Obtaining Complex Conjugates

Now, we go to complex conjugates which are extremely important, especially in applications to the quantum wave function (y) for which probability amplitudes are obtained using the complex conjugate (y*)  of the function.

 
To obtain the complex conjugate is very simple, for a complex number of the form

z = x + iy

The complex conjugate is written: z* = x – iy


Let’s use this to obtain the complex conjugates of the complex numbers in problems 1(a)-(c).  (Previous Blog from March 3rd, 'Another Form for Complex Numbers' (2))   Thus for (a): (2 + 3i)(1 – 2i) = (8 –i)

But multiplying their complex conjugates together would mean:
(2 - 3i)(1 + 2i)

Would one obtain the complex conjugate of the original result, e.g. (8 + i) and with no need to work it out?

Check!

(2 - 3i)(1 + 2i) = 2 + i -6(i)2 = 8 + i


It works!

b)The complex conjugate for this problem would be:

(1 + i) (1 – i)*  = (1 – i) (1 + i)  = 1 +1i -  1i -(i)(i) = 1 - (-1)  = 2

So there is no difference from the original form result.

Lastly, for part(c) the result we obtained in x + iy form was -2 + 3.46i


The complex conjugate is:   -2 – 3.46i

Note the first member (x-value) never changes!


A couple more odds and ends:

For a complex number z = x + iy

The REAL part of z denoted by Re(z) is always x!

The imaginary part of z, denoted Im(z) is the real number y! (Not iy!)

Problems:  

1. Find the complex conjugates for each of the following final results:  

a) (5 + i)(4 - 3i)(1 - i)  


b) (1 + i) / (1 - i)  


c) (1 – i)³

2) A quantum wave function is expressed:   y = exp [2πi(Kx )]

Obtain the complex conjugate (y *) and hence or otherwise find the probability amplitude:

[y *y ]

where the brackets denote 'absolute value'.