A change of co-ordinates can often be used to simplify the analytic solution of the few-particle Schrodinger equation. Examples are given for the case of 2 interacting particles

Roy, Mervyn

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University of Leicester Open Journals

Journal of Special TopicsQuantum mechanics with 2 particlesMervyn RoyDepartment of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH.August 26, 2009AbstractA change of co-ordinates can often be used to simplify the analytic solution of the few-particleSchro¨dinger equation. Examples are given for the case of 2 interacting particles.It is often difficult to solve the Schro¨dinger equation for more than one interacting particle. Typically,computationally intensive numerical methods must be used however, in a few cases, there are some tricksavailable to help develop analytical and semi-analytical solutions. One trick that is particularly easy toapply to 2 interacting particles involves a simple change of co-ordinates. For example, in 1D positronium,the Schro¨dinger equation is{−h¯22m∂2∂x2p+−h¯22m∂2∂x2e+ v(xp − xe)}Ψ(xp, xe) = EΨ(xp, xe), (1)where xp is the position of the positron, xe is the position of the electron, m is the electron mass andv(xp − xe) is the Coulomb interaction. This equation can be simplified by transforming into centre ofmass (CM) and relative motion (RM) co-ordinates: xCM = xp + xe and xRM = xp − xe. Then,{ −h¯22(m/2)∂2∂x2CM+−h¯22(m/2)∂2∂x2RM+ v(xRM )}Ψ(xCM , xRM ) = EΨ(xCM , xRM ). (2)In this form the Scro¨dinger equation is clearly separable, Ψ(xCM , xRM ) = φ(xCM )ψ(xRM ), and is rela-tively easy to solve. This example is also easily generalised to 3 dimensions.Another situation where CM are RM co-ordinates are useful is when two interacting particles aretrapped within a parabolic potential. This situation is particularly common because, near a minimum,all potentials are approximately parabolic. For example, for two electrons confined in a one dimensionalelectrostatic potential on a carbon nanotube the Schro¨dinger equation is approximately{−h¯22m∂2∂x21+−h¯22m∂2∂x22+12mω2x21 +12mω2x22 + v(x1 − x2)}Ψ(x1, x2) = EΨ(x1, x2). (3)Here, x1 and x2 are the co-ordinates of each of the 2 electrons and the Coulomb interaction is oftenapproximated as v(x1 − x2) = e2/[(4pi0r)(|x1 − x2|+R)]. Experimentally, 1D nanotube quantum dotshave been engineered, and typically h¯ω ∼ 10 meV, r ∼ 4 and R ∼ 1 nm. Again, by rewriting in CMand RM co-ordinates this equation can be shown to be separable. The CM part of the equation is easilysolved and the RM part can be solved approximately after Taylor expanding the RM potential to secondorder about its minimum.p-1

Quantum mechanics with 2 particles

https://journals.le.ac.uk/ojs1/index.php/pst/article/download/946/878

Quantum mechanics with 2 particles

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