Perturbation Theory Energy Shift Calculator

Perturbation Theory Energy Shift Calculator computes first-order energy correction for quantum systems, aiding quantum mechanics and atomic physics studies.

Unperturbed Energy (eV):

Perturbation Strength (λ):

Matrix Element (eV):

Formulas Used in Perturbation Theory Energy Shift Calculator

The calculator uses the following formulas for first-order non-degenerate perturbation theory:

First-Order Energy Correction:

\\[ E_n^{(1)} = \lambda \langle n | V | n \rangle \\]

Perturbed Energy:

\\[ E_n = E_n^{(0)} + E_n^{(1)} \\]

Where:

\\( E_n^{(1)} \\): First-order energy correction (eV)
\\( \lambda \\): Perturbation strength (dimensionless)
\\( \langle n | V | n \rangle \\): Matrix element of perturbation operator (eV)
\\( E_n^{(0)} \\): Unperturbed energy (eV)
\\( E_n \\): Perturbed energy (eV)

Example Calculations

Example 1: Small Perturbation, Low-Energy State

Input: Unperturbed Energy = 10 eV, Perturbation Strength = 0.1, Matrix Element = 0.5 eV

\\[ E_n^{(1)} = \lambda \langle n | V | n \rangle = 0.1 \cdot 0.5 = 0.05 \, \text{eV} \\] \\[ E_n = E_n^{(0)} + E_n^{(1)} = 10 + 0.05 = 10.05 \, \text{eV} \\]

Result: First-Order Energy Correction: 0.05 eV, Perturbed Energy: 10.05 eV

Example 2: Negative Matrix Element

Input: Unperturbed Energy = 5 eV, Perturbation Strength = 0.2, Matrix Element = -1 eV

\\[ E_n^{(1)} = 0.2 \cdot (-1) = -0.2 \, \text{eV} \\] \\[ E_n = 5 + (-0.2) = 4.8 \, \text{eV} \\]

Result: First-Order Energy Correction: -0.2 eV, Perturbed Energy: 4.8 eV

Example 3: Large Perturbation Strength

Input: Unperturbed Energy = 20 eV, Perturbation Strength = 0.5, Matrix Element = 2 eV

\\[ E_n^{(1)} = 0.5 \cdot 2 = 1 \, \text{eV} \\] \\[ E_n = 20 + 1 = 21 \, \text{eV} \\]

Result: First-Order Energy Correction: 1 eV, Perturbed Energy: 21 eV

Formulas Used in Perturbation Theory Energy Shift Calculator

Example Calculations

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