Calculation of Energy Correction in Non-Degenerate Stationary State Perturbation Theory

Introduction

Perturbation theory provides a framework for approximating the energy levels of a quantum system by expanding the Hamiltonian in terms of a small perturbation. In non-degenerate stationary state perturbation theory, the perturbation does not cause degeneracy in the energy levels, allowing for a straightforward calculation of energy corrections. This approach is particularly useful in scenarios where the perturbation is weak and the original Hamiltonian is well-defined. The energy correction, denoted as $ \Delta E $, quantifies the deviation from the original energy levels due to the perturbation.

The core challenge in non-degenerate perturbation theory lies in the derivation of the first-order energy correction, which is given by the equation:
$$ \Delta E^{(1)} = \langle \psi_0 | \hat{H}' | \psi_0 \rangle, $$
where $ \psi_0 $ is the unperturbed ground state wavefunction, and $ \hat{H}' $ is the perturbation operator. This formula assumes that the perturbation does not mix states with different energies, a condition satisfied in non-degenerate cases.

Mathematical Framework

In non-degenerate perturbation theory, the eigenstates of the unperturbed Hamiltonian $ \hat{H}_0 $ are non-degenerate, meaning each energy level corresponds to a unique state. The perturbation $ \hat{H}' $ introduces small variations to the system, and the wavefunctions $ \psi_n $ (the unperturbed states) are used to construct the perturbed wavefunctions $ \psi_n^{(1)} $ and $ \psi_n^{(2)} $, which are the first-order and second-order corrections, respectively.

The energy correction $ \Delta E^{(1)} $ is calculated by projecting the perturbation onto the unperturbed state:
$$ \Delta E^{(1)} = \sum_{n \neq 0} \frac{|\langle \psi_0^{(n)} | \hat{H}' | \psi_0^{(n)} \rangle|^2}{E_0^{(n)} - E_0^{(0)}}. $$
This formula arises from the Rayleigh-Ritz ratio, which ensures that the perturbation is applied to the unperturbed states. The denominator $ E_0^{(n)} - E_0^{(0)} $ accounts for the energy difference between the unperturbed and perturbed states, while the numerator captures the squared magnitude of the perturbation's projection onto the unperturbed state.

Derivation Details

The derivation of the first-order energy correction begins with the assumption that the perturbation $ \hat{H}' $ is small and does not induce transitions between states. The unperturbed states $ \psi_0^{(n)} $ are used to construct the perturbed states $ \psi_n^{(1)} $, which satisfy the Schrödinger equation:
$$ \left( \hat{H}_0 + \hat{H}' \right) \psi_n^{(1)} = E_n^{(0)} \psi_n^{(1)}. $$
The energy correction $ \Delta E^{(1)} $ is determined by solving the secular equation:
$$ \det \left( \begin{bmatrix} E_0^{(0)} - E_n^{(0)} & \langle \psi_0^{(1)} | \hat{H}' | \psi_0^{(1)} \rangle \ \langle \psi_0^{(1)} | \hat{H}' | \psi_0^{(1)} \rangle & E_0^{(0)} - E_n^{(0)} \end{bmatrix} \right) = 0. $$
This equation ensures that the perturbed states are orthogonal and non-degenerate. The first-order energy correction is then obtained by solving for the eigenvalues of the matrix, which simplifies to the formula:
$$ \Delta E^{(1)} = \langle \psi_0^{(0)} | \hat{H}' | \psi_0^{(0)} \rangle. $$
This result highlights the direct relationship between the perturbation and the energy correction, emphasizing that the correction is proportional to the expectation value of the perturbation in the unperturbed state.

Application and Examples

The non-degenerate perturbation theory is widely applied in quantum chemistry and solid-state physics to calculate energy levels of complex systems. For instance, in the hydrogen atom, the first-order energy correction due to the electric dipole perturbation is calculated using the formula above. The dipole moment $ \hat{d} $ is the perturbation, and the unperturbed state is the ground state $ \psi_0 $. The energy correction $ \Delta E^{(1)} $ is found by evaluating $ \langle \psi_0 | \hat{d} | \psi_0 \rangle $, which yields the shift in the energy levels due to the perturbation.

Another example is the Stark effect, where the perturbation is the electric field $ \hat{E} $, and the energy correction is calculated using the same framework. The first-order correction is proportional to the square of the dipole moment, demonstrating the importance of the perturbation's strength and the system's symmetry. These examples illustrate how the non-degenerate perturbation theory provides a systematic approach to calculating energy corrections in systems where the perturbation does not mix states.

Comparison with Degenerate Cases

In contrast to non-degenerate perturbation theory, degenerate cases involve states with the same energy, leading to more complex corrections. When the unperturbed states are degenerate, the perturbation can mix different states, resulting in a more intricate eigenvalue problem. The energy correction in degenerate cases is typically calculated using the degenerate perturbation theory, which involves diagonalizing the perturbation matrix within the degenerate subspace. The first-order correction in degenerate cases is given by:
$$ \Delta E^{(1)} = \frac{1}{2} \sum_{n \neq 0} \frac{|\langle \psi_0^{(n)} | \hat{H}' | \psi_0^{(n)} \rangle|^2}{E_0^{(n)} - E_0^{(0)}}. $$
This formula accounts for the mixing of states and the resulting energy shifts, which are more pronounced in degenerate systems. The non-degenerate case simplifies the calculation by avoiding the complexities of state mixing, making it more efficient for practical applications.

Conclusion

The calculation of energy correction in non-degenerate stationary state perturbation theory is a fundamental aspect of quantum mechanics, providing a robust framework for approximating energy levels in weakly perturbed systems. By leveraging the unperturbed states and the first-order perturbation formula, the energy correction is directly determined, ensuring accuracy and efficiency. The application of this theory in various physical systems, such as the hydrogen atom and the Stark effect, underscores its practical relevance. While degenerate cases require more advanced techniques, the non-degenerate approach remains a cornerstone of perturbation theory, offering a clear and systematic method for energy correction calculations. Future research may explore higher-order corrections and the integration of perturbation theory with other quantum mechanical principles to further enhance the accuracy of energy level predictions.