Variational method quantum mechanics, Study notes of Physics Fundamentals

Download Variational method quantum mechanics and more Study notes Physics Fundamentals in PDF only on Docsity! Variational Method for the Helium Ground State The Hamiltonian for Helium is given by H = 1 2m ( P 21 + P 2 2 ) − Ze2 ( 1 r1 + 1 r2 ) + e2 r12 , (1) where r12 = |~r1 − ~r2|. To obtain an estimate function for the ground state consider neglecting the r12. The remaing contribution to the Hamiltonian is then given by, using spherical coordinates, H = 2∑ i=1 [ − h̄ 2 2m 1 r2i ∂ ∂ri ( r2i ∂ ∂ri ) − Ze 2 ri ] . (2) Since the Hamiltonian is symmetric with respect to particle one and two in this form it can be seen that the ground state functions will be of the same form (assuming that the two electrons couple their spins to total spin = 0, which means that their spatial wave function will be symmetric). In this respect, the ground state (lowest energy) solution of the two particle wave function can be found in the form |ψ0(r1, r2) >= 2∏ i=1 u(ri). (3) With this functional form of the ground state the Schrodinger equation, H|ψ0(r1, r2) >= E|ψ0(r1, r2) >, leads to the equation 1 r2i ∂ ∂ri ( r2i ∂u ∂ri ) + 2m h̄2 ( Ei + Ze2 ri ) u = 0, (4) for u(ri) for i = 1, 2 and E = E1 + E2 = 2E1. The solution of the differential equation is of the form u(r) = Ae−ar. (5) It is easily seen that a = Z/a0, where a0 = h̄ 2/me2, and Ei = −(Ze)2/(2a0). From this it is readily clear that |ψ0(r1, r2) >= A2e− Z a0 (r1+r2). (6) The normalization will invoke the use of a symmetry argument during the 1 calculation. With this in mind the normalization follows. 1 =< ψ0(r1, r2)|ψ0(r1, r2) > = A4 ∫ r1 ∫ r2 e− 2Z a0 (r1+r2)d3r1d 3r2 = A4 [∫ ∞ 0 ∫ Ω e− 2Zr a0 r2drdΩ ]2 = A4(4π)2 [∫ ∞ 0 e− 2Zr a0 r2dr ]2 1 = A4 ( πa30 Z3 )2 . (7) From this result the value of A is given by A = √ Z3 πa30 . (8) The normalized ground state wave function to be used is now provided by |ψ0(r1, r2) >= Z3 πa30 e− Z a0 (r1+r2) (9) while the energy is given by E = −(Ze)2/a0 = 8EH = −108.8 eV. This is considerably more negative than the experimentally known binding energy of He, -78.6 eV. Since the ground state function for non-interactions has been found it can now be used to estimate the general state with interaction terms. In general the variational method is given by the form < E >= < ψ|H|ψ > < ψ|ψ > . (10) Since the wave functions being used have been normalized the form of < E > is then reduced to < E >=< ψ|H|ψ >. Consider the argument that in the presence of another electron, each of the electrons are influenced by a decreased charge from the nucleus. With this in mind let Z → Z − σ, where σ is the screening charge. Under this change the ground state wave function becomes |ψ0(r1, r2) >= (Z − σ)3 πa30 e− Z−σ a0 (r1+r2). (11) The Hamiltonian, equation (1), can be seen in the following form. H = 1 2m ( P 21 + P 2 2 ) − Ze2 ( 1 r1 + 1 r2 ) + e2 r12 H = 2∑ i=1 [ P 2i 2m − (Z − σ)e 2 ri ] − σe2 ( 1 r1 + 1 r2 ) + e2 r12 . (12) 2