**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.**

*complex conjugates*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'.

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