Quantum mechanics is full of complex numbers, numbers involving
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) |
You can think of the real and imaginary parts of a complex number
as the components of a two-dimensional vector:
Complex numbers can be manipulated pretty much in the same way as
ordinary numbers can. A relation to remember is:
(2.2) |
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) |
You can get the magnitude of a complex number by multiplying
with its complex conjugate and taking a square root:
(2.4) |
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
polar formas
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.
Multiply out and then find its real and imaginary part.
Show more directly that 1 .
Multiply out and then find its real and imaginary part.
Find the magnitude or absolute value of .
Verify that is still the complex conjugate of if both are multiplied out.
Verify that is still the complex conjugate of after both are rewritten using the Euler formula.
Verify that 2 .
Verify that .