Position eigenfunctions

If the property B has a continuous range of eigenvalues (for example, position Section 7.7), the summation in the expansion (7.66) of is replaced by an integration over the values of b  

We derived the eigenfunctions of the linear-momentum and angular-momentum operators. We now ask What are the eigenfunctions of the position operator x  

The operator x is multiplication by x. Denoting the position eigenfunctions by ga(x), we write 

Moreover, since an eigenfunction that is zero everywhere is unacceptable, we have 

These conclusions make sense. If the state function is an eigenfunction of x with eigenvalue fl, = ga(x), we know (Section 7.6) that a measurement of x is certain to give the value a. This can be true only if the probability density I Fp is zero for x a, in agreement with (7.79). 

Before considering further properties of ga(jc), we define the Heaviside step function H(x) by (see Fig. 7.4) 

We next define the Dirac delta function 5(jc) as the derivative of the Heaviside step function  

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In dealing with systems containing only two electrons we have not been troubled with the exclusion principle, but have accepted both symmetric and antisymmetric positional eigenfunctions for by multiplying by a spin eigenfunction of the proper symmetry character an antisymmetric total eigenfunction can always be obtained. In the case of two hydrogen atoms there are three... 

P (2) — p (1) a (2). The last is required to make the symmetric positional eigenfunction of Equation 29a conform to Pauli s principle, and the first three for the antisymmetric 4>H2- Since the a priori probability of each eigenfunction is the same, there... 

Fig. 4.—A diagram representing the electronic configurations of certain complex ions. Each circle represents a single-electron positional eigenfunction, each dot an electron.

The properties (7.88) of the Dirac delta function agree with the properties (7.78) and (7.79) of the position eigenfunctions g (jr). We therefore tentatively set... 

Theorem 2 shows that /x2, and hence is a continuous function of a for a > 0. Let O be the quasi-positive eigenfunction corresponding to the dominant eigenvalue of Equation (14) and be the positive adjoint... 

The positions of the (—1) terms in the diagonal indicate which of the electronic eigenfunction flips sign upon tracing the closed contour under... 

However, in order to give an unambiguous answer to the question of how one is to calculate the probability of finding a Klein-Gordon particle at some point x at time t, we must first find a hermitian operator that can properly be called a position operator, and secondly find its eigenfunctions. It is somewhat easier to determine the latter since these should correspond to states wherein the particle is localized at a given point in space at a given time. Now the natural requirements to impose on localized states are ... 

We next turn our attention to the problem of finding the position operator q, of which the localized state (t) is an eigenfunction with eigenvalue y. That this operator is not given by q = Vk in the momentum representation becomes dear upon noting that the operator... 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

Strictly speaking, the complete eigenfunction for the molecule should be made antisymmetric before the charge densities at the various positions are calculated. It is easily shown, however, that this further refinement in the treatment does not alter the results obtained. 

Equation (3.30) may be used as a criterion for completeness. If an eigenfunction tpn with a non-vanishing coefficient an were missing from the summation in equation (3.27), then the series would still converge, but it would be incomplete and would therefore not converge to /. The corresponding coefficient a would be missing from the left-hand side of equation (3.30). Since each term in the summation in equation (3.30) is positive, the sum without a would be less than unity. Only if the expansion set ipf in equation (3.27) is complete will (3.30) be satisfied. 

Eventually this procedure produces an eigenfunction S -kfi, k being a positive integer, such that 0 <( — k) 1. The next step in the sequence would give a function 5 t-i,o or 5 20 with A = ( - -l) 0, which is not allowed. Thus, the sequence must terminate with the condition... 

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