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# helium(1) - Major Project Ground state of the He atom PHZ...

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Major Project: Ground state of the He atom PHZ 5156 This problem combines what we have learned about the technique of Monte-Carlo simulation with atomic physics. In particular, we will use Monte-Carlo techniques to estimate the ground-state wave function and energy of a helium atom. Remember a He atom has two electrons, two protons, and two neutrons. While the nucleus we can regard as a single particle for our purposes, the three body problem (nucleus and two electrons) does not allow for an exact analytical solution. The Hamiltonian for the problem in a set of units with | e | = 1 (i.e. the electron and proton charge), m = 1 and = 1 is, ˆ H = - 1 2 2 r 1 - 1 2 2 r 2 - 2 r 1 - 2 r 2 + 1 r 12 where r 1 = | r 1 | , r 2 = | r 2 | , and r 12 = | r 1 - r 2 | . The ground-state solution satisfies ˆ H Ψ 0 ( r 1 , r 2 ) = E 0 Ψ 0 ( r 1 , r 2 ) where E 0 is the ground-state energy. For any trial wave function Φ T ( r 1 , r 2 ) different from the ground-state wave function Ψ 0 ( r 1 , r 2 ), the energy expectation value given by E = ... Φ * T ( r 1 , r 2 ) ˆ H Φ T ( r 1 , r 2 ) d 3 r 1 d 3 r 2 ... Φ * T ( r 1 , r 2 T ( r 1 , r 2 ) d 3 r 1 d 3 r 2 satisfies E > E 0 . We can approach the problem of obtaining the ground-state wave

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helium(1) - Major Project Ground state of the He atom PHZ...

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