Quantum Physics

A.6 Appendix A Mathematical Review5(y 2) y 86y 18y 3x y 2 1Alternate Solution Multiply each term in (1) by the factor 2 and add the result to (2):10x 2y 162x 2y 412x 12x 1y x 2 3y543x – 2y = –1(5, 3)211 2 3 4 5 6 xx – y = 2Figure A.3Two linear equations with two unknowns can also be solved by a graphicalmethod. If the straight lines corresponding to the two equations are plotted in aconventional coordinate system, the intersection of the two lines represents the solution.For example, consider the two equationsx y 2x 2y 1These are plotted in Figure A.3. The intersection of the two lines has the coordinatesx 5, y 3. This represents the solution to the equations. You should checkthis solution by the analytical technique discussed above.EXERCISESSolve the following pairs of simultaneous equations involving two unknowns:1.2.3.x y 8x y 298 T 10aT 49 5a6x 2y 68x 4y 28ANSWERSx 5, y 3T 65, a 3.27x 2, y 3G. LogarithmsSuppose that a quantity x is expressed as a power of some quantity a:x a y[A.11]The number a is called the base number. The logarithm of x with respect to thebase a is equal to the exponent to which the base must be raised in order to satisfythe expression x a y :y log a x[A.12]Conversely, the antilogarithm of y is the number x:x antilog a y[A.13]In practice, the two bases most often used are base 10, called the common logarithmbase, and base e 2.718 ..., called the natural logarithm base. Whencommon logarithms are used,y log 10 x (or x 10 y )[A.14]