Quantum Tunneling and Barrier Penetration The probability of tunneling across a potential barrier decreases exponentially as the height of the barrier increases. When the energy of a particle is much smaller than the magnitude of the potential, there is significant penetration across the barrier.
Particle in a Box A simpler example compared to previous discussions. A particle with positive energy inside an impenetrable box experiences no barrier penetration. The wave function within this box has specific boundary conditions that determine its form.
Wave Function Matching at Boundaries The wave functions must match at boundaries, but due to infinite barriers, their derivatives do not need to be continuous. This leads to simplified forms for psi 2 and determination of normalization constant A.
'n' Values and Energy Levels 'n' values are determined by setting alpha equal to n pi by L where n takes integer values excluding zero; these correspond to different energy levels E sub n = (n^2 * pi^2 * h cross squared) / (2mL^2). Each 'n' corresponds to an Eigen state with corresponding wave function given by root(2/L) sin(n*pi*x/L).