Physical Chemistry CHEM 120A
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This 4 page Class Notes was uploaded by Selena Trantow on Thursday October 22, 2015. The Class Notes belongs to CHEM 120A at University of California - Berkeley taught by W. Miller in Fall. Since its upload, it has received 16 views. For similar materials see /class/226748/chem-120a-university-of-california-berkeley in Chemistry at University of California - Berkeley.
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Date Created: 10/22/15
Complex Numbers David WH Swenson Despite the name7 imaginary numbers are very important in the physical sciences Many problems in quantum mechanics require complex numbers in order to solve them7 and for most problems complex numbers provide us With more direct techniques to approach t em 1 The Complex Plane 11 Real and Imaginary Parts as Cartesian Coordinates One very important idea about complex numbers is that of the complex plane You ve probably already seen the real numbers represented as a number line77 Each real number is described by a single coordinate We can then describe a complex number 2 With two coordinates one for the real part and one for the imaginary part We usually represent this With the real part along the zaxis7 and the imaginary part along the y axis So the complex number 3 4239 is equivalent to the point 34 32 2 a 239 b a b 31152 Figure l The Complex Plane Exercise 1 What Cartesian point is equivalent to the complex number 6239 What about 72 12 Modulus and Argument as Polar Coordinates In the standard 2D plane7 polar coordinates use a distance from the origin 7 and an angle 9 from the zaxis Just as we used Cartesian coordinates to represent the complex plane in the previous subsection we will now use polar coordinates The distance T is called the modulus or magnitude and is represented as for the complex number 2 The angle is called the argument of the number and is frequently referred to as arg2 Exercise 2 For the complex number 2 a ib what is in terms of a and b Hintz think back to trigonometry Exercise 3 For 2 a ib what is arg2 in terms of a and b For the special case of a real number I 0 what is arg2 Exercise 4 For a complex number 2 with modulus T and argument 9 what are a and I such that 2 aib7 Complex numbers can be used to do Euclidean plane geometry To learn more about this look into Tristan Needhamls book Visual Complex Analysis 2 Complex Conjugates The conjugate of a complex number 2 is denoted by either 2 or 2 It is the number such that 22 l2 2 There is a very simple rule to nd the complex conjugate of any complex number simply put a negative sign in front of any i in the number Thus 34i has 37 4239 as its complex conjugate and the complex conjugate of e M3 is ei7r 3 Exercise 5 Show that a ib a 7 ib a2 122 Exercise 6 Show that T i97 e49 T2 Exercise 7 What is the complex conjugate of a real number Exercise 8 Take a point in the complex plane In the Cartesian picture how does the act of taking the complex conjugate move the point What about in the polar coordinate picture 3 Euler s Formula Eulerls formula is a very important relation which connects the Cartesian com plex plane to the polar complex plane Eulerls formula is em cost9 i sint9 1 This equation can be derived from the power series expansions for the functions 6 sinz and cosz Exercise 9 Advanced Prove Eulerls formula Hint what s the Taylor series or MacLaurin series actually of 6 So what if you replace 1 by 219 remembering that i2 71 Now what are the series expansions for cost9 and sint9 If we multiply each side of Euler s formula be T 2 we get Teie Tcost9 isint9i The right size should look familiar from exercise 4 So we can use T6 to represent any complex number 2 with modulus T and argument argz 9 As we showed in exercise 6 when we multiply the com plex number T 619 by its complex conjugate we get 7 2 which is independent of 9 Since all the dependence of the argument 6 angle in the polar plane of 2 T 619 is contained in the 619 term we refer to this term as a phase factor77 Exercise 10 Using Eulerls Formula show that the simple rule for complex conjugation gives the same results in either realimaginary form or modu lusargument formi Hintz take a complex number 2 T619 and de ne a and I such that Teie a ibi Then take the complex conjugate Exercise 11 Two other formula are often grouped in with Eulerls formulai They are 1 cost9 E 619 6719 2 and 1 sint9 i 6 7 6719 3 22 Prove these using Eulerls formula as given in equation 1 Hintz sin7z 7 sinz and cos7z coszi Exercise 12 Advanced Therels a famous formula in mathematics which combines several of the most important mathematical constants 6 7r 239 and 1 Construct a formula which is equal to zero using each of those constants once in your expressioni Hint remember that 9 in 619 is in radians 4 Powers and Roots of Complex Numbers Although explicit formula for powers and roots exist for complex numbers writ ten as the sum of their real and imaginary parts it is often easier to calculate them using Eulerls formulai Namely to nd 2 we rst write 2 as T e and then use 21 T 6w Txeiex When looking for nth roots remember that 3 21 and use the same procedure Exercise 13 What is the square root of 239 Exercise 14 Prove de Moivre s formula 0086 isint9n cosm9 isinm9 where 9 E R and n E N Hint 6bC elm Exercise 15 Advanced The technique described above can be used to nd many trigonometric identitiesi By rst taking the trig function then using the formulae given by equations 2 and 3 doing some math With the result then converting them back to trigonemetric forms you can rather easily obtain many results from trigonometryi As an example try sin2 9 cos2 9 1 To the real showoffs try fdz sin2 ax cos2az 7 sin4az g
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