1.1 Complex Numbers

Quantum mechanics is full of complex numbers, numbers involving

$${\rm i}=\sqrt{-1}.$$

Note that $\sqrt{-1}$ is not an ordinary, “real”, number, since there is no real number whose square is $-1$; the square of a real number is always positive. This section summarizes the most important properties of complex numbers.

First, any complex number, call it $c$, can by definition always be written in the form

$$c = c_r+{\rm i}c_i$$ (1.1)

where both $c_r$ and $c_i$ are ordinary real numbers, not involving $\sqrt{-1}$. The number $c_r$ is called the real part of $c$ and $c_i$ the imaginary part.

You can think of the real and imaginary parts of a complex number as the components of a two-dimensional vector:

$$\begin{picture}(200,70)(-100,-20) \thinlines \put(-5... ...(2,1){70}} \put(0,23){\makebox(0,0)[tl]{$c$}} \end{picture}$$

The length of that vector is called the “magnitude,” or “absolute value” $\vert c\vert$ of the complex number. It equals

$$\vert c\vert = \sqrt{c_r^2+c_i^2}.$$

Complex numbers can be manipulated pretty much in the same way as ordinary numbers can. A relation to remember is:

$$\frac{1}{{\rm i}} = -{\rm i}$$ (1.2)

which can be verified by multiplying top and bottom of the fraction by ${\rm i}$ and noting that by definition ${\rm i}^2=-1$ in the bottom.

The complex conjugate of a complex number $c$, denoted by $c^*$, is found by replacing ${\rm i}$ everywhere by $-{\rm i}$. In particular, if $c=c_r+{\rm i} c_i$, where $c_r$ and $c_i$ are real numbers, the complex conjugate is

$$c^* = c_r - {\rm i}c_i$$ (1.3)

The following picture shows that graphically, you get the complex conjugate of a complex number by flipping it over around the horizontal axis:

$$\begin{picture}(200,100)(-100,-50) \thinlines \put(-... ...1){70}} \put(0,-23){\makebox(0,0)[bl]{$c^*$}} \end{picture}$$

You can get the magnitude of a complex number $c$ by multiplying $c$ with its complex conjugate $c^*$ and taking a square root:

$$\vert c\vert = \sqrt{c^* c}$$ (1.4)

If $c=c_r+{\rm i}c_r$, where $c_r$ and $c_i$ are real numbers, multiplying out $c^*c$ shows the magnitude of $c$ to be

$$\vert c\vert = \sqrt{c_r^2+c_i^2}$$

which is indeed the same as before.

From the above graph of the vector representing a complex number $c$, the real part is $c_r=\vert c\vert\cos\alpha$ where $\alpha$ is the angle that the vector makes with the horizontal axis, and the imaginary part is $c_i=\vert c\vert\sin\alpha$. So you can write any complex number in the form

$$c = \vert c\vert\left(\cos\alpha+{\rm i}\sin\alpha\right)$$

The critically important Euler formula says that:

$$\cos\alpha + {\rm i}\sin\alpha = e^{{\rm i}\alpha} %$$ (1.5)

So, any complex number can be written in “polar form” as

$$c = \vert c\vert e^{{\rm i}\alpha} %$$ (1.6)

where both the magnitude $\vert c\vert$ and the phase angle (or argument) $\alpha$ are real numbers.

Any complex number of magnitude one can therefore be written as $e^{{\rm i}\alpha}$. Note that the only two real numbers of magnitude one, 1 and $-1$, are included for $\alpha=0$, respectively $\alpha=\pi$. The number ${\rm i}$ is obtained for $\alpha=\pi/2$ and $-{\rm i}$ for $\alpha=-\pi/2$.

(See note {A.6} if you want to know where the Euler formula comes from.)

Key Points

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Complex numbers include the square root of minus one, ${\rm i}$, as a valid number.

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All complex numbers can be written as a real part plus ${\rm i}$ times an imaginary part, where both parts are normal real numbers.

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The complex conjugate of a complex number is obtained by replacing ${\rm i}$ everywhere by $-{\rm i}$.

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The magnitude of a complex number is obtained by multiplying the number by its complex conjugate and then taking a square root.

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The Euler formula relates exponentials to sines and cosines.

1.1 Review Questions

1

Multiply out $(2+3{\rm i})^2$ and then find its real and imaginary part. Answer

2

Show more directly that $1/{\rm i}=-{\rm i}$. Answer

3

Multiply out $(2+3{\rm i})(2-3{\rm i})$ and then find its real and imaginary part. Answer

4

Find the magnitude or absolute value of $2+3{\rm i}$. Answer

5

Verify that $(2-3{\rm i})^2$ is still the complex conjugate of $(2+3{\rm i})^2$ if both are multiplied out. Answer

6

Verify that $e^{-2{\rm i}}$ is still the complex conjugate of $e^{2{\rm i}}$ after both are rewritten using the Euler formula. Answer

7

Verify that $\left(e^{{\rm i}\alpha}+e^{-{\rm i}\alpha}\right)/2=\cos\alpha$. Answer

8

Verify that $\left(e^{{\rm i}\alpha}-e^{-{\rm i}\alpha}\right)/2{\rm i}=\sin\alpha$. Answer