Quantum Chemistry/Complex numbers

Complex numbers

Complex numbers are expressed as a combination of a real and imaginary part, in the form $a+bi$ . The variables $a$ and $b$ are both real integers but $a$ is the real component and $bi$ is imaginary due to the presence of the imaginary unit $i$ . It is defined as $i$ ² $=-1$ and therefore $i$ $={\sqrt {-1}}$ .

Working with the square root of negative numbers was impossible until the introduction of this concept and since its inception it has found numerous industrial and theoretical applications.

For example, electrical engineers use it to calculate impedance $(Z)$ , the resistance of an alternating current circuit as it varies with frequency, which an important analysis tool. The relation goes as follows, where $i=j$ , $R$ is resistance and $X$ is reactance in electrical engineering.

Z=R+jX

In this case resistance is real while reactance is an imaginary component.

In quantum chemistry the probability distribution of a quantum mechanical (QM) particle, is obtained by multiplying the wavefunction, $\psi$ by it's complex conjugate $\psi$ ^*.

Hydrogen Electron Orbitals as Predicted by Quantum Mechanics

The Probability Distribution of a QM particle

$[\psi (x,y,z)]$ ² $=\psi (x,y,z)$ ^*• $\psi (x,y,z)$

This is an expression of the likelihood of a quantum mechanical being at a certain point in space over time. In combination with the Schrödinger equation it has allowed us to map the density of electrons around the nucleus. These distributions form highly complex shapes which can be depicted and understandable to the human mind. The knowledge of molecules at the atomic level which this equation has brought to mankind is immense.

Euler's Relation

When working with imaginary numbers there are several shortcuts to simplify the task. An example of a simple yet useful relation is $(a+bi)$ · $(a-bi)=a$ ² $+b$ ².

In 1748 Leonhard Euler published his first work. It contained a relation which showed the fundamental relationship between the imaginary exponential, $e^{ix}$ , and the combination of a cosine curve and an imaginary sine curve.

Illustration of Euler's Relation

Euler's Relation

$e^{ix}=\cos(x)+i\sin(x)$

It demonstrated the point at which algebra and geometry unified and changed mathematics and all disciplines descended from it forever. It was groundbreaking, so much so that Richard Feynman, a renowned physicist called it "one of the most remarkable, almost astounding, formulas in all of mathematics"^[1]

Sample Question

Use Euler's relation to simplify the following expression, which is relevant to the phase factor in quantum wavefunctions:

$e^{i\pi /4}\cdot e^{-i\pi /3}$

Express your answer in the form a+bi, where a and b are real numbers.

Answer

Both halves of the expression can be written in trigonometric form using Euler's formula.

$e^{i\pi /4}=\cos(\pi /4)+i\sin(\pi /4)$

$e^{-i\pi /3}=\cos(-\pi /3)+i\sin(-\pi /3)$

They can then be plugged back into the expression.

$(\cos(\pi /4)+i\sin(\pi /4))\cdot (\cos(-\pi /3)+i\sin(-\pi /3))$

Special Right Triangle for 45 degree angle (π/4)

Special Right Triangle for 30 (π/6) and 60 (π/3) degree angles

These sine and cosines have real integer answers. They can be found using a calculator or by using shortcuts such as the special right angle triangles and "SOHCAHTOA" which relates trigonometric expressions to the opposite, adjacent and hypotheneuse of triangles.

$\left({\frac {adj}{hyp}}(\pi /4)+i{\frac {opp}{hyp}}(\pi /4)\right)\cdot \left({\frac {adj}{hyp}}(-\pi /3)+i{\frac {opp}{hyp}}(-\pi /3)\right)$

$\left({\frac {1}{\sqrt {2}}}+i{\frac {1}{\sqrt {2}}}\right)\cdot \left({\frac {-1}{2}}-i{\frac {\sqrt {3}}{2}}\right)$

The two parentheses can then be multiplied by one another. Note here that when $i$ is multiplied by itself it equals -1.

${\frac {-1}{2{\sqrt {2}}}}-i{\frac {\sqrt {3}}{2{\sqrt {2}}}}-i{\frac {1}{2{\sqrt {2}}}}+{\frac {\sqrt {3}}{2{\sqrt {2}}}}$

The last step is collecting like terms.

${\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}-i{\frac {({\sqrt {3}}+1)}{2{\sqrt {2}}}}$

Which can also be expressed with decimals.

$0.269-0.966i$

↑ Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew L. (2011). The Feynman lectures on physics (in Eng) (New millennium ed.). New York: Basic Books. ISBN 978-0-465-02414-8. OCLC 671704374.{{cite book}}: CS1 maint: unrecognized language (link)

[1] Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew L. (2011). The Feynman lectures on physics (in Eng) (New millennium ed.). New York: Basic Books. ISBN 978-0-465-02414-8. OCLC 671704374.{{cite book}}: CS1 maint: unrecognized language (link)

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