Is the particle in a 1-D infinite well doubly degenerate?

[dyn]

Hi. I'm confused about degeneracy in I-D As far as I understand it ( and please tell me if I'm wrong ) ; a free particle is doubly degenerate with a continuous energy spectrum with eigenfunctions e^ikx and e^-ikx. A particle on a ring in I-D is doubly degenerate but this time the energy is quantized.
My main question concerns a particle in a 1-D infinite well with infinite walls at x=0 and x=a. The eigenfunction is given by ψ = sin (nπx/a) where n = 1,2,3,... These eigenfunctions are non-degenerate as n only takes positive values but sin kx can be written as sin kx = ( e^ikx - e^-ikx) / 2i which is a superposition of 2 waves traveling in opposite directions. Would this not make the particle in a box doubly degenerate ?

 

[hilbert2]

There's not much room for choice of ##A## and ##B## if you want the function ##\psi (x) = Ae^{ikx}+Be^{-ikx}## to be zero at x=0 and x=a. For most values of k it's not possible at all (except if A and B are both zero).

 

[Nugatory]

but sin kx can be written as sin kx = ( e^ikx - e^-ikx) / 2i which is a superposition of 2 waves traveling in opposite directions. Would this not make the particle in a box doubly degenerate ?

Neither of those functions are solutions to the eigenvalue equation for the particle-in-a-box Hamiltonian.

 

[dyn]

The solution to the particle in a box Hamiltonian is sin kx where k = nπ /a and n takes positive integer values. This shows that the energy eigenvalues are non-degenerate. But the part that confuses me is that sin kx can be written as a linear combination of e^ikx and e^-ikx which implies doubly degenerate ?

 

[hilbert2]

The solution to the particle in a box Hamiltonian is sin kx where k = nπ /a and n takes positive integer values. This shows that the energy eigenvalues are non-degenerate. But the part that confuses me is that sin kx can be written as a linear combination of e^ikx and e^-ikx which implies doubly degenerate ?

The function ##e^{ikx}## is not zero for any value of x. Its absolute value is 1 everywhere.

 

[PeterDonis]

sin kx can be written as a linear combination of e^ikx and e^-ikx which implies doubly degenerate ?

No, because, as Nugatory said, ##e^{ikx}## and ##e^{-ikx}## are not solutions of the eigenvalue equation, whereas ##\sin kx## is. Writing a function that is a solution as a linear combination of functions that are not tells you nothing at all about degeneracy; degeneracy implies that you are talking only about functions which are solutions. For each ##k## there is only one such function, hence no degeneracy.

 

[hilbert2]

Yeah, even the harmonic oscillator ground state ##\psi (x) = Ae^{-kx^2}## can be written as ##\psi (x) = A\left(\frac{1}{2}e^{-kx^2}+\sin kx\right) + A\left(\frac{1}{2}e^{-kx^2}-\sin kx\right)## if you want to, but neither of the functions

##\psi (x) = \frac{1}{2}e^{-kx^2}+\sin kx##, or
##\psi (x) =\frac{1}{2}e^{-kx^2}-\sin kx##,

is a solution to the SHO Schrödinger equation.

 

[dyn]

Thanks everyone. Much appreciated !