DEMO: Perturbation Theory

1. Energy Levels¶

• The unperturbed energy levels are given by:

  $$E_n^{(0)} = \frac{n^2 \pi^2 \hbar^2}{2 m L^2}, \quad n = 1, 2, 3, \dots$$

  (1)

• The first-order correction to the energy is computed as:

  $$E_n^{(1)} = \langle \psi_n^{(0)} | V | \psi_n^{(0)} \rangle = \int_0^L \psi_n^{(0)}(x) V(x) \psi_n^{(0)}(x) \, dx$$

  (2)

• The second-order correction to the energy accounts for interactions with other states:

  $$E_n^{(2)} = \sum_{m \neq n} \frac{|\langle \psi_m^{(0)} | V | \psi_n^{(0)} \rangle|^2}{E_n^{(0)} - E_m^{(0)}}$$

  (3)

• In the left panel, the energy levels are plotted for:

  • $E_n^{(0)}$: Unperturbed energy (blue line).

  • $E_n^{(0)} + E_n^{(1)}$: First-order corrected energy (orange line).

  • $E_n^{(0)} + E_n^{(1)} + E_n^{(2)}$: Second-order corrected energy (green line).

2. Wavefunctions¶

• The unperturbed wavefunction for state $n$ is:

  $$\psi_n^{(0)}(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right)$$

  (4)

• The first-order correction adds contributions from other states:

  $$\psi_n^{(1)}(x) = \sum_{m \neq n} \frac{\langle \psi_m^{(0)} | V | \psi_n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} \psi_m^{(0)}(x)$$

  (5)

• The second-order correction further refines the wavefunction by including higher-order terms:

  $$\psi_n^{(2)}(x) = \sum_{m \neq n} \sum_{k \neq m} \frac{\langle \psi_k^{(0)} | V | \psi_m^{(0)} \rangle \langle \psi_m^{(0)} | V | \psi_n^{(0)} \rangle}{(E_n^{(0)} - E_m^{(0)})(E_m^{(0)} - E_k^{(0)})} \psi_k^{(0)}(x)$$

  (6)

• In the right panel, the wavefunctions are plotted for:

  • $\psi_n^{(0)}(x)$: Unperturbed wavefunction (blue line).

  • $\psi_n^{(0)}(x) + \psi_n^{(1)}(x)$: First-order corrected wavefunction (orange line).

  • $\psi_n^{(0)}(x) + \psi_n^{(1)}(x) + \psi_n^{(2)}(x)$: Second-order corrected wavefunction (green line).

Interactive Features¶

• Magnitude of the Perturbation ($a$):

  • Increasing $a$ increases the influence of the perturbing potential, making corrections to the energy levels and wavefunctions more pronounced.

• Quantum Number ($n$):

  • Higher $n$ states exhibit larger deviations due to their higher energy levels and larger overlaps with the perturbing potential.

Define the problem¶

First-order correction¶

Second-order correction¶

Visualizing corrections¶

Insights into Molecular Orbtials and Bonding from pertrubation theory perspective¶

Roles of Bonding and Antibonding:

• Bonding energy $E_-$: Stabilized by $-|H_{AB}|^2 / \Delta E$.

• Antibonding energy $E_+$: Destabilized by $+|H_{AB}|^2 / \Delta E$.