A quantum system is described by the Hamiltonian
\(H=-J \sigma_z+\lambda(t) \sigma_x\),
where σ (i = x, y, z) are Pauli matrices, J and λ are positive constants (J >> λ) and
\(\lambda(t)=\left\{\begin{array}{lll} 0 & \text { for } & t<0 \\ \lambda & \text { for } & 0
At t < 0, the system is in the ground state. The probability of finding the system in the excited state at t >> T, in the leading order in λ is
1
\(\frac{\lambda^2}{8 J^2} \sin ^2 \frac{J T}{h}\)
2
\(\frac{\lambda^2}{J^2} \sin ^2 \frac{J T}{\hbar}\)
3
\(\frac{\lambda^2}{4 J^2} \sin ^2 \frac{J T}{h}\)
4
\(\frac{\lambda^2}{16 J^2} \sin ^2 \frac{J T}{\hbar}\)