6.2 Finite square well potential

The finite square well potential is a fundamental model in quantum mechanics, bridging the gap between idealized infinite wells and real-world systems. It introduces bound states with discrete energy levels below the well's depth and continuous states above it.

This model showcases key quantum phenomena like energy quantization and wave function decay outside the well. It also sets the stage for understanding quantum tunneling, where particles can penetrate classically forbidden regions, a concept with wide-ranging applications in physics and technology.

Bound States and Energy Spectrum

Characteristics of Bound States in Finite Square Wells

• Bound states represent confined particles within the finite square well potential
• Energy levels of bound states fall below the well's potential energy barrier height
• Wavefunctions of bound states decay exponentially outside the well, ensuring localization
• Number of bound states depends on well depth and width, deeper and wider wells accommodate more states
• Quantum confinement effects become more pronounced as well dimensions approach particle's de Broglie wavelength

Energy Spectrum and Quantization

• Energy spectrum in finite square wells consists of discrete energy levels for bound states
• Quantization arises from boundary conditions and wave nature of particles
• Energy levels are not equally spaced, spacing increases for higher energy states
• Lowest energy level (ground state) always exists in finite square wells, unlike infinite square wells
• Higher energy states may not exist if well is too shallow or narrow
• Continuous spectrum of energies exists above the well depth, representing unbound states

Transcendental Equation and Solutions

• Transcendental equation determines allowed energy levels in finite square wells
• Derived from matching wavefunctions and their derivatives at well boundaries
• General form: $\tan(ka) = \sqrt{\frac{V_0}{E} - 1}$ where $k$ is wave number, $a$ is well width, $V_0$ is well depth, and $E$ is energy
• No closed-form analytical solution exists, requires numerical or graphical methods
• Solutions correspond to intersections of $\tan(ka)$ and $\sqrt{\frac{V_0}{E} - 1}$ curves
• Even and odd parity solutions alternate, representing symmetric and antisymmetric wavefunctions

Numerical Approaches and Approximations

• Numerical methods (Newton-Raphson, bisection) solve transcendental equation for energy levels
• Graphical solutions involve plotting both sides of equation and finding intersection points
• WKB approximation provides analytical estimates for energy levels in deep wells
• Perturbation theory applies for shallow wells, treating finite well as perturbed infinite well
• Variational method offers upper bounds on ground state energy
• Computational techniques (finite difference, matrix methods) calculate energy levels and wavefunctions simultaneously

Quantum Tunneling

Fundamentals of Quantum Tunneling

• Quantum tunneling describes particles penetrating potential barriers classically forbidden
• Occurs due to wave-like nature of particles in quantum mechanics
• Probability of tunneling depends on barrier height, width, and particle energy
• Tunneling probability decreases exponentially with barrier width and square root of barrier height
• Plays crucial role in various phenomena (alpha decay, scanning tunneling microscopy, nuclear fusion in stars)

Evanescent Waves and Barrier Penetration

• Evanescent waves represent wavefunctions inside classically forbidden regions
• Characterized by exponential decay rather than oscillatory behavior
• Amplitude of evanescent waves decreases with distance into barrier
• Decay constant depends on particle mass, barrier height, and particle energy
• Continuity of wavefunction and its derivative at barrier boundaries ensures smooth transition

Transmission Coefficient Analysis

• Transmission coefficient quantifies probability of particle tunneling through barrier
• Calculated as ratio of transmitted to incident probability current densities
• For rectangular barrier: $T = \frac{4E(V_0-E)}{4E(V_0-E) + V_0^2\sinh^2(\kappa L)}$ where $E$ is particle energy, $V_0$ is barrier height, $L$ is barrier width, and $\kappa = \sqrt{2m(V_0-E)}/\hbar$
• Exhibits oscillatory behavior for energies above barrier height due to resonance effects
• Approaches unity for very thin barriers or high-energy particles

Reflection Coefficient and Conservation

• Reflection coefficient represents probability of particle bouncing back from barrier
• Complementary to transmission coefficient, sum of two always equals unity (R + T = 1)
• Calculated as ratio of reflected to incident probability current densities
• For rectangular barrier: $R = 1 - T = \frac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E) + V_0^2\sinh^2(\kappa L)}$
• Approaches unity for thick barriers or low-energy particles
• Interplay between transmission and reflection crucial for understanding quantum transport phenomena (resonant tunneling diodes, quantum dots)