Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 35B / ii

Table of contents
- Solution ii

Solution

The characteristic equation is

(E_{-} - E) (E_{+} - E) - λ^{2} = 0.

Hence the imaginary-coupled two-level Hamiltonian has exact energies

E_{L, H} = \frac{E _{-} + E _{+}}{2} \mp \frac{Δ ^{2}}{4} + λ^{2} .

Choose a real angle

θ

satisfying

tan (2 θ) = \frac{2 λ}{Δ}, θ \to 0 as λ \to 0.

Normalized exact eigenstates are

∣ L ⟩ = cos θ ∣ - ⟩ - i sin θ ∣ + ⟩, ∣ H ⟩ = - i sin θ ∣ - ⟩ + cos θ ∣ + ⟩ .

Since

θ = \frac{λ}{Δ} + O (λ^{3}), \frac{Δ ^{2}}{4} + λ^{2} = \frac{Δ}{2} + O (λ^{2}),

these states and energies reproduce exactly the linear-order perturbative results from part (i).

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version