Smoothness of W matrix elements and vanishing of W(t) at infinity

[Sam Wong]

The Quantum Theory of Fields, Steven Weinberg, P.120

above 3.3.24, it says,

If the matrix elements of W between H0 - eigenstates are sufficiently smooth functions of energy, then matrix elements of W(t) between smooth super-positions of energy eigenstates vanish for t→±∞

I can't get this. Could anyone please show this explicitly?

 

[Bill_K]

Well I'm not sure, but here's a stab.. Weinberg has W(t) ≡ e^iH₀t W e^-iH₀t

Take the matrix elements of both sides, between energy eigenstates Ψ_α and Ψ_β.

(Ψ_α,W(t)Ψ_β) = e^iE_αt (Ψ_α,WΨ_β) e^-iE_βt = e^{i(E_α - E_β)t} (Ψ_α,WΨ_β)

Now what he wants us to look at is matrix elements between smooth superpositions of energy eigenstates. So, um, take ∫∫f₁(E_α) f₂(E_β) dE_α dE_β of both sides, where f₁, f₂ are arbitrary smooth functions. The point now is that as t → ±∞ the RHS is basically the high-frequency limit of a Fourier transform. But the Fourier transform of a sufficiently smooth function goes to zero at ω = ±∞, so we have to conclude that the RHS is zero in the limit, and hence so is the LHS!

 

[Sam Wong]

Thanks for your reply!

I was not sure about what he meant by smooth superposition.

I think your understanding make sense.

 

[Sam Wong]

Well I'm not sure, but here's a stab.. Weinberg has W(t) ≡ e^iH₀t W e^-iH₀t

Take the matrix elements of both sides, between energy eigenstates Ψ_α and Ψ_β.

(Ψ_α,W(t)Ψ_β) = e^iE_αt (Ψ_α,WΨ_β) e^-iE_βt = e^{i(E_α - E_β)t} (Ψ_α,WΨ_β)

Now what he wants us to look at is matrix elements between smooth superpositions of energy eigenstates. So, um, take ∫∫f₁(E_α) f₂(E_β) dE_α dE_β of both sides, where f₁, f₂ are arbitrary smooth functions. The point now is that as t → ±∞ the RHS is basically the high-frequency limit of a Fourier transform. But the Fourier transform of a sufficiently smooth function goes to zero at ω = ±∞, so we have to conclude that the RHS is zero in the limit, and hence so is the LHS!

I read your explanation again and I think your explanation proves that the matrix element of W(t) is diagonal only. It is not guaranteed vanishing for α=β