By Eugene P. Wigner, J. J. Griffin

Crew concept And Its software To The Quantum Mechanics Of Atomic Spectra

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Subtracting, we see that if Ε φ F , ( ψ , ψ ) must be zero. Likewise, the discrete eigenfunctions are orthogonal t o all the eigendifferentials, and the Ε Ε 38 GROUP T H E O R Y A N D ATOMIC SPECTRA eigendifferentials are orthogonal to one another, provided the regions t o which they belong do not overlap. More than one linearly independent eigenfunction m a y belong to one eigenvalue of, say, the discrete spectrum. " Every possible linear combination o f degenerate eigenfunctions is also an eigenfunction with the same "eigenvalue.

We have seen that every operator which can be applied to a finite-dimensional vector is equivalent to a matrix. The infinitedimensional operators also have a matrix form, but it is often strongly singular. 1) '. It transforms the vector ψ into the vector q ^ , with components QM i 2 -"Xf) x x = 2 = Σ Χχ-Xf tai^i - ι ΧχΧι χ * i ^¥2 δ - ν( Ί Χ χ/ ö " ' 'f) x Xf*t Χ Μ χ ι · ' · */)· This vector is exactly the function x y into which ψ is transformed by the operation "multiplication by x . " The matrix which corresponds to the operator "differentiation with respect to x " is denoted by (i'/^)Pi since (Hji)djdx corresponds to p x t x 1 \ / x — x\ 1 f x' x ·" χ/ Δ _ > x ϋ It transforms the vector ψ into 2 χ '»·χ/ χ Hm i {Δ ΐΑ, > ΧΙ+ ΧΙ ί^-ΐΔ,^') δ χχ 2 ô 2 Ψ( ν χ x x/ f X 2> " ' > x f) ^ = lim — Δ—0 Δ (ψ(χ χ + £Δ, χ, 2 · · · , x) f — ψ(χ χ — |Δ, # , · · · , 2 x )) f and this is precisely the derivative of ψ with respect to x .

IF (Φ, Φ) = 1, then Ψ is said t o be normalized. I f the integral oo is finite, then Φ can always be normalized b y multiplication b y a constant ^1/c in the case above, since ^— ,— J = i j . T w o FUNCTIONS are orthogonal if their scalar product is zero. The scalar product given in the Eq. 7) is constructed by considering the functions Φ(χ · · · X ), G(X · · · # / ) of X X , · · · , X as vectors, whose components are labeled by / continuous indices. The function vector Φ{χ · · · X ) is defined in an /-fold infinite-dimensional space.