Perturbation Theory

Perturbation Theory: The Art of Almost-Perfect Answers

The Big Idea in One Sentence

When a problem is too hard to solve exactly, we solve a simpler version first, then add small corrections step by step.

🎭 The Universal Analogy: The Recipe Adjustment

Imagine you have a perfect recipe for chocolate cake. You’ve made it 100 times. You know exactly how it turns out.

Now, your friend asks: “What if we add just a tiny bit of cinnamon?”

You don’t need to start from scratch! You think:

• First guess: It will taste mostly like your regular cake
• Small correction: Plus a hint of cinnamon flavor
• Tiny adjustment: Maybe the texture changes slightly

This is perturbation theory. Start with what you know, then add small fixes.

🕐 Time-Independent Perturbation

What Does “Time-Independent” Mean?

The problem doesn’t change over time. It just sits there, steady and calm.

Real-life example: A guitar string vibrating at a fixed note. The note doesn’t change by itself.

The Setup

We have:

1. A simple problem we CAN solve (the unperturbed system)
2. A small extra thing that makes it harder (the perturbation)

Full Problem = Simple Problem + Small Extra Thing
      H      =      H₀        +       λH'

• H₀ = The easy part (we know the answer)
• H’ = The small disturbance
• λ = A tiny number (shows how small the disturbance is)

Example: Particle in a Box with a Bump

Simple problem: A ball bouncing in a flat-bottomed box.

Perturbation: Someone puts a small hill in the middle.

We already know how the ball moves in a flat box. The hill is small. So we figure out the small changes the hill causes.

🎯 Non-Degenerate Perturbation Theory

What is “Non-Degenerate”?

Degenerate means several things have the same energy.

Non-degenerate means each energy level is unique. No ties. No sharing.

Think of it like a race:

• Non-degenerate: Everyone finishes at different times. Clear 1st, 2nd, 3rd.
• Degenerate: Some runners finish at exactly the same time. A tie!

Why Does This Matter?

When there are no ties, the math is simpler. We know exactly which state to start with.

The Simple Formula

For the energy correction:

E_n = E_n⁰ + λ⟨n|H'|n⟩ + higher terms...

Translation:

• New energy = Old energy + (how much the disturbance affects this state)

📊 Perturbation Corrections

First-Order Correction

The first fix. The biggest adjustment.

Energy correction (1st order):

E_n^(1) = ⟨n|H'|n⟩

This says: “How much does the disturbance H’ affect state n directly?”

Example: If you add a small magnet near an atom, the first correction tells you how much the atom’s energy changes.

Second-Order Correction

The fine-tuning. Smaller but still important.

Energy correction (2nd order):

E_n^(2) = Σ |⟨m|H'|n⟩|² / (E_n⁰ - E_m⁰)
       m≠n

This says: “State n borrows a bit from all other states m.”

The Pattern

graph TD
    A["Original Energy E₀"] --> B["Add 1st Correction"]
    B --> C["Add 2nd Correction"]
    C --> D["Add 3rd Correction..."]
    D --> E["Closer and Closer to True Answer"]

Each correction is smaller than the last. Eventually, we’re close enough!

🎭 Degenerate Perturbation Theory

The Problem with Ties

Remember our race? Now imagine two runners finish at exactly the same time.

If you want to give out medals, you have a problem. Which one gets gold? Which gets silver?

In quantum mechanics, when two states have the same energy, we can’t tell them apart. The perturbation might mix them together!

The Solution

Before adding corrections, we must first figure out which combinations of the tied states are “stable.”

Step 1: Find all states with the same energy.

Step 2: Build a small matrix just for those states:

H' matrix = | ⟨1|H'|1⟩  ⟨1|H'|2⟩ |
            | ⟨2|H'|1⟩  ⟨2|H'|2⟩ |

Step 3: Find the eigenvalues of this matrix. These are the first-order energy corrections!

Example: Two Equal Springs

Imagine two identical springs attached to a wall. They vibrate at the same frequency.

Now connect them with a weak rubber band.

The rubber band slightly couples them. One combination vibrates together (in phase). The other vibrates opposite (out of phase).

The degeneracy is lifted! They now have different energies.

graph TD
    A["Two States: Same Energy"] -->|Add Perturbation| B{Which Combo is Stable?}
    B --> C["In-Phase Mode: Lower Energy"]
    B --> D["Out-of-Phase Mode: Higher Energy"]

⏰ Time-Dependent Perturbation Theory

Now Things Are Changing!

The perturbation turns on and off. It wiggles. It oscillates.

Example: Shining a light on an atom. The light wave oscillates. The electric field goes up and down.

The Big Question

If a system starts in one state, what’s the probability it jumps to another state?

Transition Amplitude

The probability of jumping from state |i⟩ to state |f⟩:

c_f(t) = -(i/ℏ) ∫₀ᵗ ⟨f|H'(t')|i⟩ e^(iω_fi t') dt'

Where: ω_fi = (E_f - E_i)/ℏ

Simple Example: A Sudden Push

Imagine a child on a swing. You give one quick push.

• If you push at the right moment: Big swing!
• If you push at the wrong moment: Weak response.

The timing matters. The frequency of your push compared to the swing’s natural frequency determines the result.

Sinusoidal Perturbation

If the perturbation oscillates like H’(t) = V cos(ωt):

graph TD
    A["Light Wave Hits Atom"] --> B{Does Frequency Match?}
    B -->|Yes: Resonance| C["Strong Transition"]
    B -->|No: Off-Resonance| D["Weak Effect"]

Maximum effect happens when the driving frequency ω matches the transition frequency ω_fi.

✨ Fermi’s Golden Rule

The Crown Jewel

This is the most famous result. It tells us the transition rate when:

1. The perturbation keeps going for a long time
2. There are many final states to jump into

The Formula

Γ = (2π/ℏ) |⟨f|H'|i⟩|² ρ(E_f)

Let’s decode this:

• Γ = Transition rate (how many jumps per second)
• |⟨f|H’|i⟩|² = How strongly the perturbation connects initial and final states
• ρ(E_f) = Density of states (how many final states are available at that energy)

Why “Golden”?

Enrico Fermi called this his “golden rule” because it’s so useful. It applies everywhere:

• Atoms absorbing light
• Radioactive decay
• Electrons scattering

Example: Flashlight on Hydrogen

Shine light on a hydrogen atom.

The electron starts in the ground state.

Using Fermi’s Golden Rule:

1. Calculate the matrix element ⟨f|H’|i⟩ for the light-atom interaction
2. Find the density of states ρ(E)
3. Multiply and get the absorption rate!

graph LR
    A["Photon arrives"] --> B["Initial quantum state"]
    B --> C{Interaction present?}
    C --> D["Available final states"]
    D --> E["Transition probability"]

Real-World Applications

🎁 Wrapping Up: The Perturbation Toolkit

When to Use What?

graph TD
    A["Is the problem changing in time?"]
    A -->|No| B["Time-Independent"]
    A -->|Yes| C["Time-Dependent"]
    B --> D{Are energy levels unique?}
    D -->|Yes: No ties| E["Non-Degenerate Theory"]
    D -->|No: Ties exist| F["Degenerate Theory"]
    C --> G{Long time? Many final states?}
    G -->|Yes| H[Fermi's Golden Rule]
    G -->|No| I["Calculate c_f directly"]

The Key Insight

You don’t need perfect answers. Good approximations, built step by step, can take you incredibly far.

That’s the beauty of perturbation theory. Start simple. Add corrections. Get closer and closer to truth.

Like adjusting a recipe, one pinch at a time, until it’s just right.

🌟 Quick Summary Table

Remember: Every expert was once confused. Perturbation theory is your friend for when exact solutions run away. Embrace the approximate! 🚀