Group Theory: And its Application to the Quantum Mechanics by Eugene P. Wigner, J. J. Griffin

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.

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