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Unformatted text preview: Physics 3220 – Quantum Mechanics 1 – Fall 2008 Problem Set #3 Due Wednesday, September 10 at 2pm Problem 3.1 : Practice with complex numbers. (20 points) Every complex number z can be written in the form z = x + iy where x and y are real; we call x the real part of z , written x = Re z , and likewise y is the imaginary part of z , y = Im z . We further define the complex conjugate of z as z * ≡ x − iy . a) Prove the following relations that hold for any complex numbers z , z 1 and z 2 : Re z = 1 2 ( z + z * ) , (1) Im z = 1 2 i ( z − z * ) , (2) Re ( z 1 z 2 ) = (Re z 1 )(Re z 2 ) − (Im z 1 )(Im z 2 ) , (3) Im ( z 1 z 2 ) = (Re z 1 )(Im z 2 ) + (Im z 1 )(Re z 2 ) . (4) b) The modulus-squared of z is defined as | z | 2 ≡ z * z . What is Im | z | 2 , and what is Im z 2 ? In doing quantum mechanics confusing z 2 and | z | 2 is very common; be careful! c) Any complex number can also be written in the form z = Ae iθ , where A and θ are real and θ is usually taken to be in the range [0 , 2 π ); A and θ are called the modulus and the phase of z , respectively. Use Euler’s relation (which is provable using a Taylor expansion), e ix = cos x + i sin x , (5) to find Re z , Im z , z * and | z | in terms of A and θ . d) Use the above relations on e i ( α + β ) = e iα e iβ to derive trigonometric identities for sin( α + β ) and cos( α + β ). e) The second-order differential equation, d 2 dx 2 f ( x ) = − k 2 f ( x ) , (6) has two linearly independent solutions. These can be written in more than one way, and two convenient forms are f ( x ) = Ae ikx + Be- ikx , f ( x ) = a sin( kx ) + b cos( kx ) . (7) Verify that both are solutions of (...
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This note was uploaded on 02/27/2012 for the course PHYSICS 3220 taught by Professor Stevepollock during the Fall '08 term at Colorado.
- Fall '08