguish between both states
We have the "+k" and "-k" quantum numbers
The plane waves are not normalizable
We integrate to calculate the norm
The norm diverges
Plane waves are artificial elements
The "sin" and "cos" solutions that we found for the infinite quantum wells are made from plane waves
Plane waves are useful tools
The plane waves are momentum eigenvalues and hamiltonian eigenvalues at the same time
Bye Bye
