Question

Al. Practice with complex numbers: Every complex number z can be written in the form z r + iy where r and y are real; we call r the real part of z, written Re z, and likewise y is the imaginary part of z, y - Im z We further define the compler conjugate of z aszT-iy a) Prove the following relations that hold for any complex numbers z, 21 and 22: 2i Re (2122)(Re z) (Re z2) - (Im z) (Im22 Im (2122)(Rez (Im z2) +(Im z) (Re 22 b) The modulus-squared of z is defined as lz12-z"z. What is lmlzP, and what is Imz2? In doing quantum mechanics confusing 22 and 2 is very common; be careful! c) Any complex number can also be written in the form z-Ae , where A and θ are real and θ is usually taken to be in the range 10, 27); A and θ are called the modulus and the phase of z, respectively. Prove Euler's relation (use a Taylor expansion), d) Use (5) to find Rez, Im z, z* and Iz in terms of A and θ e) Use the above relations on ei(ai β) eiaeiß to derive trigonometric identities for sin (α+β) and cos(α + β)

Answer 1

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