Quantum Mechanics for Scientists and Engineers. David Miller

Transcription

1 Quantum Mechanics for Scientists and Engineers David Miller

2 Background mathematics Basic mathematical symbols

3 Elementary arithmetic symbols Equals Addition or plus 3 5 Subtraction, minus or less 3 or Multiplication or / Division 63 6/3 or ( numerator) dividend ( demonimator) divisor quotient

4 Relational symbols Is equivalent to x x/ y y or Is approximately equal to Is proportional to a x x Is greater than 3 Is greater than or equal to x Is less than 3 Is less than or equal to Is much greater than Is much less than x 00 00

5 Greek characters used as symbols Upper case Lower case Name Roman equiv. alpha a a beta b b gamma g g delta d d epsilon e e zeta z z eta (e) h theta th q iota i i kappa k k lambda l l mu m m Upper case Lower case Name Roman equiv. Keyboard Keyboard nu n n xi x x omicron (o) o pi p p rho r r sigma s s tau t t upsilon u u phi ph f chi ch c psi psy y omega o w

6 Background mathematics Basic mathematical operations

7 Conventions for multiplication For multiplying numbers We explicitly use the multiplication sign 3 6 For multiplying variables We can use the multiplication sign But where there is no confusion We drop it ab c might be simply replaced by ab c

8 Use of parentheses and brackets When we want to group numbers or variables We can use parentheses (or brackets) (34) 74 For such grouping, we can alternatively use square brackets or curly brackets When used this way, there is no difference in the mathematical meaning of these brackets

9 Associative property Operations are associative if it does not matter how we group them e.g., addition of numbers is associative a b cabc e.g., multiplication of numbers is associative abcabc But division of numbers is not associative 8/4 / / but 8/ 4/ 8/ 4

10 Distributive property Property where terms within parentheses can be distributed to remove the parentheses abc abac Here, multiplication is said to be distributive over addition Many other conceivable operations are not distributive, however E.g., addition is not distributive over multiplication

11 Commutative property Property where the order can be switched round e.g., addition of numbers is commutative a bba e.g., multiplication of numbers is commutative abba But e.g., subtraction is not commutative e.g., division is not commutative 6/3 3/6

12 Background mathematics Algebra notation and functions

13 Parentheses and functions A function is something that relates or maps One set of values Such as an input variable or argument x To another set of values which we could think of as an output For example, the function f x x 4 f x 3 0 x

14 Parentheses and functions Conventionally, we say f of x when we read Here obviously f x is not f times x Most commonly Only parentheses are used around the argument x not square [ ] or curly { } brackets f x f x 3 0 x

15 Parentheses and functions For a few very commonly used functions Such as the trigonometric functions The parentheses are optionally omitted when the argument is simple sin instead of sin Note, incidentally, sin sin sin

16 Parentheses and functions For a few very commonly used functions Such as the trigonometric functions The parentheses are optionally omitted when the argument is simple cos instead of cos Note, incidentally cos cos cos

17 r or hypotenuse angle x, base or adjacent side y, height or opposite side Sine, cosine, and tangent Defined using a right-angled triangle sin y r cos Natural units for angles in mathematics are radians radians in a circle radian ~ 57.3 degrees x r tan tan y x sin cos

18 r or hypotenuse angle x, base or adjacent side y, height or opposite side Cosecant, secant, and cotangent Cosecant r cosec csc y Secant sec r x cos sin Cotangent x cos cotan cot y tan sin

19 Inverse sine function The inverse sine function sin a or arcsine function Pronounced arc-sine works backwards to give the angle from the sine value If a sin then arcsin a asin a sin a Note sin a does not mean /sina 0 The - here means inverse function not reciprocal sin a a

20 sin and cos functions sin However sin sinsin sin Not sinsin Similarly cos cos Only trigonometric functions and their close relatives commonly use this notation 0 cos 0

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