Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.11 PHYSICAL APPLICATIONS OF GROUP THEORY However, we have so far allowedxi,yito be completely general, and we must now identi ...

REPRESENTATION THEORY M M M Figure 29.6 A circular drumskin loaded with three symmetrically placed masses.
A circular drumskin ...

29.12 EXERCISES as the sum of two one-dimensional irreps and, using the reasoning given in the previous example, are therefore s ...

REPRESENTATION THEORY (d) Complete the character table using orthogonality properties, and check the summation rule for each irr ...

29.12 EXERCISES Use this to show that the character of the rotation in a representation based on the orbital wavefunctions is gi ...

REPRESENTATION THEORY the corresponding coordinate and 2 cosθfor the two orthogonal coordinates. If the rotation is followed by ...

29.13 HINTS AND ANSWERS (a) Make an initial sketch showing an arbitrary small mass displacement from, say, vertexC. Draw the res ...

REPRESENTATION THEORY (because of orthogonality) have characters 1, 1, 1,−1. The summation rules and orthogonality require the o ...

30 Probability All scientists will know the importance of experiment and observation and, equally, be aware that the results of ...

PROBABILITY iiiiii iv A B S Figure 30.1 A Venn diagram. be (we will return to the probability of an outcome in due course) but r ...

30.1 VENN DIAGRAMS A B S 1 (^23) 4 5 6 Figure 30.2 The Venn diagram for the outcomes of the die-throwing trials described in the ...

PROBABILITY A B C S 1 2 3 (^45) 6 7 8 Figure 30.4 The general Venn diagram for three events is divided into eight regions. not b ...

30.1 VENN DIAGRAMS gets beyond three it becomes impossible to draw a simple two-dimensional Venn diagram, but this does not chan ...

PROBABILITY SinceA∩X=Xwe must haveX⊂A. Now, using the second equality in (30.4) in a similar way, we find A∪X=A∪[A∩(A∪B)] =(A∪A) ...

30.2 PROBABILITY times then we expect that a six will occur approximatelyN/6 times (assuming, of course, that the die is not bia ...

PROBABILITY This is particularly useful for problems in which evaluating the probability of the complement is easier than evalua ...

30.2 PROBABILITY However, we may write Pr(A 1 ∩B)as Pr(A 1 ∩B)=Pr[A 1 ∩(A 2 ∪A 3 )] =Pr[(A 1 ∩A 2 )∪(A 1 ∩A 3 )] =Pr(A 1 ∩A 2 )+ ...

PROBABILITY
Find the probability of drawing from a pack a card that has at least one of the following properties: A,itisanace; ...

30.2 PROBABILITY ace from a pack of cards from which one has already been removed, given that eventA, the card already removed w ...

PROBABILITY Two eventsAandBarestatistically independentif Pr(A|B)=Pr(A) (or equiva- lently if Pr(B|A)=Pr(B)). In words, the prob ...