Exploring perturbation theory: examples
To conclude my journey with time independent perturbation theory (the previous posts are here and here), I now present an in-depth exploration of how constant and linear perturbations impact quantum systems. This discussion builds on concepts and introduces new computational strategies for dealing with perturbed quantum systems.
Constant perturbation in quantum mechanics
In cases of constant perturbation, the Hamiltonian perturbation \mathbf{H}_p is defined as a constant E_c. The first-order correction to the energy is simply the expectation value of \mathbf{H}_p within the state, yielding:
E^{(1)} = \langle \psi_n | \mathbf{H}_p |\psi_n\rangle = E_c
The fascinating aspect here is that the correction to the wavefunction for the first order vanishes, indicating no change in the state’s structure at this order. It’s a reminder of how symmetrical and well-behaved quantum systems can sometimes be under straightforward disturbances.
def first_order_correction(E_c):
return E_c # Returns the energy correction as it equals the constant perturbation
Exploring linear perturbations
When introducing a linear perturbation, such as a potential varying linearly with position, we face a more complex scenario. Consider a perturbation represented by f(\xi-\frac{1}{2}) in a symmetric quantum well. The first order energy correction, surprisingly, results in zero due to the symmetrical properties of the wave functions and the odd nature of the perturbation across the well.
E^{(1)} = 0
However, the coefficients for the correction to the wavefunction are non-trivial and are obtained by integrating over the product of eigenfunctions, reflecting the orthogonality and interaction between different states.
def wavefunction_correction(i, n, psi_n, E_n, E_i):
# Integration for wavefunction correction coefficient
integral = 0 # Assume computation of integral
a_i = (2 * f * integral) / (E_n - E_i)
return a_i
Insights from second-order corrections
Second-order corrections provide further insights into the system’s response to perturbations. These corrections often reveal subtle and intricate dependencies on the properties of both the perturbation and the unperturbed system. For instance, the correction for energy can be expressed as:
E^{(2)} = \sum_{i\ne n} \frac{\left| \langle \psi_i | \mathbf{H}_p | \psi_n\rangle \right|^2}{E_n - E_i}
This formula shows that the perturbation’s influence accumulates over all other states, weighted by their energy differences from the state considered. The second-order correction is crucial for understanding long-term stability and shifts in quantum systems.
def second_order_energy_correction(psi_n, states):
E_n = states[psi_n-1]['energy']
total_correction = 0
for state in states:
if state['index'] != psi_n:
contribution = (state['matrix_element']**2) / (E_n - state['energy'])
total_correction += contribution
return total_correction
For more insights into this topic, you can find the details here.