D O C . 5 1 6 S C H R Ö D I N G E R ’ S WAV E M E C H A N I C S 5 1 3 … (3) holds, where signifies the second covariant spatial differential quotient of ψ in configuration space, . … (3a) We want to denote as the “tensor of the ψ-curvature,” as the scalar of this tensor, as the “scalar of the ψ-curvature.”[5] Our first task is then to assign unambiguously n directions to the tensor of the ψ-curvature. If is a unit vector, which accordingly satisfies the equation , … (4) then this direction vector together with defines the scalar . We ask about the directions for which becomes an extremum. Lagrange’s method produces for this the conditions[6] . … (5) They involve the determinant equation … (5a) which has n ¢real positive² roots . If all of these are real and differ from one another, then the equations (5) define n directions, i.e., n unit vectors except for the sign (principal directions). These directions are perpendicular to each other, as one recognizes if for the equations one multiplies the first by , the second by , and then substracts them both from each other. These n directions determine an orthogonal local system of coor- dinates of the ’s, in which the metric (at the origin) is Euclidean ( ). In general, we shall denote with an overline quantities that refer to the local coor- dinate system. In this local coordinate system the tensor of the ψ-curvature can be decom- posed into summands, each of which is allocated to one of the principal directions, as, according to (3), . … (3b) Δψ gαβψαβ = ψαβ ψαβ ∂2ψ ∂qα∂qβ ----------------- αβ σ ¯ ¿∂qσ ® ¾ ­ ½∂ψ –= ψαβ Δψ Aα gμνAμAν 1= ψαβ ψμνAμAν ψA = A(μα) ψA ψμν λgμν)Aν – ( 0= ψμν λgμν – 0, = λα) ( A(μα) ψμν λ(α)gμν)A(να) – ( 0= ψμν λ(β)gμν)A(β) – ( ν 0= A(β) μ A(μα) ξα gμν δμν = ψαβ Δψ α ¦ψαα =