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This is the Solved Past Paper of Optics and Modern Physics which includes Transparent Sphere of Index, Principle Points, Transform Matrix, Ray Incident, Total Internal Reflection, Index of Refraction etc. Key important points are: Magnitude of Complex Number, Exponential Form, Imaginary Part, Complex Number Representation, Real Part, Ground State, Potential Height, Kinetic Energy of Electrons, Photoelectric Effect
Typology: Exams
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a) Working directly with exponentials, show that the magnitude of z˜ = Ae iωt is A.
Recall that the magnitude of a complex number ˜z is given by
|z˜| = [˜z ∗ z˜]
1 / 2
Using the exponential form, as instructed in the problem, gives
|z˜| =
A e −iωt A e iωt
2
Since it is assumed that A is a real number when we write complex numbers in exponential form, we
can then simplify this to
|z˜| =
2
b) Show that the imaginary part of z˜ is given by (˜z − ˜z ∗ )/ 2 i. You do not need to use exponentials for (b).
It is probably easiest here to use the complex number representation ˜z = a + bi where a is the real part
and b is the imaginary part. Putting this into the expression above gives
(a + bi) − (a − bi)
2 i
(a − a) + (bi + bi)
2 i
2 bi
2 i
= b
Given the definition of this form of complex number representation, b is the imaginary part of ˜(z).