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(84) Particle in Finite Square Potential Well

(84) Particle in Finite Square Potential Well - Particle in...

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Particle in Finite Square Potential Well Consider a particle of mass trapped in a one-dimensional, square, potential well of width and finite depth . Suppose that the potential takes the form (1179) Here, we have adopted the standard convention that as . This convention is useful because, just as in classical mechanics, a particle whose overall energy, , is negative is bound in the well (i.e., it cannot escape to infinity), whereas a particle whose overall energy is positive is unbound (Fitzpatrick 2012). Because we are interested in bound particles, we shall assume that . We shall also assume that , in order to allow the particle to have a positive kinetic energy inside the well. Let us search for a stationary state (1180) whose stationary wavefunction, , satisfies the time independent Schrödinger equation, ( 1141 ). Solutions to ( 1141 ) in the symmetric [i.e., ] potential ( 1179 ) are either totally symmetric [i.e., ] or totally antisymmetric [i.e., ]. Moreover, the solutions must satisfy the boundary condition (1181) otherwise they would not correspond to bound states. Let us, first of all, search for a totally symmetric solution. In the region to the left of the well (i.e., ), the solution of the time independent Schrödinger equation that satisfies the boundary condition as is
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(1182) where (1183) and is a constant. By symmetry, the solution in the region to the right of the well (i.e.,
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