Parity switching wave functions for a parity invariant hamiltonian?

[VortexLattice]

Hi guys, I'm reading Shankar and he's talking about the Variational method for approximating wave functions and energy levels.

At one point he's using the example $V(x) = λx^4$, which is obviously an even function. He says "because H is parity invariant, the states will occur with alternating parity".

I believe him because I remember this from the Harmonic oscillator and the infinite square well, and I see why in each of those examples they alternate, from a mathematical standpoint.

But is there a better physical explanation? Right now all I can really tell is, mathematically, for those two examples, the solutions have to have alternating parity. So how can he say this for sure of all even potentials?

Thanks!

 

[VortexLattice]

Anyone? I'm very curious.

 

[Bill_K]

See Messiah, Vol I, Chapter III, "One-Dimensional Quantized Systems", Sect 12, "Number of Nodes of Bound States". If one arranges the eigenstates oin the order of increasing energies, the eigenfunctions likewise fall in the order of increasing number of nodes; the nth eigenfunction has (n-1) nodes between each of which the following eigenfunctions all have at least one node.

Regarding parity, each eigenfunction is either even or odd. If you have an even/odd number of nodes, you must have an even/odd eigenfunction.