### 2.1 Complex Numbers

Quantum mechanics is full of complex numbers, numbers involving

Note that 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 , can by definition always be written in the form

 (2.1)

where both and are ordinary real numbers, not involving . The number is called the real part of and the imaginary part.

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

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

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

 (2.2)

which can be verified by multiplying the top and bottom of the fraction by and noting that by definition 1 in the bottom.

The complex conjugate of a complex number , denoted by , is found by replacing everywhere by . In particular, if , where and are real numbers, the complex conjugate is

 (2.3)

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

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

 (2.4)

If , where and are real numbers, multiplying out shows the magnitude of to be

which is indeed the same as before.

From the above graph of the vector representing a complex number , the real part is where is the angle that the vector makes with the horizontal axis, and the imaginary part is . So you can write any complex number in the form

The critically important Euler formula says that:
 (2.5)

So, any complex number can be written in polar form as
 (2.6)

where both the magnitude and the phase angle (or argument) are real numbers.

Any complex number of magnitude one can therefore be written as . Note that the only two real numbers of magnitude one, 1 and 1, are included for 0, respectively . The number is obtained for ​2 and for ​2.

(See derivation {D.7} if you want to know where the Euler formula comes from.)

Key Points
Complex numbers include the square root of minus one, , as a valid number.

All complex numbers can be written as a real part plus times an imaginary part, where both parts are normal real numbers.

The complex conjugate of a complex number is obtained by replacing everywhere by .

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

The Euler formula relates exponentials to sines and cosines.

2.1 Review Questions
1.

Multiply out and then find its real and imaginary part.

2.

3.

Multiply out and then find its real and imaginary part.

4.

Find the magnitude or absolute value of .

5.

Verify that is still the complex conjugate of if both are multiplied out.

6.

Verify that is still the complex conjugate of after both are rewritten using the Euler formula.

7.

Verify that ​2 .

8.

Verify that .