Math Review for Physical Chemistry

Transcription

1 Chemistry 362 Spring 27 Dr. Jean M. Stanar January 25, 27 Math Review for Physical Chemistry I. Algebra an Trigonometry A. Logarithms an Exponentials General rules for logarithms These rules, except where note, apply to both log (base ) an ln (base e = ). ln ( a b) = ln a + ln b ln a = ln a ln b b ln ( a b ) = b ln a For natural logs only, ln ( e x ) = x (since ln e = ). Note that ln ( a + b) ln a + ln b. This is a common mistake. General rules for exponentials e a e b = e a+b e a e b = e a b ( e b ) m = e m b B. Trigonometry Definitions base on a right triangle y r sinθ = cosθ = opposite hypotenuse = y r ajacent hypotenuse = x r x θ tanθ = sinθ cosθ = opposite ajacent = y x

2 2 Other trigonometric function efinitions cotθ = secθ = cscθ = tanθ cosθ sinθ = cosθ sinθ Trigonometric Ientities sin 2 θ + cos 2 θ = sin2θ = 2sinθ cosθ cos2θ = cos 2 θ sin 2 θ II. Calculus [More information may be foun in Appenix A of your textbook.] A. Derivatives Derivatives of common functions x x n x eax = n x n = a e ax x ln x = x sin x = cos x x cos x = sin x x General rules for manipulation of erivatives x c f x [ ( )] = c f ʹ ( x) (c is a constant) x [ f ( x) + g( x) ] = x f ( x) + x g( x) x [ f ( x) g( x) ] = f x ( ) g ʹ ( x) + g( x) f ʹ ( x) (the Prouct Rule) x f u x ( ( )) = f u u x (the Chain Rule)

3 3 B. Integrals Integrals of common functions Note that since these are inefinite integrals, they all shoul inclue an overall constant of integration. x n x = n + x n+ e bx x = b ebx x x = ln x sin x x = cos x cos x x = sin x General rules for manipulation of integrals c f ( x) x = c f ( x) x (c is a constant) [ f ( x) + g( x) ] x = f ( x) x + g x ( ) x

4 4 Some More Definite an Inefinite Integrals. e bx x = b 2. x n e bx x = n! b n+ 3. e bx 2 x = 2 π b 2 4. x e bx 2 x = 2b 5. x 2 e bx 2 x = π 4b b 2 6. sin 2 bx x = x 2 sin2bx 4b 7. x sinbx x = sinbx b 2 x cosbx b 8. x sin 2 bx x = x 2 4 x sin2bx 4b 9. sin 3 bx x = cosbx 3b [ sin 2 bx + 2] cos2bx 8b 2. sinbx cosbx x = sin2 bx 2b. cos 2 bx x = x 2 + sin2bx 4b

5 5 III. A Guie to Complex Numbers General Definitions All complex numbers have at their root the imaginary number i, Complex numbers are written as a real part an an imaginary part, i =. () z = a + i b, (2) where z is a complex number an a an b are real numbers. The number a is referre to as the real part of the complex number, while the number b is referre to as the imaginary part since it is multiplie by i. A function may also contain imaginary numbers. The simplest types of such functions can be ivie into real an imaginary parts, h( x) = f ( x) + i g( x). (3) In this equation, f ( x) an g( x) are real functions. As for the complex numbers efine in Equation (2), f ( x) is referre to as the real part of the function h( x) an g( x) is referre to as the imaginary part of the function g( x). Euler s Relation Functions that contain imaginary numbers may not always be easily separate into real an imaginary parts. However, a typical function use in quantum mechanics has the imaginary number in the exponent, f ( x) = e i k x, (4) where k is a constant. Even this function may be separate into real an imaginary parts using Euler s relation, e i k x = cos kx + i sin kx. (5) Complex Conjugates An important quantity when ealing with complex numbers an functions is the complex conjugate. The complex conjugate of a number or function that contains an imaginary part is obtaine by replacing i by i where it appears. A complex conjugate is enote by an asterisk. For example, for a complex number z, the complex conjugate is z*. If z = a + i b, then the complex conjugate is z * = a i b. (6) The complex conjugate of a function such as the one in Equation (3) is efine similarly, An, for the function given in Equation (4), the complex conjugate is h * ( x) = f ( x) i g( x). (7) f * ( x) = e i k x. (8)

6 6 Absolute Squares of Complex Variables An important property of the complex conjugate of a number or a function is that when the complex conjugate is multiplie by the original number or function, the result is always real an positive. For example, consier the prouct of a complex number z an its complex conjugate, z z *, which is known as the absolute square, z z * = ( a + i b) ( a i b) = a 2 + i ab iab i 2 b 2 = a 2 i 2 b 2 z z * = a 2 + b 2. (9) The above relation simplifies using the result that i 2 =. The complex conjugate multiplie by the original also yiels a real an positive result for functions. For example, consier the function given in Equation (4), f ( x) f * ( x) = e i k x e i k x = e f ( x) f * ( x) =. () More information relate to complex variables may be foun in Appenix A of your textbook.