Bound State Problems

🌊 Quantum Mechanics: Bound State Problems

The Story of Tiny Prisoners

Imagine you have a bouncy ball. In your everyday world, you can throw it anywhere—up, down, left, right. It goes wherever you want. But what if you put that ball in a box? Now it can only bounce inside!

Quantum particles are like magical bouncy balls. Sometimes they’re free to roam everywhere. Sometimes they’re trapped. And when they’re trapped, they do something VERY surprising—they can only bounce in certain special ways!

Let’s meet our quantum characters and their prisons.

🏃 The Free Particle: No Walls, No Rules

What Is It?

A free particle is like a bird in the open sky. No cage. No limits. It can fly anywhere it wants!

🐦 ~~~~~~~~~~~~~~~~~~~~~~~~>
     Flying freely forever

The Simple Truth

When a particle has no walls around it, it can have any energy it wants. Think of it like this:

• You can walk slowly (low energy)
• You can jog (medium energy)
• You can sprint (high energy)

Any speed works! There are no rules.

The Wave Picture

A free particle behaves like a wave stretching forever in both directions:

graph TD
    A["Free Particle"] --> B["No Boundaries"]
    B --> C["Any Energy Allowed"]
    C --> D["Wave Extends Forever"]
    D --> E["Continuous Energy Spectrum"]

Key Takeaway 🎯

Free = Any energy allowed. Like choosing any radio station, not just preset channels.

📦 Particle in a Box: The Simplest Prison

The Setup

Now let’s trap our particle! Imagine putting a ball inside a shoebox:

|         |
|  🔵     |
|         |
|_________|
  The Box!

The walls are infinitely hard—the ball can NEVER escape. This is called the infinite square well.

The Magic Rule

Here’s where quantum mechanics gets weird and wonderful!

When you trap a particle in a box, it cannot have just any energy. It can only have certain special energies—like how a guitar string can only vibrate in certain ways.

Why Only Special Energies?

Think of a jump rope held by two kids:

Mode 1:  ___/‾‾‾\___    (1 hump)
Mode 2:  __/‾\_/‾\__    (2 humps)
Mode 3:  _/‾\_/‾\_/‾\_  (3 humps)

The rope is fixed at both ends (the kids’ hands). It can only make whole numbers of humps. You can’t have 1.5 humps!

Particles in a box work the same way!

The Energy Ladder

The allowed energies form a ladder:

         ════════════ E₄ (Level 4)

         ════════════ E₃ (Level 3)

         ════════════ E₂ (Level 2)

         ════════════ E₁ (Level 1) ← Ground state
         ══════════════════════════
              FLOOR (Zero energy)

Each step up requires MORE energy than the last. The gaps get bigger!

The Ground State Rule

The particle can NEVER sit still at zero energy. The lowest level (E₁) is above zero!

Analogy: It’s like the particle is always nervously jiggling, even when it’s as calm as possible. Scientists call this the zero-point energy.

graph TD
    A["Particle in Box"] --> B["Walls are infinite"]
    B --> C["Wave must fit exactly"]
    C --> D["Only certain wavelengths work"]
    D --> E["Quantized energy levels!"]

Simple Example

Imagine an electron trapped in a tiny box just 1 nanometer wide (about 10 atoms!):

• Level 1: Electron gently waves (lowest energy)
• Level 2: Electron waves faster (4× more energy!)
• Level 3: Electron waves even faster (9× more energy!)

The pattern? Energy grows like n² (1, 4, 9, 16, 25…)

♾️ The Infinite Square Well: Deep Dive

What Makes It “Infinite”?

The walls have infinite potential energy. This means:

• Inside the box: particle is free to wave around
• At the walls: ABSOLUTE STOP! Wave must be zero.
• Outside the box: particle has ZERO chance of being there

OUTSIDE | INSIDE  | OUTSIDE
   ∞    |   0     |    ∞      ← Potential Energy
        |  🌊     |
________|         |________
        |_________|
        0         L

The Boundary Conditions

The wave function (the particle’s “wave shape”) must equal zero at both walls. Why? Because the particle CAN’T be in the wall!

This is like: A vibrating string tied at both ends. It must have zero movement at the tie points.

The Wave Solutions

Only waves that fit perfectly are allowed:

n=1: |  ∿  |     One half-wave
n=2: | ∿∿ |     Two half-waves
n=3: |∿∿∿|     Three half-waves

The Famous Energy Formula

For a box of width L:

Eₙ = n² × (h²/8mL²)

Where:
n = 1, 2, 3, ... (quantum number)
h = Planck's constant
m = particle mass
L = box width

Key Insights 🎯

🎚️ The Finite Square Well: Softer Walls

Real Life Is Softer

In real life, walls aren’t infinitely hard. They have some “give.” The finite square well has walls with finite height.

     _____________
    |             |  ← Wall height V₀
____|             |____
    |    🔵      |
    |_____________|

The Surprising Difference

With softer walls, something magical happens: the particle can leak into the walls!

Wait, what?! The particle can be found where it “shouldn’t” be?

Understanding the Leak

The particle’s wave doesn’t suddenly stop at the wall. It gently fades away into the wall region. This is called quantum tunneling (we’ll meet it properly with the step potential!).

              INSIDE      |    WALL
                 🌊       |     🌊
                          |      🌊  ← Fading away
                          |       🌊
                          |        ·

Finite vs Infinite: Quick Compare

Why Finite Wells Matter

Real atoms and real traps are finite wells! The electrons in atoms can “tunnel” through barriers—this is how:

• Tunnel diodes work
• Radioactive decay happens
• The Sun fuses hydrogen

graph TD
    A["Finite Well"] --> B["Walls have finite height"]
    B --> C["Wave can leak into walls"]
    C --> D["Tunneling is possible!"]
    D --> E["Fewer bound states than infinite well"]

🪜 The Step Potential: Climbing a Cliff

The Setup

Imagine walking and suddenly hitting a cliff—a wall you need to climb!

         |‾‾‾‾‾‾‾‾
         |         ← Height V₀
_________|
    ← 0 →
 Region 1   Region 2

Region 1: Ground level (V = 0) Region 2: Raised cliff (V = V₀)

Classical vs Quantum

Classical Ball:

• If ball has energy > V₀: It climbs over!
• If ball has energy < V₀: It bounces back. Always.

Quantum Particle:

• If energy > V₀: It goes over (but some reflects!)
• If energy < V₀: Mostly reflects… BUT some tunnels through!

The Mind-Bending Part

Even when a particle DOESN’T have enough energy to climb the cliff, there’s a chance it appears on the other side!

    Incoming →   🌊        |
                           |
    Reflected ←  🌊        |    🌊 → Transmitted!
                           |         (Tunneling)
___________________________|_______________

Reflection and Transmission

When a wave hits the step:

1. Some reflects (bounces back)
2. Some transmits (continues forward)

The amounts depend on:

• Particle’s energy
• Height of the step

High Energy Case (E > V₀)

The particle has enough energy! But unlike a ball that just goes over, the quantum particle still partially reflects!

Incoming:    |=========>
             |
Reflected:   |<====
Transmitted: |        ====>
             |
_____________|________________

Think of light hitting glass—some reflects, some goes through!

Low Energy Case (E < V₀)

The particle SHOULDN’T be able to climb… but quantum magic allows a tiny bit to tunnel through!

Incoming:    |=========>
             |
Reflected:   |<========
Transmitted: |        · · · · ·→ (exponentially small)
             |
_____________|________________

The wave fades exponentially in the “forbidden zone.”

graph TD
    A["Step Potential"] --> B{Particle Energy?}
    B -->|E > V₀| C["Goes Over"]
    C --> D["But some still reflects!"]
    B -->|E < V₀| E["Mostly Reflects"]
    E --> F["But some tunnels through!"]

🌟 The Big Picture

Comparison Table

Why This Matters

These simple systems are the building blocks for understanding:

• Atoms = electrons in finite wells
• Quantum dots = particles in boxes (used in TVs!)
• Semiconductors = electrons meeting step potentials
• Nuclear decay = particles tunneling through barriers

The Quantum Philosophy

Confinement creates quantization. The more you trap a particle, the more limited its options become—but each option is perfectly precise!

graph TD
    A["Quantum Bound States"] --> B["Free Particle"]
    A --> C["Particle in Box"]
    A --> D["Finite Well"]
    A --> E["Step Potential"]
    B --> F["Any energy: continuous"]
    C --> G["Only special energies: quantized"]
    D --> H["Quantized + tunneling into walls"]
    E --> I["Reflection even when E > V₀"]

🎓 Summary: What We Learned

1. Free Particle 🏃

   • No walls = no restrictions
   • Any energy is allowed (continuous spectrum)

2. Particle in a Box 📦

   • Infinite walls trap the particle
   • Only specific “wave-fitting” energies work
   • Energy levels: E₁, 4E₁, 9E₁, 16E₁…

3. Infinite Square Well ♾️

   • Wave = exactly zero at walls
   • Zero chance outside the box
   • Energy formula: Eₙ = n²h²/8mL²

4. Finite Square Well 🎚️

   • Softer walls with finite height
   • Wave leaks into walls (tunneling!)
   • Fewer bound states than infinite well

5. Step Potential 🪜

   • One region higher than another
   • Particles can tunnel into forbidden zones
   • Even high-energy particles partially reflect

🚀 You Made It!

You now understand how quantum particles behave when they’re trapped! These aren’t just abstract ideas—they explain:

• Why atoms have specific energy levels
• How electrons move in materials
• Why the sun can fuse atoms
• How quantum computers trap information

Remember the magic: When you confine a particle, you quantize its options. Nature loves neat, whole-number solutions!

Next time you see a colorful neon sign or use your phone’s screen, remember: you’re seeing quantum particles jumping between their allowed energy levels inside tiny boxes!

🎉 Congratulations, Quantum Explorer! 🎉