Symmetries in QM

🔮 SYMMETRIES IN QUANTUM MECHANICS

The Secret Rules That Nature Follows

Imagine you’re playing with a snow globe. You shake it, put it down, and watch the snow fall. Now here’s a magical question: what if the snow fell the same way no matter where you placed the globe in your room?

That’s symmetry — when something stays the same even when you change something about it.

In quantum mechanics, symmetries are like nature’s secret rulebook. They tell us what can and cannot happen. And the coolest part? Every symmetry comes with a gift — a quantity that never changes, called a conservation law.

Let’s explore this magical world together! 🌟

🎭 SYMMETRY PRINCIPLES

“If nothing looks different, nothing changes”

Think about a perfect circle. You can spin it any amount, and it looks exactly the same. That’s rotational symmetry.

Now imagine you have a magic box. If you do something to the box (like moving it or turning it) and you can’t tell the difference, that’s a symmetry.

The Big Idea

Symmetry = Doing something that changes nothing observable.

In quantum mechanics, when a system has symmetry, the Hamiltonian (the energy equation) doesn’t change.

If H stays the same after transformation → Symmetry exists!

Simple Example:

• Put a marble in an empty box
• Spin the box 90 degrees
• The marble doesn’t care — physics works the same way
• This box has rotational symmetry!

Why Does This Matter?

Because of Noether’s Theorem — one of the most beautiful ideas in physics:

Every symmetry gives you something that stays constant forever.

Like finding a treasure chest with every new symmetry!

⚖️ CONSERVATION LAWS

“The Gifts That Symmetries Give Us”

Remember how every symmetry comes with a gift? Here’s the treasure map:

graph TD
    A["SYMMETRY"] --> B["CONSERVATION LAW"]
    C["Translation in Space"] --> D["Momentum Conserved"]
    E["Translation in Time"] --> F["Energy Conserved"]
    G["Rotation"] --> H["Angular Momentum Conserved"]
    I["Parity"] --> J["Parity Conserved*"]

The Magic Connection

*Parity is special — it’s not always conserved! We’ll see why later.

Simple Example:

• Two kids on roller skates push each other apart
• Kid A goes left, Kid B goes right
• Total momentum before = 0
• Total momentum after = 0
• Momentum is conserved! 🛼

➡️ TRANSLATION SYMMETRY

“The Universe Doesn’t Care Where You Are”

Imagine you have a science experiment. You do it in your kitchen. Then you pack everything up and do the exact same experiment in your friend’s kitchen.

What happens? You get the same results!

This is translation symmetry — physics works the same way everywhere in space.

The Math (Made Simple)

When you move (translate) by a distance a, we use an operator:

T(a) = e^(-i p·a / ℏ)

This looks scary, but here’s what it means:

• T(a) = “move everything by distance a”
• p = momentum (the conserved quantity!)
• ℏ = Planck’s constant (a tiny number)

The Gift: Momentum Conservation

Because space looks the same everywhere:

• The translation operator commutes with the Hamiltonian
• [T, H] = 0
• This means momentum is conserved!

Simple Example:

• A ball rolling on a flat, infinite table
• No matter where the ball is, it feels the same forces
• So its momentum stays constant!

Types of Translation Symmetry

1. Continuous Translation: Move any distance
2. Discrete Translation: Move by specific amounts (like in crystals)

graph TD
    A["Translation Symmetry"] --> B["Continuous"]
    A --> C["Discrete"]
    B --> D["Free particles in space"]
    C --> E["Electrons in crystals"]

🔄 ROTATION SYMMETRY

“Spin It Any Way — Physics Doesn’t Mind”

Hold a basketball. Now spin it. Does gravity suddenly work differently? No!

That’s rotation symmetry — physics is the same no matter which direction you face.

The Rotation Operator

For rotating around the z-axis by angle θ:

R_z(θ) = e^(-i L_z θ / ℏ)

Where:

• L_z = angular momentum around z-axis
• θ = rotation angle
• This works for any axis!

The Gift: Angular Momentum Conservation

Because empty space has no preferred direction:

• [R, H] = 0 for isolated systems
• Angular momentum is conserved!

Simple Example:

• Ice skater spinning
• Pulls arms in → spins faster
• Pushes arms out → spins slower
• But angular momentum stays the same! ⛸️

Rotations Are Special

Unlike translations, rotations don’t commute:

• Rotating around x, then y ≠ rotating around y, then x
• Try it with a book! Rotate it differently and see.

This leads to the beautiful angular momentum algebra:

[L_x, L_y] = i ℏ L_z
[L_y, L_z] = i ℏ L_x
[L_z, L_x] = i ℏ L_y

🪞 PARITY OPERATOR

“Looking in Nature’s Mirror”

Stand in front of a mirror. Raise your right hand. Your reflection raises its left hand!

Parity (P) is the quantum version of looking in a mirror. It flips all spatial coordinates:

P: (x, y, z) → (-x, -y, -z)

What Parity Does

graph TD
    A["Apply Parity P"] --> B["Position: r → -r"]
    A --> C["Momentum: p → -p"]
    A --> D["Angular Momentum: L → L"]

Wait — why doesn’t angular momentum flip?

Because L = r × p. Both r and p flip, so the cross product stays the same! It’s like flipping both a pancake and the pan — the pancake is still on top.

Parity Eigenvalues

Apply parity twice: P² = 1

This means parity eigenvalues are:

• +1 (even parity) — wave function unchanged
• -1 (odd parity) — wave function flips sign

P ψ(x) = ψ(-x)

Even parity: ψ(-x) = +ψ(x)   Example: cos(x)
Odd parity:  ψ(-x) = -ψ(x)   Example: sin(x)

The Shocking Discovery! ⚡

For decades, physicists assumed parity was always conserved.

Then in 1957, Wu’s experiment showed:

Weak nuclear force violates parity!

This was like discovering that your mirror reflection does something different from you. Mind-blowing!

Simple Example:

• Cobalt-60 atoms emit electrons
• If parity were conserved, equal electrons go up and down
• But MORE electrons go one way!
• Nature’s mirror is slightly broken for weak interactions

⏰ TIME REVERSAL SYMMETRY

“Can We Run the Movie Backwards?”

Imagine filming a bouncing ball, then playing the video in reverse. Would it look weird? No! The ball just bounces the other way.

This is time reversal symmetry — physics equations work the same forwards and backwards in time.

The Time Reversal Operator (T)

Unlike parity, time reversal is anti-unitary. This means:

T i T⁻¹ = -i

The imaginary unit flips sign! This is strange and special.

What Time Reversal Does

Think about it:

• If you reverse time, velocity reverses
• So momentum (p = mv) reverses too!

Time Reversal and Quantum States

For a spinless particle:

T ψ(r, t) = ψ*(r, -t)

The star (*) means complex conjugate. Time reversal takes the mirror image in the complex plane!

For particles with spin, things get more interesting. Time reversal includes spin flipping.

Kramer’s Degeneracy 🎁

A beautiful consequence for systems with half-integer spin:

If the system has time reversal symmetry, energy levels come in pairs!

This is called Kramer’s degeneracy. It’s like nature giving you a “buy one, get one free” deal for electrons.

Simple Example:

• Electron in an atom (no magnetic field)
• Time reversal symmetry exists
• Spin-up and spin-down states must have same energy
• That’s why we can fit 2 electrons per orbital!

Breaking Time Reversal

Time reversal symmetry breaks when:

• Magnetic fields are present (B → -B under T)
• This removes Kramer’s degeneracy
• Zeeman splitting separates spin states!

graph TD
    A["Time Reversal Symmetry"] --> B{Magnetic Field?}
    B -->|No| C[Kramer's Degeneracy]
    B -->|Yes| D["Degeneracy Broken"]
    C --> E["Paired Energy Levels"]
    D --> F["Split Energy Levels"]

🏆 PUTTING IT ALL TOGETHER

The Symmetry Family Portrait

The Hierarchy of Symmetries

graph TD
    A["ALL SYMMETRIES"] --> B["Continuous"]
    A --> C["Discrete"]
    B --> D["Translation"]
    B --> E["Rotation"]
    C --> F["Parity P"]
    C --> G["Time Reversal T"]
    C --> H["Charge Conjugation C"]

The Ultimate Symmetry: CPT

Even though P and T can be broken, there’s a deeper truth:

CPT symmetry is ALWAYS conserved in any quantum field theory!

• C = Charge conjugation (particle ↔ antiparticle)
• P = Parity (mirror flip)
• T = Time reversal

If you apply all three together, physics stays the same. It’s the universe’s deepest known symmetry!

💡 KEY TAKEAWAYS

1. Symmetry = Invariance: When something looks the same after a transformation

2. Noether’s Theorem: Every continuous symmetry → Conservation law

3. Translation Symmetry → Momentum conservation

   • Space is the same everywhere

4. Rotation Symmetry → Angular momentum conservation

   • Space has no preferred direction

5. Parity (P): Mirror reflection operator

   • Broken by weak nuclear force!

6. Time Reversal (T): Run physics backwards

   • Creates Kramer’s degeneracy for fermions
   • Broken by magnetic fields

7. CPT Theorem: The ultimate unbreakable symmetry

🎓 FINAL THOUGHT

Symmetries are like the grammar rules of the universe. Just as grammar tells you how to build correct sentences, symmetries tell nature what physics is allowed.

And just like some languages break grammar rules for special words, nature breaks some symmetries (like parity) for special forces (like the weak force).

But at the deepest level, CPT symmetry stands strong — the ultimate rule that even the universe doesn’t break.

You now understand one of the most beautiful ideas in all of physics! 🌌

“Symmetry is what we see at a glance; based on the fact that there is no reason for any difference.” — Blaise Pascal