Symmetry - Multiparticle Wavefunctions and Symmetry Michael...

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Multiparticle Wavefunctions and Symmetry Michael Fowler University of Virginia Two Electrons in a One Dimensional Well So far, we have used Schrödinger’s equation to see how a single particle, usually an electron, behaves in a variety of potentials. If we are going to think about atoms other than hydrogen, it is necessary to extend the Schrödinger equation so that it describes more than one particle. As a simple example of a two particle system, let us consider two electrons confined to the same one-dimensional infinite square well, V ( x ) = 0, - L /2 < x < L /2 V ( x ) = ∞ otherwise. To make things even simpler, let us assume that the electrons do not interact with each other—we switch off their electrostatic repulsion. Then by analogy with our construction of the Schrödinger equation for a single electron, we write ( ) ( ) ( ) 22 12 ,, xxt E xxt mx ψψ ψ ∂∂ −−=  inside the well, with the wave function going to zero for x 1 or x 2 equal to L /2 or - L /2. On looking at this equation, we see it is the same as the Schrödinger equation for a single electron in a two dimensional square well, and so can be solved in the same way, by separation of variables. For example, the wave function we plotted for the two dimensional rectangular well is in the square case: ( ) ( ) ( ) ( ) / 1 2 2,3 , , sin 2 / cos 3 / iEt x x t A x L x Le ππ = has the same energy ( ) 2 23 2 E mL π = as the physically distinct wavefunction: ( ) ( ) ( ) ( ) / 2 1 3,2 , , sin 2 / cos 3 / iEt A =
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2 Interpreting the Wavefunction We have already discussed how the above wavefunctions are to be interpreted if they are regarded as two-dimensional wavefunctions for a single electron: ( ) 2 12 , x x dx dx ψ is the probability of finding the electron in a small area dx 1 dx 2 at ( x 1 , x 2 ).
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Symmetry - Multiparticle Wavefunctions and Symmetry Michael...

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