dm qm background mathematics lecture 1

8/10/2019 Dm Qm Background Mathematics Lecture 1

1/21

Quantum Mechanics for

Scientists and Engineers

David Miller


2/21

Background mathematics 1

Basic mathematical symbols


3/21

Elementary arithmetic symbols

Equals

Addition or plus

Subtraction, minus or less

Multiplication

Division

2 3 5

3 2 1

/or 6 3 2 6 / 3 2

2 3 6 or 2 3 6

or

( )

( )

dividendnumeratorquotient

demonimator divisor 6 23

6 23

6

3 2


4/21

Relational symbols

Is equivalent to

Is approximately equal to

Is proportional to

Is greater thanIs greater than or equal to

Is less than

Is less than or equal toIs much greater than

Is much less than

/ x

x y

y

a x x

or 1

0.333

3 221 1x

2 3

2

1 1 x 100 1

1 100


5/21

Greek characters used as symbols

Uppercase

Lowercase

Name Romanequiv.

Key-board

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

Uppercase

Lowercase

Name Romanequiv.

Key-board

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


6/21


Basic mathematical operations


7/21

Conventions for multiplication

For multiplying numbers

We explicitly use the multiplication sign

For multiplying variables

We can use the multiplication signBut where there is no confusion

We drop it

might be simply replaced by

2 3 6

a b c

ab c


8/21

Use of parentheses and brackets

When we want to group numbers or variables

We can use parentheses (or brackets)

For such grouping, we can alternatively use

square brackets

or curly brackets

When used this way, there is no difference inthe mathematical meaning of these brackets

2 (3 4) 2 7 14

2 3 4 2 7 14

2 3 4 2 7 14


9/21

Associative property

Operations are associative if it does not matter

how we group theme.g., addition of numbers is associative

e.g., multiplication of numbers is associative

But

division of numbers is not associative

but

a b c a b c

a b c a b c

8 / 4 / 2 2 / 2 1 8 / 4 / 2 8 / 2 4


10/21

Distributive property

Property where terms within parentheses can

be distributed to remove the parentheses

Here, multiplication is said to be

distributive over additionMany other conceivable operations arenot distributive, however

E.g., addition is not distributive over

multiplication

a b c a b a c

3 2 5 13 3 2 3 5 40


11/21

Commutative property

Property where the order can be switched round

e.g., addition of numbers is commutative

e.g., multiplication of numbers is commutative

But

e.g., subtraction is not commutative

e.g., division is not commutative

a b b a

a b b a

5 3 2 3 5 2

126 / 3 2 3 / 6


12/21


Algebra notation and functions


13/21

Parentheses and functions

A function is something that

relates or mapsOne set of values

Such as an inputvariable or argument x

To another set of values

which we could think ofas an output

For example, the function

1

4f x x

2 1 0 1 2

2

1

1

2

3

x

f x


14/21


Conventionally, we say

f of x when we readHere obviously

is not f times x

Most commonly

Only parentheses are usedaround the argument x

not square [ ] or curly { }brackets

2 1 0 1 2

2

1

1

2

3

x

f x f x

f x


15/21


For a few very commonly used

functionsSuch as the trigonometricfunctions

The parentheses areoptionally omitted whenthe argument is simple

instead of

Note, incidentally,

sin

sin sin

1

1

sin sin


16/21

1

1


For a few very commonly used

functionsSuch as the trigonometricfunctions

The parentheses areoptionally omitted whenthe argument is simple

instead of

Note, incidentally

cos

cos cos

cos cos


17/21

Sine, cosine, and tangent

Defined using a right-angled

triangle

Natural units for angles inmathematics are radians

2 radians in a circle1 radian ~ 57.3 degrees

x, base oradjacent side

y, height oropposite

side

rorhypotenuse

sin y

r cos

x

r tan

y

x

angle

sintancos


18/21

Cosecant, secant, and cotangent

Cosecant

Secant

Cotangent

x, base oradjacent side

y, height oropposite

side

rorhypotenuse

1cosec cscsin

ry

1sec

cosx

r

1 cos

cotan cot tan siny

x

angle


19/21

Inverse sine function

The inverse sine function

or arcsine functionPronounced arc-sine

works backwards to give the

angle from the sine valueIf then

Note does not mean

1 0 1

2

2

1arcsin asin sina a a

sina

1sin a 1sin a

a

1sin a 1/sin a

The -1 here means inverse function not reciprocal


20/21

sin2 and cos2 functions

However

Not

Similarly

Only trigonometric functionsand their close relatives

commonly use this notation

22sin sin sin sin

sin sin 0

12sin

22

cos cos

0

12cos


21/21

dm qm background mathematics lecture 1

Documents