PHYS_342_hmwrk1_math - PHYS 342 Fall Semester 2011 Homework...

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1 PHYS 342 Fall Semester 2011 Homework No. 1: Math Review Due Date: August 29, 2011 1. When we discuss the Schrödinger wave equation for the H atom, we will make extensive use of the polar co-ordinate system. Anticipating this development, the Figure to the right illustrates a point P in both rectangular and polar coordinates. If the coordinates of a point P in rectangular coordinates is (x,y,z), what are the equivalent coordinates (r, θ , φ ) in spherical polar coordinates? Write down expressions for an area element dA and a volume element dV using both coordinate systems. 2. We will use complex notation regularly throughout the semester and it is a good idea to make sure that you you have a good idea what the imaginary number i=√(-1) is all about. When dealing with complex numbers (ie numbers that have a real and an imaginary part), it is convenient to draw a new co- ordinate system as shown. Note that we no longer use (x,y) to label the axis, but instead use (real, imaginary). As an example, the position of a point specified by the co-ordinates (3, 2 ) i is shown. Using a complex number co- ordinate system is quite useful and you must understand how it is different than the conventional (x,y) system that you use all the time. First, we need a definition. When a real number is multiplied by i, what actually happens? It is useful to think of “multiplication by i” as an operation that just rotates a “real” number counter clockwise by 90 o . Normally, we associate multiplication with an operation that changes the original value of a number to a new value which is different. Not so when you multiply by I - the magnitude of the number stays the same, but the location of the number rotates. z y P(x, y, z) or P(r, ϑ , ϕ ) x ϑ ϕ r 0 1 2 3 4 Real Imaginary 4 3 2 0 i i i i (3, 2 ) i
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2 a) Using this operator idea, what happens when a real number like 3 is multiplied by i two times? Make a plot to show the original point and the final point. Does this plot indicate that i 2 is equivalent to -1? b) Using this operator idea, state in words what happens when a real number is multiplied by i three times? c) When a real number is multiplied by –i, what type of rotation is imposed? d) Using the rotation operator idea, does it make sense that i i i i × × = - ? e) What type of rotation is imposed when you divide a real number like 3 by i? f) What happens when you multiply the complex number (3, 2 ) i by i? What is the angular separation between the two points (3, 2 ) i and ( 2, 3 ) i - ? g) What type of operator describes the complex conjugation of a complex number like (3, 2 ) i ? A few important points to remember:
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This note was uploaded on 12/08/2011 for the course PHYS 342 taught by Professor Staff during the Fall '08 term at Purdue University-West Lafayette.

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PHYS_342_hmwrk1_math - PHYS 342 Fall Semester 2011 Homework...

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