Exploring first and second order corrections in perturbation theory
To continue on my journey on perturbation theory here, I have explored the first and second order corrections in quantum mechanics with a focus on their mathematical formulations and applications. This exploration is crucial for those interested in the theoretical aspects of quantum physics and seeks to provide an understanding of how perturbations affect systems at a quantum level.
First order correction
The first step in this analysis involves computing the first order energy correction, which is given by:
E^{(1)} = \langle \psi_n | \mathbf{H}_p |\psi_n\rangle
This formula illustrates the adjustment in energy due to the perturbation, \mathbf{H}_p, and is critical for understanding how external influences alter the quantum state.
Further, the correction to the wavefunction |\phi^{(1)} \rangle at the first order can be expanded in the basis set {|\psi_i\rangle} as follows:
|\phi^{(1)} \rangle = \sum_{i \ne n} a_i^{(1)} | \psi_i \rangle, \quad a_i^{(1)} = \frac{\langle \psi_i | \mathbf{H}_p | \psi_n\rangle}{E_n - E_i}
This expression highlights the changes in the wavefunction’s components outside the initial state, thereby showing how the system’s quantum state evolves due to perturbations.
Second order correction
Moving to the second order corrections, I focused on both the energy and the wavefunction adjustments. The second order energy correction is calculated by:
E^{(2)} = \langle \psi_n | \mathbf{H}_p | \phi^{(1)}\rangle
This calculation demonstrates the additional shifts in energy resulting from the first order changes in the wavefunction.
The correction to the wavefunction |\phi^{(2)}\rangle at the second order is:
|\phi^{(2)}\rangle = \sum_{i \ne n} a_i^{(2)} |\psi_i\rangle, \quad a_i^{(2)} = \sum_{i \ne n} \frac{a^{(1)}\langle \psi_i |\mathbf{H}_p |\psi_i\rangle}{E_n - E_i} - \frac{E^{(1)} a_i^{(1)}}{E_n - E_i}
This aspect is particularly interesting as it underscores the complex interplay between different order corrections and their cumulative effect on the quantum state.
Throughout this exploration, my aim was to elucidate the theoretical underpinnings of perturbation theory in quantum mechanics, making the content accessible and informative. The mathematical rigor involved in deriving these corrections provides a robust framework for understanding more complex quantum systems and their behavior under various perturbative influences.
For more insights into this topic, you can find the details here.