Question

A two-level quantum system has energy eigenvalues \( E_1 \) and \( E_2 \). A perturbing potential \( H' = \lambda \Delta \sigma_x \) is introduced, where \( \Delta \) is a constant having dimensions of energy, \( \lambda \) is a small dimensionless parameter, and \( \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \).

The magnitudes of the first and the second order corrections to \( E_1 \) due to \( H' \), respectively, are:

• A. 0 and \( \frac{\lambda^2 \Delta^2}{|E_1 - E_2|} \)
• B. \( |\lambda \Delta|^2 \) and \( \frac{\lambda^2 \Delta^2}{|E_1 - E_2|} \)
• C. \( |\lambda \Delta| \) and \( \frac{\lambda^2 \Delta^2}{|E_1 - E_2|} \)
• D. 0 and \( \frac{1}{2} \lambda^2 \Delta^2 \left| E_1 - E_2 \right| \)

Hint:

In perturbation theory, the first-order correction is zero if the perturbing Hamiltonian does not couple the state to itself. The second-order correction involves the energy difference between the states.

Answer 1

The Correct Option is A

Step 1: The first-order energy correction in perturbation theory is given by the expectation value of the perturbing Hamiltonian in the unperturbed state. Since \( \sigma_x \) connects the two states, the first-order correction to the energy is zero. 
Step 2: The second-order correction is non-zero and is given by the formula: \[ E_1^{(2)} = \frac{\lambda^2 \Delta^2}{|E_1 - E_2|} \] This is the second-order energy correction due to the perturbation \( H' = \lambda \Delta \sigma_x \).