ХИМИЧЕСКАЯ ФИЗИКА,, 2014, том 33, № 2, с. 19-26
ЭЛЕМЕНТАРНЫЕ ФИЗИКО-ХИМИЧЕСКИЕ ПРОЦЕССЫ
УДК 544.147.5; 544.183.4
COMPLEX COORDINATE SCALING AND THE SCHRÖDINGER EQUATION
© 2014 R. J. Buenker
Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Germany
E-mail: bobwtal@yahoo.de Received 15.01.2013
The complex rotation method (CRM) for the description of resonance states is critically analyzed by noting that quantum mechanical wave functions and properties are not affected by a change in spatial coordinates, complex or otherwise. It is shown by means of the Cauchy-Coursat Theorem that equivalent approximate solutions of the Schrödinger equation for a complex-rotated Hamiltonian H(Q) can be obtained without loss of accuracy by using the un-rotated Hamiltonian H(0) in its place. Despite the fact that the latter operator is hermitean, it is possible to obtain a complex symmetric matrix representation for it by following a few simple rules: a) the square-integrable basis functions must have complex exponents, i.e. with non-zero imaginary components and b) the symmetric scalar product must be employed to compute matrix elements of H(0). The approximate wave functions obtained by diagonalization of the latter matrix should satisfy the stationary principle as closely as possible. This objective can optimally be achieved by individually scaling the complex exponents in the basis functions. The nature of this approximation is investigated by means of explicit calculations which are based on diabatic RKR potentials for the B1E+—vibronic resonance states of the CO molecule.
Keywords: complex coordinate rotation method, resonance states, complex symmetric matrix representation, square-integrable basis functions, approximate wave functions, diagonalization of matrix, vibronic resonance states of CO molecule.
DOI: 10.7868/S0207401X14020022
1. INTRODUCTION
The original tests of the Schrodinger equation [1, 2] dealt exclusively with bound states for a variety of potentials. Such states are characterized by time-independent charge distributions and thus have infinite decay lifetimes. Gamov [3] and Siegert [4] later showed that there could be other solutions of the Schrodinger equation corresponding to meta-stable states with a finite lifetime. The condition for this type of (resonance) state is that it must possess a complex energy eigenvalue E=Er + iE. In that case the product of the time-dependent part of the wave function ®(t) = exp(—iEt) with its complex conjugate is no longer constant and thus would in principle correspond to a charge distribution which decays over time. One of the outgrowths of this idea was the common belief that only a non-hermitean Hamiltonian operator can have eigenfunctions that have other than real energy eigenvalues. This conclusion in turn led to the complex rotation method (CRM) for the description of resonance states [5—8], which is based on earlier studies of the convergence properties of time-dependent perturbation series for atomic resonances by Balsev, Aguilar and Combes [9, 10] and Simon [11—13]. Accordingly, the Hamilto-nian of such a theoretical treatment must differ from the standard operator employed for bound states by
rotating the original real spatial coordinates into the complex plane with the mapping: R ^Rexp(i9). The working hypothesis of the CRM is therefore that for each resonance state there is a unique (real) value of the scaling parameter 9 that defines a Hamiltonian H(9) of which the state is an eigenfunction. The corresponding complex energy eigenvalue Ee can then be used to determine both the energy position of the resonance as well as its lifetime. The question that will be considered in the following is whether this hypothesis of a unique value of the scaling parameter in the CRM is correct or if in fact the desired characteristics of resonances can be obtained directly by solving the Schrodinger equation on the real axis with the un-scaled Hamiltonian H(0).
2. COMPLEX ROTATION OF RESONANCE EIGENFUNCTIONS
According to the CRM, the following time-independent Schrodinger equation holds for the complex rotated Hamiltonian H(9) described above:
H (0) ¥ (0) = E^ (0). (1)
The key assumption is that if the eigenfunction T(9) corresponds to a resonance, the parameter 9 must
have a unique real value and the energy eigenvalue Ee must be complex, i.e. it must contain a non-zero imaginary component E. To test this assertion it is useful to apply a further complex-coordinate scaling to each of the quantities in this equation: R ^ Rexp(i9'). This means that the original Hamiltonian operator H(0) defined on the real axis is rotated by the sum of the two angles, i.e. by 9 + 9'. We can therefore express the result of the combined rotation as H(9 + 9'). Since the wave function T(9) in (1) also depends on the spatial coordinates R, it follows that it is also changed by the complex scaling and the result can be similarly expressed as T(9 + 9'). However, the eigenvalue Ee is by definition a constant and therefore its value is not changed by the additional scaling. We are thus led to the following result which also has the form of a Schrodinger equation with the same eigenvalue as before:
H (0 + 0') ¥ (0 + 0') = (0 + 0'). (2)
All that has been done is to systematically change the coordinates used to define a differential equation and thereby obtain a different but equivalent form for it. This is a standard procedure that is used as a matter of course in the field of differential equations. In short, the assertion of the CRM [5—8] that the Hamiltonian of the Schrodinger equation corresponding to a given resonance is uniquely defined by a specific complex rotation angle is patently false. Any rotation angle will lead to the same complex energy eigenvalue, including the special case of 9 + 9' = 0. If(1) is correct for a given resonance, it therefore must be possible to obtain both its energy eigenvalue and wave function by solving the corresponding Schrodinger equation on the real axis, or for any other value of 9 + 9'.
In this connection it is interesting to consider an argument that is commonly given to counter the above conclusion. It starts by introducing a prototype resonance function F(9, r) = exp(—e'er) defined on the real axis with n/2 < 9 < n. It is therefore an unbound or "non-L2" function since cos 9 < 0 in this region, i.e. it does not vanish in the limit as r approaches to. If the
'9'
coordinate is changed by the complex scaling r = e r',
the result is F (9 + 9', r') = exp(-e'(0+0)r'). It is then argued that if 9' is chosen so that 0 < 9 + 9' < n/2, the rotated function F(9 + 9', r') becomes a bound or L2 function. This claim is based on the fact that the angle 9 + 9' has been changed relative to that in the original function F(9, r), but it overlooks the fact that the coordinate has
also been changed. Substitution of r' = er in the rotated function shows that F(9 + 9', r') = F(9, r) and therefore that the coordinate rotation does not in fact convert the former into a bound function.
It should not come as a great surprise that the physical content of (1) is not affected by complex scaling. From a purely mathematical point of view, there is no essential difference between multiplying coordinates
with a complex constant or a real one. In the latter case it is clear that one would merely be altering the unit of distance in the theoretical formulation, changing from bohr to Á, for example. Once one has obtained a solution to a given Schrödinger equation, it is a straightforward matter to express the resulting wave function in any other set of units, complex or otherwise. Since the energy eigenvalue itself is not a function of the spatial coordinates, it is obvious that it will not change as the result of any such scaling procedure. However, the above argument shows in addition that no expectation value for any other quantum mechanical operator will be changed either. The conclusion is therefore that if a solution of (1) exists for a complex energy eigenvalue for any value of 9, exactly the same physical information must result from solution of the corresponding Schrödinger equation for #(0).
3. USE OF STANDARD MATRIX TECHNIQUES TO DESCRIBE RESONANCES
The above conclusion needs to be reconciled with the fact that the Hamiltonian defined on the real axis, i.e. H(0), is hermitean. One often reads in the literature that the eigenvalues of any hermitean matrix are real, for example, in Goldstein's treatise on classical mechanics [14]. This statement might seem to contradict the conclusion that the H(0) operator can possess complex energy eigenvalues, but that is not the case. The reason it is not is that the hermitean property for a linear operator only holds in general for a specific class of functions. In the case of quantum mechanical operators this set consists exclusively of square-integrable functions. As a result, there is no a priori reason to conclude that the spectrum of such an operator only consists of real eigenvalues. In the present case, the her-mitean character of H(0) merely requires that any and all eigenfunctions with complex energy eigenvalues do not belong to the class of square-integrable functions. This is a clear distinction between conventional bound states on the one hand, and resonances on the other.
The most effective means of obtaining approximate solutions for bound states is to represent them as linear combinations of square-integrable functions, and thus one would like to employ similar techniques for their resonance counterparts. However, in order to accomplish this goal in an optimal manner, it is necessary to deal with the fact that in this case the exact wave functions are themselves unbounded at infinity. Moiseyev and Corcoran [5] and McCurdy and Rescigno [7, 8] have designed a computational method that has been quite successful in describing resonances in terms of a basis of square-integrable func
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