Perturbation Theory Calculator

Unperturbed Energy E⁽⁰⁾ (eV or a.u.)

First-order matrix element ⟨ψ⁽⁰⁾|Ĥ′|ψ⁽⁰⁾⟩

Off-diagonal element ⟨ψ⁽⁰⁾|Ĥ′|ψ⁽¹⁾⟩ (for 2nd order)

Unperturbed energy of excited state E₁⁽⁰⁾

About the Perturbation Theory Calculator

The Perturbation Theory Calculator is a precision tool that instantly computes first- and second-order energy corrections using Rayleigh-Schrödinger perturbation theory—the gold standard in quantum mechanics. Perfect for students, researchers, and professors solving the time-independent Schrödinger equation for systems with small perturbations (e.g., Stark effect, Zeeman effect, anharmonic oscillators). Results are accurate to 6 decimal places. Explore more physics tools at Agri Care Hub.

What is Perturbation Theory?

Perturbation Theory approximates solutions to quantum systems where the Hamiltonian is split into a solvable part Ĥ₀ and a small perturbation Ĥ′:

Ĥ = Ĥ₀ + λĤ′

Energy and wavefunction are expanded in powers of λ:

E = E⁽⁰⁾ + λE⁽¹⁾ + λ²E⁽²⁾ + …
ψ = ψ⁽⁰⁾ + λψ⁽¹⁾ + λ²ψ⁽²⁾ + …

Formulas Used

First-order energy correction:

E⁽¹⁾ = ⟨ψ⁽⁰⁾|Ĥ′|ψ⁽⁰⁾⟩

Second-order energy correction:

E⁽²⁾ = Σₖ₌₁∞ |⟨ψₖ⁽⁰⁾|Ĥ′|ψ⁽⁰⁾⟩|² / (E⁽⁰⁾ − Eₖ⁽⁰⁾)

For a two-level system, this reduces to:

E⁽²⁾ = |H′₁₂|² / (E⁽⁰⁾ − E₁⁽⁰⁾)

Why This Calculator Matters

Manual second-order calculations involve infinite sums and complex algebra. This tool automates everything, validates inputs, and displays LaTeX-quality results—saving hours in quantum chemistry labs, university assignments, and research papers.

User Guidelines

Enter the unperturbed ground-state energy E⁽⁰⁾.
Input the first-order matrix element ⟨ψ⁽⁰⁾|Ĥ′|ψ⁽⁰⁾⟩.
For second-order, provide the off-diagonal element and the excited-state energy.
Click “Calculate Energy Corrections”.
Copy results for reports or publications.

When to Use This Tool

Solving Stark/Zeeman splitting problems
Computing polarizability of atoms
Analyzing anharmonic corrections in molecular vibrations
Teaching quantum mechanics courses

Real-World Examples

Hydrogen atom in electric field (Stark): E⁽²⁾ = −(9/2)a₀³ε²
Helium ground state: E⁽¹⁾ = 5/4 Z (a.u.)
Quantum harmonic oscillator with x⁴ term: E⁽²⁾ = 3ħ²λ/(4mω²)

Advantages

Instant, accurate, mobile-friendly. No software installation. SEO-optimized for “Perturbation Theory Calculator”.

Limitations

Assumes non-degenerate states and |λ| ≪ 1. For degenerate cases, use degenerate perturbation theory.

Conclusion

The Perturbation Theory Calculator brings textbook quantum mechanics to your browser in 3 seconds. Bookmark it for every QM problem. For more free tools, visit Agri Care Hub.

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