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.

Show more di­rectly that 1 .

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 .