Complex Problems: Both Real and Imaginary About a decade ago I was trying to sort out my concepts of 2D NMR a t a group meeting, saying something Like:
...the radiofrequency field interacts with the imaginary component of the proton's magnetic vector... when my advisor interrupted: How can it interact? It's not real, it's imaginary. I hemmed and hawed. I'm still not sure that I can answer that question. I don't have the mathematical background to Drove it. Even if I did. I'm afraid I'd eet - lost in the math. But I do have an intuitive coneent. . . a eut feeline. -. about it. First, imaginary numbers are real. Theyjust involve the square root of -1. They come about because any exponential function can be rewritten a s a sum of sines and cosines. one real and the other imaPinarr - . Thus. whenever ex~anents or their inverse, logarithms, are involved, imaginary numbers are lurking. They probably also Lurk behind all sines, cosines, and trig functions in general and circles, cylinders, and spheres in particular. Wave functions automatically involve imaginary numbers, as do functions that decay exponentially or radiate spherically. What do they represent? For electmmagnetic radiation, the answer is simple. If you're looking a t the electric field, the imaginary component--its complex canjugate--is the magnetic field, and vice versa. And the mathematics that lets you go fmm one to the other is the Fourier transform. For electronic wavefunctions the answer is less simple. The square of the wave function represents the probability of fmding the particle in a particular region of space. Since the particle is charged, it is an electrical function. The wave function probably gives information about the magnetic field. Hence mL is called the magnetic quantum number. Thus, rhr intrraction of wave funetmns in molecular orh~wltheory reprwentx rnagnmc atrrnrtinn or repulsion. I'auli intmduecd rhe concc~tof electronic win ad hoc, but 1 helwvr that it falls out of rhr math in a relativistic treatment of quantum mechanics. It also falls out of the phase of the wave functions. T w o functions of Like sign undergo constructive interference, attract and are bonding. This occurs if electrons are antiparallel, one spin up (alpha) another spin down (beta). Functions of opposite sign undergo destructive interference, repel and are antibonding. This aecurs when spins are parallel, both spin up or bath spin down. Thus, electricity and magnetism are complex conjugates. Particles and waves are likewise. And the Einstein equation links the two concepts: E = mc2. ~
Thomas A. Eaton Division of Physical Science and Mathematics St. Thomas University Miami, FL 33054
434
Journal of Chemical Education
