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Homework Statement
Show that for a 1d potential V (-x)=-V (x), the eigen functions of the Schrödinger equation are either symmetric/ anti-symmetric functions of x.
Homework Equations
The Attempt at a Solution
I really don't know how to do it for odd potential.
Let me show you how I am doing it for even potential.
V (x)=V (-x)
- (
Making x to -x transformation we get
- (
where i use V (-x)=V (x)
Comparing (1) and (2) we see that ψ (x) and ψ(-x) eigenfunctions belongs to the same
energy E.
For a non degenerate state, ψ (-x ) must be a multiple of ψ (x):
Ψ (-x) = λψ (x).
Clearly ψ (x)= λψ (-x)=λ2ψ (x).
λ2=1 or λ=+/- 1
So ψ(-x)=+/- ψ(x)
Now if i use odd potential in (2) eigenfunctions no longer belong to same energy E.
The hamiltonian becomes weird for negative potential.
Please help.