Operators and Waves

John Snygg

The relationship between the density operator and operator solutions of Schr\"odinger's equation is discussed. We designate operator solutions of Schr\"odinger's equation as "operator wave functions" and show how they can be constructed from ordinary wave functions.

John C. Morrison

Adriana Moreo, Elbio R. Dagotto

A novel minimal-size pairing operator ${\mathrm{\ensuremath{\Delta}}}_{D0}^{\ifmmode\dagger\else\textdagger\fi{}}$ with $d$-wave symmetry in ${\mathrm{CuO}}_{2}$ planes is introduced. This pairing operator creates on-site Cooper pairs at the four oxygens that surround a copper atom. Via the time evolution of ${\mathrm{\ensuremath{\Delta}}}_{D0}^{\ifmmode\dagger\else\textdagger\fi{}}$, an additional interorbital pairing operator ${\mathrm{\ensuremath{\Delta}}}_{Dpd}^{\ifmmode\dagger\else\textdagger\fi{}}$ with $d$-wave symmetry is generated that pairs fermions located in a Cu and its four surrounding O atoms. The subsequent time evolution of ${\mathrm{\ensuremath{\Delta}}}_{Dpd}^{\ifmmode\dagger\else\textdagger\fi{}}$ generates an intraorbital $d$-wave pairing operator ${\mathrm{\ensuremath{\Delta}}}_{Dpp}^{\ifmmode\dagger\else\textdagger\fi{}}$ involving the four O atoms that surround a Cu atom, as well as the $d$-wave operator ${\mathrm{\ensuremath{\Delta}}}_{D}^{\ifmmode\dagger\else\textdagger\fi{}}$ traditionally used in single-band models for cuprates. Because we recover the larger size operators extensively used in the three-orbital Hubbard model, we suggest that long-range order using the canonical extended operators occurs together with long-range order in the new minimal operators. However, our minimal $d$-wave operators could be more practical to study $d$-wave superconductivity because in the finite-size relatively small systems accessible to computational techniques it is easier to observe long-range order using local operators. Moreover, an effective model with the usual tight-binding hopping of the ${\mathrm{CuO}}_{2}$ planes supplemented by an attractive potential $V$ in the $d$-wave channel is introduced. Using mean-field techniques, we show that a paired ground state is stabilized for any finite value of $V$. We observed that the values of $V$ that lead to gap sizes similar to those in the cuprates are smaller for $d$-wave pairing operators that include Cu $d$ orbitals than those that include only $p$ orbitals. In all cases the gap that opens in the spectrum has standard $d$-wave symmetry. Finally, a simpler effective model is introduced to study the phenomenology of multiorbital $d$-wave superconductors, similar to how the negative-$U$ Hubbard model is used for properties of $s$-wave superconductors.

John Morrison

AUGUST STERN

George Frederick James Temple

Abstract The operational wave equation used in this paper is a generalisation of the linear wave equation introduced by Dirac. The generalisation consists in replacing Dirac’s matrices by any four linear operators (A1, A2, A3, A4), which satisfy Dirac’s conditions, ½ (AmAn + AnAm) = 0 if m ≠ n, (1.1) 1 if m = n. The operational wave equation for free space is therefore.