ABSTRACT

The Hamiltonian for a system of n particles that are identical, that is, particles that can be substituted for each other with no physical change, must be completely symmetric under any interchange of its arguments. Let Ψ(1, 2, …, n) be any solution of the Schrödinger equation, depending on the coordinates (spatial and spin) of the n identical particles 1 to n, and let P be any permutation of the n numbers 1 to n. Then PΨ(1, 2, …, n) ≡ Ψ(P1, P2, …, Pn) is a function which depends on the coordinates of particle Ψ in the same manner as the original function Ψ depended on the coordinates of particle i. Then using the symmetry of H, we can show that the operator P commutes with the Hamiltonian () H ( P Ψ ) = P ( H Ψ ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429493645/abe8c7ec-36ac-49e5-aba9-71ecf8f2ef67/content/eq89.tif"/>