Wave Functions, Tunneling & Schrödinger Equation Explained | Chapter 40 of University Physics

Chapter 40 dives into the core of quantum mechanics: wave functions, Schrödinger’s equation, particle confinement in potential wells, quantum tunneling, and the quantum harmonic oscillator. Whether you're studying atomic systems or advanced quantum devices, this breakdown will clarify these fundamental concepts.

Watch the full video summary on YouTube to see animated solutions of the Schrödinger equation and tunneling demonstrations.

Wave Functions & Schrödinger’s Equation

The wave function Ψ(x,t) encodes all information about a quantum system, with probability density given by |Ψ|². Its evolution follows the time-dependent Schrödinger equation:

−(ħ²/2m) ∂²Ψ/∂x² + U(x)Ψ = iħ ∂Ψ/∂t.

Superpositions of eigenstates form wave packets that describe localized particles, but they spread over time due to the uncertainty principle (Δx·Δp ≥ ħ/2).

Particle in a Box (Infinite Square Well)

For a particle trapped between impenetrable walls at x=0 and x=L, the boundary conditions Ψ(0)=Ψ(L)=0 yield quantized energy levels:

• Eₙ = n²h² / (8mL²) (n = 1,2,3…)
• ψₙ(x) = √(2/L) · sin(nπx/L)

No zero-energy state exists, in line with the uncertainty principle. Probability densities |ψₙ|² show distinct standing-wave patterns for each n.

Finite Potential Wells & Quantum Tunneling

A finite well of depth U₀ confines particles more loosely. Inside (0 < x < L) the wave function is sinusoidal; outside it decays exponentially. Bound-state energies are found by solving transcendental equations. Importantly, nonzero amplitude outside the well allows quantum tunneling through barriers, with transmission coefficient approximated by:

T ≈ G · e^(−2κL), κ = √[2m(U₀−E)]/ħ.

Tunneling underlies α-decay, scanning tunneling microscopy, and many nanoscale devices.

Quantum Harmonic Oscillator

The quantum analog of a mass on a spring has potential U(x)=½kx² and energy levels:

• Eₙ = (n + ½)ħω (n = 0,1,2…)
• Ground state energy E₀ = ½ħω prevents collapse to x=0.

Wave functions involve Hermite polynomials, and even classically forbidden regions have nonzero probability. As n increases, behavior approaches classical motion (correspondence principle).

Measurement & Wave-Function Collapse

Measuring an observable causes the wave function to collapse into one of its eigenstates. Prior to measurement, a system may exist in a superposition of states. Repeated measurements of an energy eigenstate always yield the same result, but measuring momentum of a mixed state selects a single outcome, reflecting the probabilistic nature of quantum mechanics.

Conclusion

Chapter 40 builds the foundation of quantum theory, showing how Schrödinger’s equation and wave mechanics govern particles at atomic scales. From confined "particles in a box" to tunneling and quantized oscillators, these concepts are key to semiconductors, lasers, and modern nanotechnology.

If you found this breakdown helpful, be sure to subscribe to Last Minute Lecture for more chapter-by-chapter textbook summaries and academic study guides.