Wavefunctions imaginary numbers

Appendix 9 for now, though, we just note that the product of a function with its own complex conjugate gives a real (nonimaginary) value that is the magnitude of the wavefunction squared the wavefunctions may contain imaginary numbers, but the density will be real and so observable. 

In Appendix 6 we made a quick note that our wavefunctions may turn out to contain imaginary numbers, i.e. they may contain i = V. For the electron wavefunction problems studied in this appendix, imaginary numbers become more important, and so this section will give a brief overview to show how they should be handled. 

Eq. (14.3) is the iterative equivalent of the TDSE. It does not contain the imaginary number i. If x, t) is just the real part of a complex wavefunction, then the equation will propagate forward, completely exactly, this real part of the complete wavefunction. The advantage of this is both one of storage and of computer time. Only half the computer storage is required for the real part of a wavefunction as compared for the complete complex wavefunction. The multiplication of two complex numbers requires four times as many operations as the multiplication of two real numbers. 

Note that the Is function given above is not the only function which satisfies the Schrbdinger equation for E = Ei multiplication of the Is wavefunction by any number c, positive or negative, real, imaginary or complex, yields a new function which is also a solution of the time-independent Schrodinger equation, equation (1.9), with the same energy ... 

In summary, the band theory of solids can be used to explain the structure, spectroscopy and electrical properties of one-dimensional solids, such as the linear chain tetracyanoplatinates. However, most crystalline solids have translational symmetry in more than one dimension. Can band theory be extended to two and even three dimensions The answer is, of course, yes. However, the shapes of the bands become significantly more complicated as the number of dimensions increases. Let us just consider two dimensions for the moment and an imaginary lattice composed of only H atoms. In two dimensions, the wavefunction for the Bloch orbitals is given by Equation (I 1.27) ... 

When we report numbers we can measure in the laboratory, we do expect them to be pure real numbers, but we never directly measure wavefunctions. We allow wavefunctions to be complex functions, having both real and imaginary components, because the imaginary component can carry information about the interference properties of wavefunctions. This is just a more sophisticated treatment of the phase angle 0. A more general way to write a function that has a well-defined kinetic energy uses Euler s equation (Eq. 2.8),... 

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