Bemerkung zur Rayleigh-Schrödinger-Störungstheorie

The time-independent Hamiltonians ? ₀ and ?=? ₀ + V have a discrete spectrum, eigenvalues, and eigenvectors E _s ^(o) , ¦s〉^(o) resp. E _s, ¦s〉. If the RS perturbation theory can be applied here then an operator (mathfrak{p}) with the property $$left| s rightrangle ^{(n + 1)} = frac{1}{{n + 1}}mathfrak{p}left| s rightrangle ^{(n)} , E_s^{(n + 1)} = frac{1}{{n + 1}}mathfrak{p}E_s^{(n)} $$ exists where ¦s〉⁽ⁿ⁾ and E _s ⁽ⁿ⁾ denote the n-th order corrections of perturbation theory if E _s ^(o) is nondegenerate. In the case of degeneracy the operation (mathfrak{p}) remains defined and can always be used todetermine perturbation corrections of quantum mechanical expressions which are invariant in zerothorder under transformations of the basis in degenerate subspaces of ? ₀. The equations $$left| s rightrangle = sumlimits_n^{0,infty } {left| s rightrangle ^{(n)} = e^mathfrak{p} left| s rightrangle ^{(0)} } , E_s = sumlimits_n^{0,infty } {E_s^{(n)} } = e^mathfrak{p} E_s^{(0)} $$ correspond to a basis transformation where nondegenerate eigenvectors ¦s∝> ^(o) and eigenvalues E _s ^(o) of ? ₀ transform into eigenvectors ¦s∝> and eigenvalues E _s of ?. Examples show the usefulness of this formulation.