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HW3 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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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 i θ , 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 i α e i β to derive trigonometric identities for sin( α + β ) and cos( α + β ). e) The second-order di ff erential 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 (6). Since both are equally good solutions, we must be able determine a and b in terms of A and B ; do so.
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