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PIB3D - Physical Chemistry Course Number C362 7 Particle...

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Physical Chemistry Course Number: C362 7 Particle inside a three-dimensional box: The time-independent Schr¨odinger Equation in three dimensions Consider the figure of the metal porphyrinin in problem 3-27. The electrons are delocalized and one might consider these to be spread over an entire plane. Can we use the particle-in-a-box idea to study such problems? One could consider the electrons in this case as particles in a two-dimensional box. To study such problems let us consider the general case of a particle in a three-dimensional box. [The two-dimensional box is a special case of the three-dimensional box.] ( x,t ) = ¯ h 2 2 m 2 + V ψ ( vector r,t ) = ( x,t ) (7.1) The Hamiltonian for a particle whose position is given by a vector vector r is H = ¯ h 2 2 m 2 + V (7.2) In Cartesian coordinates: 2 = 2 ∂x 2 + 2 ∂y 2 + 2 ∂z 2 (7.3) The time-independent Schr¨odinger Equation of the Particle-in-a-box in 3D would then look like: ¯ h 2 2 m 2 ∂x
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