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particle in a box


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A system in quantum mechanics used to illustrate important features of quantum mechanics, such as quantization of energy levels and the existence of zero-point energy. A particle with a mass m is allowed to move between two walls having the coordinates x = 0 and x = L. The potential energy of the particle is taken to be zero between the walls and infinite outside the walls. The time-independent Schrödinger equation is exactly soluble for this problem, with the energies En being given by n2h2/8mL2, n = 1,2,…, and the wave functions ψn being given by ψn = (2/L)½ sin(nπx/L), where n are the quantum numbers that label the energy levels and h is the Planck constant. The particle in a box can be used as a rough model for delocalized electrons in a molecule or solid. Thus it can be used to explain the colours of molecules with conjugated double bonds. It can also explain the colour imparted to crystals by F-centres. The problem of a particle in a box can also be solved in two and three dimensions, enabling relations between symmetry and degenerate energy levels to be seen.

Subjects: Chemistry.


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