An Introduction to Complex Numbers and Their Role in Quantum Mechanics
Complex Numbers Basics Complex numbers extend the real number system by introducing the imaginary unit i, where i² = -1. A complex number is written as: z = a + bi Here, a is the real part, b is the imaginary part, and i represents the square root of -1. Complex numbers can be visualized on a 2D plane, called the complex plane, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part. Operations with Complex Numbers 1. Addition and Subtraction: Operate component-wise. (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) - (c + di) = (a - c) + (b - d)i 2. Multiplication: Use the distributive property and simplify using i² = -1. (a + bi)(c + di) = (ac - bd) + (ad + bc)i 3. Division: Multiply numerator and denominator by the conjugate of the denominator. (a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)] = [(ac + bd) + (bc - ad)i] / (c² + d²) 4. Magnitude (Modulus): The distance from the origin in the complex plane. |...