MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps

... use the approximations sin 6$ N S+, sin 6Cp N 6Cp (10. 96) cos6$ N 1, cos6Cp _N 1 to find 10 R1R2 = [ 6+ 1 =R2R,. (10.97) 6$ -64 Note that in terms of the generators ... choosing three mutually orthogonal straight lines. A point is defined by giving its coordinates, (q,z2,q), or by using the position vector 163 188 COORDINATES AND TENSORS Hence aa and ... 1 0 - sin $ cosCp sin4cos$ cosdsin+ -sin4 cos~cos$ (10.93) Reversing the order we get cos $ (10.94) sin $ sin Cp - sin $ cos Cp R2R1 = 0 cos Cp sin 4 [ sin $ - cos...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 5 ppsx

... Cancelling m on both sides and noting that (9. 166 ) (9. 167 ) and using Equation (9. 164 ) we finally write Similarly We now define the ladder operators L+ and L- for the m index of ... yiL (6) by 9 .6. 3 Spherical harmonics are defined as Ladder Operators for the Spherical Harmonics Using Equation (9. 162 ) we write Using (9. 162 ) (9. 163 ) (9. 164 ) (9. 165 ) 1 46 STURM-LIOUVILLE ... exists a minimum value, mmin, thus determining X as X = mmin (9.143) To find mmin we equate the two expressions [Eqs. (9.141) and (9.143)] for X to obtain (9.144) mmin = -1....

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 8 pps

... AND ITS LIE ALGEBRA 245 and and introduce the parameters (1 1. 166 ) 8= vf8:+8;+8,2 and (11. 167 ) (1 1. 168 ) so that we can summarize these results as L = X .68 + v-pp (11. 169 ) ... that remain invariant under similarity transformations. One such 254 CONTINUOUS GROUPS AND REPRESENTATIONS Y’ Fig. 11 .6 (z,y,z) and the (z‘,y’,z‘) coordinates we obtain (Fig. 11 .6) L,= ... iei(*-@)/' sin $12. (1 1.358) ( 1 1.359) 2 36 CONTINUOUS GROUPS AND REPRESENTATIONS 11 .6 SPECIAL UNITARY GROUP su(2) In quantum mechanics we are particularly interested in SU(2), a...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 9 pps

... useful in finding solutions of Laplace equation in two dimensions. 2. The method of analytic continuation is very useful in finding solutions of differential equations and evaluating some ... complex techniques are very helpful in certain problems of physics and engineering, which are essentially problems defined in the real domain, complex numbers in quantum mechanics appear as an ... absolutely and uniformly in all subdomains free of points of discontinuity. At the points of discontinuity it represents the arithmetic mean of the right- and the left-hand limits. 00 m=O In this...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 10 pps

... INTEGRALS Many of the definite integrals encountered in physics and engineering can be evaluated by using the complex integral theorems and analytic continuation: I. Integrals of the form ... 0 and a-lrnl # 0, then is called a singular point of order m. Definition I11 Essential singular point: If m is infinity, then a is called an essential singular point. Definition ... CLASSIFICATION OF SINGULAR POINTS 347 13.4 CLASSIFICATION OF SINGULAR POINTS Using Laurent the series we can classify singular points of a function. Definition I Isolated singular point: If...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx

... Schrodinger Equation in the Presence of Interactions 20.8 Feynman Phase Space Path Integral 61 0 61 1 61 3 61 5 61 5 61 6 61 8 62 1 62 4 62 6 63 3 63 3 63 5 64 9 65 0 65 0 65 5 65 5 ... 562 564 565 567 567 567 569 569 572 572 573 575 579 582 583 584 593 594 5 96 597 60 3 60 6 60 6 60 9 19.2.3 Compounding Propagators 60 9 67 2 INDEX Divergent series, 465 ... elements method, 65 0 methods of calculation, 64 6 Schrodinger equation, 65 8 semiclassical method, 65 0 time slice method, 64 7 Wiener path integral, 63 5 tegral, 65 9 Pauli spin matrices, 2 36 Permutation...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 2 pdf

... write x = fl in the generating function Equation (2 .65 ) we find (2.85) Expanding the left-hand side by using the binomial formula and comparing equal powers oft, we obtain 9 (1) = 1 ... part of the learning process. In a vast area like mathematical methods in science and engineering, there is always room for new approaches, new applications, and new topics. In fact, the number ... and for meticulously reading Chapters 1 and 9 with 14 and 20. I also thank Prof. N. K. Pak for many interesting and stimulating discussions, encouragement, and critical reading...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 3 docx

... that the value of the integral in Equation (3.38) can be obtained by expanding in powers of t and s and then by comparing the equal powers of tnsm with the left-hand side of Equation ... left-hand side of Equation (4 .61 ) explicitly we get 00 &e-t2+2tx -2+2sx -12 e ex (4 .62 ) = e2st (s + t) ,h, (4 .63 ) where we have defined u=x-(s+t). (4 .64 ) Expanding this in ... (4.52) 'I A This Page Intentionally Left BlankORTHOGONALiTY OF LAGUERRE POLYNOMlALS 49 Interchanging the integral and the summation signs and integrating with re- spect to x gives...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 4 pptx

... obtain the following recursion relations and First by adding and then by subtracting these equations we also obtain the relations m Jm-l(X) = -Jm(IL.) + J:,(X) (6. 52) X and (6. 53) ... equations in physics and engineering can be solved by the method of separation of variables. This method helps us to reduce a second-order partial differential equation into a set of ordinary ... (6. 55) we have used the Dirichlet boundary condition, that is, R(a) = 0. (6. 61) In terms of the Bessel functions this condition implies Jn(ka) = 0 (6. 62) and gives us the infinitely...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 7 pdf

... (r2sinog) +ae d (TaB) r2sinOaW +a (WE)] After simplifying, the Laplacian is obtained as r2sin6 dr 34 r2sin28 a4 ' (10.239) Here we have obtained a well-known formula in ... be expressed in terms of the poten- tials -A' and 4 as In the Lorentz gauge 2 and 4 satisfy and (10.373) (10.374) (10.375) (10.3 76) (10.377) respectively. Defining a four-potential ... Groups with infinite number of elements like R(n) are called infinite groups. A group with finite number of elements is called a finite group, where the number of elements in a finite group...

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