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Computational Physics - M. Jensen Episode 2 Part 3

Tham khảo tài liệu 'computational physics - m. jensen episode 2 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 12.2. VARIATIONAL MONTE CARLO FOR QUANTUM MECHANICAL SYSTEMS09 using integration by parts and the relation . . - where we have used the fact that the wave function is zero at R oo. This relation can in turn be rewritten through integration by parts to y dR V R V r R y dR r R V2 T R 0. 12.24 The rhs of Eq. 12.22 is easier and quicker to compute. However in case the wave function is the exact one or rather close to the exact one the lhs yields just a constant times the wave function squared implying zero variance. The rhs does not and may therefore increase the variance. If we use integration by part for the harmonic oscillator case the new local energy is EL x rr2 l a4 12.25 and the variance Ọ a4 l 2 1226 which is larger than the variance of Eq. 12.18 . We defer the study of the harmonic oscillator using the Metropolis algorithm till the after the discussion of the hydrogen atom. 12.2.2 The hydrogen atom The radial Schrodinger equation for the hydrogen atom can be written as h2 d2u f 2 m dr2 2mr2 u r Eu r 12.27 where m is the mass of the electron its orbital momentum taking values I 0 1 2 . and the term kê2 r is the Coulomb potential. The first terms is the kinetic energy. The full wave function will also depend on the other variables Ớ and o as well. The energy with no external magnetic field is however determined by the above equation . We can then think of the radial Schrodinger equation to be equivalent to a one-dimensional movement conditioned by an effective potential V cir r ke2 n2 l r 2mr2 12.28 When solving equations numerically it is often convenient to rewrite the equation in terms of dimensionless variables. One reason is the fact that several of the constants may be differ largely in value and hence result in potential losses of numerical precision. The other main reason for doing this is that the equation in dimensionless form is easier to code sparing one for eventual 210 CHAPTER 12. QUANTUM MONTE CARLO METHODS typographic errors. In order to do so we