Hamiltonian Symmetry: Implications for Negative Energies and Potential

[Sangoku]

Hi, let be a Hamiltonian so its energies satisfy E(n)=-E(-n) for every n >0

then my question is does this imply certain symmetry of potential ?? i believe that V(x)=-V(-x) so in absolute value positive and negative energies have the same value.

 

[Nesk]

I assume that by n you mean the quantum state number so that n=0 is the ground state. In that case, what does the negative number in E(-n) mean?

 

[Sangoku]

Here 'n' is some kind of label for the Energy levels for example:

E_{-2} = -0.6784356

E_{-1} = -0.000456

(if n is negative this means that energies will be negatives)

E_{1} = 1.23456

E_{2} =4.5676868

(if n is positive energies will be positive)

E_{0} = 0.4 (there is no ground state since if there are negative energies this means that operator H is 'unbounded' )