Prelude—Imaginary and Complex Numbers

Before we go into the mathematical framework behind wave mechanics, we will review one more mathematical concept normally seen in high school imaginary and complex numbers. As discussed in Section 1.2, for a general quadratic equation ax2 + bx+c = 

If we define i = (an imaginary number) then we can still write out two solutions for b2 4ac. In general these solutions will have both a real and an imaginary part, and are called complex numbers. For example, if x2 — 2x + 2 = 0, the solutions are x = 1 i. Complex solutions to quadratic equations are not important in chemical problems however, complex numbers themselves will prove to be important in quantum mechanics. 

The Taylor series expansion in Chapter 2 makes it possible to derive a remarkable relationship between exponentials and trigonometric functions, first found by Euler  

All of the usual properties of exponentials (Equations 1.13 and 1.14) also apply to complex exponentials. For example, the product of two exponentials is found by sum- 

FIGURE 6.1 Left representation of z = x + iy in the complex plane, showing the magnitude z and phase 0. Right the quantity e ° is always on the unit circle, and is counterclockwise hy an angle 9 from the x-axis. 

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