Question

1. - Let z equal the square root of −i. (a) Find Rez and Imz (real and imaginary parts of z). (b) Find real and imaginary parts for the square root of z ∗ . Don’t forget that every number (real or complex) has two square roots.

2. - Use the Euler Formula derived in Lecture to evaluate the real and imaginary parts of the complex wave function ψ(x) = 2e ikx for these 5 values of x: x = λ/2, λ/3, λ/4, 3λ + λ/5, 13λ/6 . You’ll have to recall the standard relation between wave vector and wavelength and evaluate some trig functions.

Answer 1

1. (a) First note that we can write -i as follows

because,

for any real number . Above formula is the Euler formula.

Now we can use above to obtain z :

where stands for two possible values of .

Similarly we can obtain real and imaginary parts of square roots of z , using again the Euler formula.

Another, more systematic, way to obtain the nth root of   is to note that

for any integer , i.e.

So, we obtain that nth root of   has n different values :

for    .

So, for fourth root of   we have n=4 and we get

for m=0, 1, 2, 3 . Note that for any other integer m , we do not get a new answer for   .

2. Recall that the wave-vector k is related to the wavelength by

So, at the given 5 values of x , we have

Using Euler formula

Use above five values of   to obtain the real and imaginary parts of the wave-function,   , at the given five values of x .

,    where