What is the meaning of En^+ or En^-

[devoured_elysium]

I see in a lot of places things like

E_n⁺

where E stands for the energy, but I don't get what that upperscript + could be. I see both + and - in the upperscript.

Thanks

 

[Bob_for_short]

Without the proper context... Maybe they are energies of propagating waves in positive and negative direction (different waveguided modes)?

 

[devoured_elysium]

Without the proper context... Maybe they are energies of propagating waves in positive and negative direction (different waveguided modes)?

The exercise is the following(note that I am not asking for a solution to the exercise, I just want to know what the + stands for!):

"An operator U is said to be unitary if U^+ U = U^- U = 1. Prove that if H is hermitian, then exp(iH) is unitary"

(Can't find at the moment something involving the En^+..that's because I saw it as a resolution to an exercise, not in the problem itself.

 

[kanato]

The + (usually dagger) in that context is the Hermitian conjugate, which, for a matrix, is the complex conjugate of the transpose of a matrix. A matrix is Hermitian if $$H = H^\dagger$$ and unitary if $$U^{-1} = U^\dagger$$.

 

[devoured_elysium]

The + (usually dagger) in that context is the Hermitian conjugate, which, for a matrix, is the complex conjugate of the transpose of a matrix. A matrix is Hermitian if $$H = H^\dagger$$ and unitary if $$U^{-1} = U^\dagger$$.

I had thought of that, but I can swear it is not a dagger, it is a +!. Although looking in the resolution of the exercise I'd say that in that case it is indeed a dagger. I'll give you another one:

"Calculate the probability that an energy measurement yields the ground state energy; the energy of the first excited state"

The resolution then starts with:

Minimum energy -> n = 1

P(n=1) = P1 = |A₁|
P₁⁺=|A₁⁺|², P₁^-=|A₁^-|²

$$\Psi$$(x,t) = $$\sum$$A_n⁺U_n⁺(x)Exp(-i/h * E_n⁺t) + $$\sum$$A_n^-U_n^-(x)Exp(-i/h * E_n^-t)

U_n⁺=$$\sqrt{2/a}$$ Cos(2n-1)PI x / a), U_n^-=$$\sqrt{2/a}$$ Cos(2n)PI x / a)