What Are Negative Energies in System ψ?

[chill_factor]

Lets say I have a system ψ with a Hamiltonian matrix H and energy eigenvalues E. Just a general system, with no particular basis given.

When I solve the eigenvalue equation for H, and get zero or negative numbers or zero for E, is that physical? If it is not physical, do the negative numbers or zero energy mean anything?

 

[The_Duck]

It doesn't mean anything if some or all of the energy eigenvalues are negative. The zero of energy is arbitrary. That is, we can decide to add 5 to all energies (by replace H with H+5) and no observable differences will arise from this. Only differences in energies are meaningful. So we needn't be concerned if some energies come out negative. If we like, we can measure all energies relative to the ground state, the state of minimum energy. Then all energies will be nonnegative.

The only physical requirement is that a ground state has to exist. That is, the set of energy eigenvalues has to be bounded from below.

 

[chill_factor]

It doesn't mean anything if some or all of the energy eigenvalues are negative. The zero of energy is arbitrary. That is, we can decide to add 5 to all energies (by replace H with H+5) and no observable differences will arise from this. Only differences in energies are meaningful. So we needn't be concerned if some energies come out negative. If we like, we can measure all energies relative to the ground state, the state of minimum energy. Then all energies will be nonnegative.

The only physical requirement is that a ground state has to exist. That is, the set of energy eigenvalues has to be bounded from below.

Thank you.

If there were just a limited amount of states, say, 3, then if 1 was negative and the others positive or zero, it doesn't matter that its negative, it'll just be the ground state.

 

[tom.stoer]

A well-known example is the hydrogen atom where the bound states are at E<0 and the continuum of free or scattering states is at E>0

 

[chill_factor]

yes, very true; the hydrogen atom is analogous to a finite potential well in that sense with bound states at E< 0 and scattering states at E > 0.

However, we solve that problem for a position basis. Is this true for a general basis?

I ask this because I was given a problem in matrix mechanics with a general basis which asked "what states are possible" and I got negative energies.

 

[tom.stoer]

Of course it is true for a general basis.

Suppose you are able to construct the energy eigenbasis exactly, like for the qm harmonic oscillator.

The eigenfunctions U_E(a) in a specific basis 'a' are nothing else but a representation, a projection on the a-basis. If you have a complete spectrum with energy values {E} and the corresponding basis vector {|E>} consisting of a discrete bound-state spectrum and a continuous scattering-state problem such a state in a-basis can be written as

u_E(a) = \langle a|E\rangle

If you chose 'a' = 'x' you get the position space basis for the energy eigenvectors; if you chose 'a' = 'p' you get the momentum space basis for the energy eigenvectors. So for the hydrogen atom you will find

u_E(x) = \langle x|E\rangle \to u_{nlm}(r,\theta,\phi)

Chosing the momentum space the wave functions will look different, but the underlying spectrum of the operator is not affected.

 

[chill_factor]

Thanks a lot! I think I understand now.