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Mathematical tools for QM 1. Complex Exponential A basic understanding of complex calculus will help enormously with grasping the ins and outs of Quantum Mechanics. In particular, the function eix plays a pivotal role in all of this. Why? Because functions of this form are general solutions of the wave equation, and Quantum Mechanics happens to be based on the Schrödinger’s Wave Equation. The complex exponential is defined as follows: eix = cos(x) + i · sin(x) (1) with i, of course, defined as i2 = −1. From (1) the wave-character of the complex exponential is clear. In fact, it consists of two waves, a Real part Re eix = cos(x), and an Imaginary part Im eix = sin(x). Importantly, these waves are always π/2 ‘out-of-phase’ with each other (remember that cos(x) = sin(x + π/2)). The Real and Imaginary parts are therefore said to be orthogonal to each other. The complex exponential is often thought of as a ‘phase function’, with an amplitude of 1 and a phase x as defined by the angle between the Real and Imaginary parts. Below are some useful expressions related to this. Square modulus ix 2 e = eix · e−ix = {cos(x) + i · sin(x)} · {cos(x) − i · sin(x)} = cos2 (x) + sin2 (x) = 1 (2) Amplitude ix q e = |eix |2 = 1 Phase Θ ≡ arctan ! Im eix sin(x) = arctan =x Re {eix } cos(x) (3) (4) Special phase settings (with n = ± 0, 1, 2, 3....) e e ei2nπ = 1 i(2n+1)π = −1 i(2n+1) π 2 = 4 (5) (6) n (−1) i (7) 2. Useful Trigonometric Identities sin2 (x) + cos2 (x) = 1 (8) sin(2x) = 2 sin(x) cos(x) cos(2x) = cos2 (x) − sin2 (x) = cos2 (x) = sin2 (x) = sin3 (x) = cos3 (x) = sin(x) = cos(x) = 1 − 2 sin2 (x) 1 {1 + cos(2x)} 2 1 {1 − cos(2x)} 2 1 {3 sin(x) − sin(3x)} 4 1 {3 cos(x) + cos(3x)} 4 eix − e−ix 2i eix + e−ix 2 (9) (10) (11) (12) (13) (14) (15) (16) 3. Table of Useful Integrals where a > 0 and n is an integer 2π Z sin2 (nx)dx = π (17) sin(ax2 )dx = (18) cos(ax2 )dx Z ∞ e−ax dx = 0 ∞ Z 0 Z ∞ 0 = 0 ∞ Z xn e−ax dx = n!/an+1 r 1 π = 2 a 1 = 2a r 1 π = 4a a 0 Z Z Z ∞ 0 ∞ 2 e−ax dx 0 ∞ {n 6= 0} r 1 π 2 2a r 1 π 2 2a 1 a 2 xe−ax dx 2 x2 e−ax dx 0 5 (19) (20) (21) (22) (23) (24) 4. Rules for Differentiation and Integration d (f g) dx dg df +g dx dx df {g(x)} dg(x) d (f {g(x)}) = dx dg(x) dx Z Z dg df f dx = f g − g dx dx dx = f 6 (25) (26) (27)