Lamb Shifts in Helium
The application of this program would be full of interest, it would mean that we are no longer restricted to analytical solutions. That is a big step forward. Calculation of the Lamb shift for other elements would be exactly what is needed, provided of course that the results are compared with the data at each step. The necessary first step is to define the wave functions and compute results in the first instance with the usual wavefunction, starting with helium. This has an exchange energy as you know. The after that use the quantization rules t goes to m power half t and r goes to r / m(r) half. I think that if the wavefunctions are correctly inputted all the output will be correct. This looks like a powerful program. The calculations for the analytical solutions play an important role in clarifying the method. I did a quick literature search and there is a Lamb shift in the ground state of helium. By now these Lamb shifts must have been measured with great accuracy.
UFT436, Sections 1 and 2
Thanks for the hints. I see that normalization has to be restricted to a volume for infinitely extended states like the oscillatory time function psi_2.
The LDA program I have can compute structures of all atoms except hydrogen, because there is no electronic exchange and correlation. Therefore the Lamb shift could only be computed for other elements. I will give some demos of effects of m theory in UFT 436, but a correct handling would require both a modified Schrödinger or Dirac equation and (possibly) a change in the LDA expressions for the total energy which depends on the charge density alone. The energy eigenvalues are the so-called one-particle energies which do not represent binding energies. I guess if the wave functions have been computed self-consistently including the m function, the charge density is that of m theory automatically and needs not to be transformed any more. What do you think?
Horst
Am 12.04.2019 um 08:26 schrieb Myron Evans:
FOR POSTING: UFT436, Sections 1 and 2
Many thanks for going through these notes.
1) The problem of the divergence was addressed in Note 436(3), by restricting the normalization to the unit sphere. The same type of problem occurs in linear motion and has to be dealt with using a limit procedure (see Atkins, "Molecular Quantum Mechanics" for example, in this case Atkins happens to be right, his treatment of the particle in a box is wrong, as we showed in UFT226 ff.).
2) The separation of variables technique used in this paper is the same as in the previous paper UFT435, and the result (8) is more general. In other words if the assumption (2) is made, Eqs. (6) and (7) result self consistently. Eq. (6) is H psi1(r) = E psi(r) and Eq. (7) is the original Schroedinger quantization of energy. Since psi2 is defined as a function of t, then its partial derivative with respect to r vanishes. Similarly psi1 is defined as a function of r so its partial derivative with respect to t vanishes. These results were used in Eqs. (3) to (5) to give the self consistent result (6) and (7). To check this conceptually, consider the hydrogenic wavefunctions. These are made up of the product ps1(r) psi2(theta, phi), so the derivative of psi with respect to theta or phi is zero, and the derivative of psi2 with respect to r is zero. 3) The quantization rules in UFT435 and UFT436 were introduced so that the wavefunction is modified, but the structure of the Schroedinger quantization remains the same:
E psi = i h bar partial psi / partial t; p psi = – i h bar del psi.Your use of this new method in a computational quantum chemistry package produced sensible results, so my suggestion is to develop it systematically in computational quantum chemistry, in the first instance to give the Lamb shift in H.
Notes 436(3) and (5) had not come through to me before. There is a problem in both. The integralintegral m(r)^(1/2) d tau
diverges since m(r) goes to 1 for r–>inf.
The problem obviously is that replacement of time in the time part of the wave function psi_2 destroys its independence of r and even the separability of the wavefunction psi = psi_1 * psi_2.A second point is the question if also the integration variable r had to be transformed by
r –> r / sqrt (m(r)^(1/2)) .
This would probably lead to the result that the integrals have the same values when taken in r or r1 space. From the standpoint of general covariance this would even be desirable. As I had pointed out earlier, it seems only to make sense for me to transform the RESULTS obtained in m space back to the configuration space.
Horst
Am 10.04.2019 um 09:18 schrieb Myron Evans:
FOR POSTING: UFT436, Sections 1 and 2
This paper is on the development of generally covariant standard solutions of quantum mechanics. In section 3, the same kind of development can be implemented with computational quantum mechanics, an important new development by co author Horst Eckardt.