Mathematical Tools for Physics
7—Operators and Matrices 173 7.3 Examples of Operators The four cases that I started with, rotation in the plane, angular moment ...
7—Operators and Matrices 174 The way to write a vector~vin components is ~v=vxˆx+vyˆy+vzˆz, or v 1 ~e 1 +v 2 ~e 2 +v 3 ~e 3 = ∑ ...
7—Operators and Matrices 175 For two vectors to be equal, the corresponding coefficients of~e 1 ,~e 2 ,etc.must match; their res ...
7—Operators and Matrices 176 The rotated~e 1 has two components, so Rα(~e 1 ) =~e 1 cosα+~e 2 sinα=R 11 ~e 1 +R 21 ~e 2 This det ...
7—Operators and Matrices 177 Pick the common rectangular, orthogonal basis and evaluate the components of this function. Equatio ...
7—Operators and Matrices 178 The integrals in Eq. ( 14 ) are simply sums this time, and the sums have only two terms. I’m making ...
7—Operators and Matrices 179 are bothonthe axis, and they have no angular momentum. In the general case as drawn, the vector~Lpo ...
7—Operators and Matrices 180 As an example, place a disk of massM and radiusRand uniform mass density so that its center is at ( ...
7—Operators and Matrices 181 This saves integration.I 11 =I 22 =MR^2 / 4. For the other term in the sum ( 18 ), you have a point ...
7—Operators and Matrices 182 7.4 Matrix Multiplication How do you multiply two matrices? There’s a rule for doing it, but where ...
7—Operators and Matrices 183 Matrix multiplication is just the component representation of the composition of two functions, Eq. ...
7—Operators and Matrices 184 and the index notation is completely general, not depending on whether you’re dealing with two dime ...
7—Operators and Matrices 185 7.6 Areas, Volumes, Determinants In the two-dimensional example of arrows in the plane, look what h ...
7—Operators and Matrices 186 so you must get the same answer. But will I get the same simple formula ( 26 ) for the answer if I ...
7—Operators and Matrices 187 Compute the components as usual. M~e 1 = ( 1 2 3 4 )( 1 0 ) = ( 1 3 ) = 1~e 1 + 3~e 2 This says tha ...
7—Operators and Matrices 188 can always picture it as a piece of a square thatisat the origin. The shaded square that is 1 / 16 ...
7—Operators and Matrices 189 This is called aneigenvalueequation. It says that for any one of these special vectors, the operato ...
7—Operators and Matrices 190 Write the problem in terms of components, and of course you aren’t yet in the basis where the matri ...
7—Operators and Matrices 191 The solutions to this quadratic areλ= 1, 3. For these values then, the apparently two equation for ...
7—Operators and Matrices 192 You can make it look more like the previous example with some further arrangement [( k 1 +k 3 −k 3 ...
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