Potential wells and barriers are key concepts in quantum mechanics, shaping particle behavior in ways that defy classical physics. They're crucial for understanding how particles move and interact at the atomic scale, where wave-like properties become significant.

In this part of the chapter, we'll see how the Schrödinger equation describes particles in wells and barriers. We'll explore energy quantization, wave functions, and the mind-bending phenomenon of quantum tunneling, where particles can pass through seemingly impassable barriers.

Particle behavior in potential wells

Potential wells and quantum mechanics

• Potential wells vary potential energy of particles in space affecting motion and quantum properties
• Classical mechanics particles oscillate between turning points in wells
• Quantum mechanics particles exhibit discrete energy levels and wave-like behavior
• Wave functions in potential wells must satisfy boundary conditions and be continuous and differentiable
• Probability density of finding particle given by square magnitude of wave function
• Energy spectrum can be discrete (bound states) or continuous (unbound states) depending on well depth and shape

Quantum tunneling and potential barriers

• Tunneling allows particles to penetrate classically forbidden potential barriers
• Particles incident on potential barriers have wave functions with incident, reflected, and transmitted components
• Transmission probability through barrier depends on height, width, and particle energy
• Rectangular potential barriers calculated using transfer matrix method or WKB approximation for thick barriers
• Tunneling probability decreases exponentially with barrier width and square root of barrier height minus particle energy
• Resonant tunneling in double barrier structures can approach unity transmission probability for specific energies

Solving the Schrödinger equation

Time-independent Schrödinger equation

• One-dimensional potential well equation: $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V(x)\psi = E\psi$
• $V(x)$ represents potential energy function
• Infinite potential well wave function vanishes at boundaries
• Quantization of energy levels for infinite well: $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$
• $n$ quantum number, $L$ well width
• Infinite well wave functions sinusoidal with nodes increasing at higher energy levels

Finite potential wells

• Wave function extends beyond well boundaries, decaying exponentially in forbidden regions
• Energy levels and wave functions solved numerically or with approximation methods
• No general analytical solution exists for finite wells
• Number of bound states depends on well depth and width
• Deeper and wider wells accommodate more bound states

Bound states and energy levels

Characteristics of bound states

• Particles confined to limited region of space, typically within potential well
• Energy quantized with discrete energy levels
• Ground state lowest energy bound state
• Higher energy levels called excited states
• Energy spacing between states increases as well narrows or energy level increases
• Normalizable wave functions with finite probability of finding particle in space

Wave function properties

• Probability of finding particle somewhere in space equals one for bound states
• Number of nodes in wave function increases with higher energy levels
• Pattern: $n-1$ nodes for nth energy level (ground state 0 nodes, first excited state 1 node)
• Wave functions must be continuous and differentiable at all points

Potential barriers and transmission probabilities

Barrier characteristics and solutions

• Potential barriers regions where potential energy exceeds particle's total energy
• Classically forbids particle penetration
• Wave function for incident particle includes incident, reflected, and transmitted components
• Transmission coefficient for rectangular barrier calculated using transfer matrix method
• WKB approximation used for thick barriers

Quantum tunneling phenomena

• Non-zero probability of transmission through barrier for particles with energy below barrier height
• Tunneling probability decreases exponentially with barrier width
• Tunneling probability also decreases with square root of difference between barrier height and particle energy
• Resonant tunneling in double barrier structures possible
• Transmission probability can approach unity for specific particle energies in resonant tunneling