Arithmetic of Complex Numbers

Arithmetic of Complex Numbers#

If we have two complex numbers \(a+bi\) and \(c+di\), then we have the following rules for the arithmetic between them:

  1. Addition:

    \[ (a+bi)(c+di) = (a+c) + (b+d)i \]

    Add the real parts and add the imaginary parts separately.

  2. Subtraction:

    \[ (a+bi)-(c+di) = (a-c) + (b-d)i \]

    Same as addition, but with subtraction.

  3. Multiplication:

    \[ (a+bi)(c+di) = (ac - bd) + (ad + bc)i \]

    Treat it as 2 sets of brackets and use FOIL. Remembering that \(i^2=-1\).

  4. Division:

    \[ \frac{a+bi}{c+di} = \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i \]

    Multiply the numerator and demoniator by the conjugate of the denominator. This makes the demoninater a real number and the numerator the multiplication of two complex numbers.

Example

Find the following:

  1. \((2+i)+(1-i)\)

  2. \((-4-4i) - (-5i)\)

  3. \((1+i)(2-3i)\)

  4. \(\frac{1-i}{2+3i}\)

Solution:

  1. \((2+i)+(1-i) = (2+1) + (1-1)i = 3\)

  2. \((-4-4i) - (-5i) = (-4-0) + (-4-(-5))i = -4+i\)

  3. \((1+i)(2-3i) = ((1\times 2) - (1\times -3)) + ((1\times -3) + (1\times 2))i = 5-i\)

  4. \(\frac{1-i}{2+3i} = \frac{(1\times 2) + (-1\times 3)}{2^2+3^2} + \frac{(-1\times 2) - (1\times 3)}{2^2+3^2} i = \frac{-1}{13} - \frac{5}{13}i\)

Naturally, the Python complex-type is capable of this arithmetic.

2 + 1j + 1 - 1j

(3+0j)

-4-4j - (-5j)

(-4+1j)

(1 + 1j) * (2 - 3j)

(5-1j)

(1 - 1j) / (2 + 3j)

(-0.07692307692307694-0.3846153846153846j)

Particle Probability

A particle is describes by a wavefunction \(\Psi\). To calculate the probability of finding this particle somewhere \(\Psi\Psi^*\) needs to be calculated. When \(\Psi\) is in the form:

\[ \Psi = a+bi \]

calculate \(\Psi\Psi^*\).

Solution: Since \(\psi = a+bi\), this means that \(\Psi^*=a-bi\).

  • Multipying them together gives: \((a+bi)(a-bi)\)

  • Expand the brackets: \(a^2 + abi - abi - (bi)^2\)

  • Simplify this: \(a^2-b^2i^2\)

  • Simplify the \(i^2\): \(a^2+b^2\).