Real wavefunctions

Wigner (1930) has shown that if time is reversible in a quantum-mechanical system, then all wavefunctions can be made real. This theorem enables us to use real wavefunctions whenever possible, which are often more convenient than complex ones. Here we present a simplified proof of Wigner s theorem, with some examples of its applications. 

In the absence of a magnetic field, the Schrodinger equation of a particle moving in an external potential U(r) is 

This equation is explicitly time-reversal invariant, because it only contains the square of the momentum p = - ihV. By taking the complex conjugate of Eq. (A.l), we have 

Obviously, if i i is a solution of Eq. (A.l), then i / is also a solution of Eq. (A.l) with the same energy eigenvalue E. Consequently, the linear combinations of i i and i j are also solutions of Eq. (A.l) with the same energy eigenvalue. If i i and i i are linearly independent (that is, they do not differ by a constant multiplier), then the following two real wavefunctions. 

Those wavefunctions represent a pair of standing-wave states with a 90° phase difference in space. 

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I have included the modulus bars in IV (r)p because wavefunctions can be complex quantities. For most of this and subsequent chapters, I will assume that we are dealing with real wavefunctions. 

The first and third terms on the right-hand side are equal for a real wavefunction. (hfferentiating the normalization condition gives... 

An analytical gradient calculation is invariably faster than a numerical one. To repeat the argument from Chapter 14, with real wavefunction Hamiltonian H (including the field terms) and parameter a (where a is a component of the external electric field)... 

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

The time-dependent wavepacket constructed in Section 4.1 is not the wavepacket that a laser with finite duration creates in the excited electronic state. It represents the wavepacket created by a pulse with infinitely narrow width in time. In order to construct the real wavefunction of the molecular system we must go back to Section 2.1. For simplicity of presentation, let us consider a diatomic molecule with internuclear separation R. We assume that the excitation takes place from the electronic ground state (index 0) to a bound upper state (index 1). The extension to a dissociative state, several coupled excited states, or several degrees of freedom is formally straightforward. 

Notice that these equations explicitly include derivatives and the complex conjugate f of the wavefunction. The expression for the momentum even includes i = f—W Complex numbers are not just a mathematical convenience in quantum mechanics they are central to the treatment. Equation 6.6 illustrates this point directly. Any measurement of the momentum (for example, by measuring velocity and mass) will of course always give a real number. But if the wavefunction is purely real, the integral on the right-hand side of Equation 6.6 is a real number, so the momentum is a real number multiplied by ih. The only way that product can be real is if the integral vanishes. Thus any real wavefunction corresponds to motion with no net momentum. Any particle with net momentum must have a complex wavefunction. 

Since this is a one-dimensional problem, there are only two choices the momentum is either pointed along +x or —x, so px = nh/2L. However, for any of the stationary states (or any other real wavefunction, as discussed earlier), Equation 6.6 gives (p) = 0. So a particle in any of the stationary states is not moving on average. Any single observation would give px = nh/2L, but the average of many observations would be zero. 

Inspection shows that T (x) = T (x), which is the expected result for a real wavefunction. 

A set of real wavefunctions Xi will be constructed such that upon taking appropriate linear combinations, the complex y>fi and having asymptotic forms given by (9a)-(9d) can be obtained. Following Kohn, real wave-functions may be constructed having the asymptotic form... 

In order to interpret these values further, it is necessary to have values for the overlap integrals (10). We obtained these as follows An approximate real wavefunction which fits the outer regions of the 5/function well (within 0.4% for af,nonlinear least squares procedure. The square of a function composed of Slater-type orbitals was fit to the probability amplitude calculated from a relativistic SCF wavefunction. The result is... 

Moreover, this term is the difference of the kinetic energy density of the actual system and of that of a system of spin-free independent particles both with identical one-particle densities />(r).For real wavefunctions or for stationary states, it is simply the difference of the definite positive kinetic energies since the (unwanted) remaining contributions cancel one another. Another attractive property of the non-von Weizsacker contribution is that it appears to be the trace of the Fisher s Information matrix[28]. 

Four of these wavefunctions are complex, because they include the term e /, but real wavefunctions can be obtained by combining pairs of complex wavefunctions with the same mj, but opposite sign. The orbitals obtained by this process are listed in Table 6.2, and their shapes are shown in Figure 6.11. Linear combinations of the complex orbitals corresponding to m, = 1 give the 3d and 3d, orbitals, whereas those with m, = 2 combine to give the 3d 2 j,2 orbitals. The 3d,j orbital,... 

Or can we A closer look reveals that we can form pure real wavefunctions by taking linear combinations of the m/ = 1 orbitals like this ... 

As can be seen from Table 11.4, wavefunctions having a nonzero value for ntf have an imaginary exponential function part. This means that the overall wavefunction is a complex function. In cases where completely real functions are desired, it is useful to define real wavefunctions as linear combinations of the complex wavefunctions, taking advantage of Eulers theorem. For example ... 

The complete set of wavefunctions is similar. Such a set can be used to define the complete space of a system. The true wavefunction for a real, that is, nonmodel, system can be written in terms of the complete set of ideal wavefunctions, just like any point in space can be written in terms of x, y, and z. Using first-order perturbation theory, any real wavefunction jeai can be written as an ideal wavefunction plus a sum of contributions of the complete set of ideal wavefunctions... 

Although the process is lengthy, it is algebraically straightforward to determine what the expansion coefficients are for the correction to the nth real wavefunction,... 

Recall that each is approximated initially by an ideal The m expansion coefficient a , for the perturbation to the th real wavefunction jeai can be defined in terms of the perturbation operator H, the th and mth ideal wavefunctions and and the energies E and of the ideal wavefunctions. Specifically,... 

Note the ordering of the terms having m and n indices in the above equation it is important to keep them straight. ,reai is still very similar to the nth ideal wavefunc-tion, but now it is corrected in terms of the other wavefunctions that define the complete set of wavefunctions for the model system. Real wavefunctions defined in this way are not normalized. They must be normalized independently, once the proper set of expansion coefficients has been determined. 

Assume that, for a real system, a real wavefunction is a linear combination of two orthonormal basis functions where the energy integrals are as follows = —15... 

Show that the two real wavefunctions determined in Example 12.12 are orthonormal. 

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