dm qm background mathematics lecture 3

8/10/2019 Dm Qm Background Mathematics Lecture 3

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Quantum Mechanics for

Scientists and Engineers

David Miller


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Background mathematics 3

Coordinates and vectors


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Coordinate axes and vectors

Ordinary geometry

Three axesx,y, andzAll at right angles

Cartesian axes

(from RenDescartes)

Lines or directions at rightangles are also called

orthogonal

x

y

z


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Right-handed axes

Using your right handThumb x

Index (first) finger y

Middle finger zNo matter how you now

rotate your whole hand

the axes remain right-

handed

x

y

z


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If you use your left hand

Thumb xIndex (first) finger y

Middle finger z

give left-handed axesNo rotation of this entireset of left-handed axes willever make it right-handed

We use right hand axes unlessotherwise stated

x

y

z


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For some point P in space

The correspondingprojections onto thecoordinate axes give

Cartesian coordinatesxP,yP, andzP,

relative to the origin ofthe axes

Sometimes written(xP,yP,zP)

P.(xP,yP,zP)

x

y

z

xP

yP

zP

origin .


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A vector is something with

a magnitudesuch as a length

and a direction

Usually written in bold fonte.g., G

Sometimes G or

And shown as an arrow

With length anddirection

G

G


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A vector could be

the distanceand

direction

you need to walk to getfrom A to BA

B

r


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A vector could be

A forcehow hard you are

pushing

andwhat direction you arepushing

F


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A vector could be

A velocityhow fast you are going

(speed)

e.g., the number onyour car speedometer

and

what direction you are

going ine.g., on a compass

v


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A vector has components

along three orthogonal axesGx, Gy, and Gz

We can also define vectors ofunit length along each axis

i unit vector alongx

j unit vector alongy

k unit vector alongz

x

y

z

G

Gx

Gy

Gz

i

j

k


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Then we can write

G=Gx i+Gyj+Gzk

x

y

z

G

Gx

Gy

Gz

Gxi

Gyj

Gzk


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Then we can write

G=Gx i+Gyj+Gzkmaking the final vector upby adding its vectorcomponents

x

y

z

G

Gx

Gy

Gz

Gxi

Gyj

Gzk


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Adding vectors

To add vectors

graphicallyconnect them head to tail in any

order

G

S

G + S

S

G

G + S


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Adding vectors

To add vectors

algebraicallyadd them component by

component

Gxi

Sxi

Gyj

Syj

Gzk

Szk

G + S

x

x

x x

z

z

y z

y

y

y z

GS

GS

G S G

S

G S S

G

ii

i j

kk

G S jj

k


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Multiplying vectors

Two kinds of multiplications or products for

geometrical vectorsDot product

Gives a scalar resultCross product

Gives a vector result

a b

a b


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Vector dot product

One formula for the dot product is

Here the modulus sign | | meanswe take the length of the vector

Note that

Also

So

aa

ab b a

cos cosab ab ba

angle

b

a

2

a a a

a a a


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Vector dot product

One formula for the dot product is

Note that

for two vectors at right angles

and

so

the dot product is zero

cos cosab ab ba

cos / 2 0

/ 2 90

a

b


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Vector dot product

The unit vectors along the coordinate

directions are all orthogonal (atright angles)

So all their dots products with oneanother are zero

Also, since these are unit length

vectors, by definition

i

j

k 0 i j 0 j k0 i k

0 j i 0 k j0 k i

1 i i 1 j j 1 k k


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Vector dot product

Since

Forming the dot productalgebraically

gives

which is an equivalent formula forthe dot product

b

a

x y z x y za ba a b b b i ja i j k k

x zzyx ya a bab b ba

0 i j 0 j k0 i k0 j i 0 k j0 k i


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Vector dot product

The components of a vector can be

found bytaking the dot product

with the unit vectors along thecoordinate directions

For example

G

i

x z xy GGG G j k iG ii


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Vector cross product

For two vectors

the vector cross product is

n is a unit vector with a directiongiven by the

right hand screw rule

x y za a a a i j k

x y zb b b b i j k

b

a

sin sinba na b ban


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Note that

If we have to turn clockwise to gofrom a to b

So the corkscrew goes in

So n points inwards

Then we have to turn anti-clockwiseto go from b to a

So the corkscrew goes outSo n point outwards

b

a

a b b a


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An equivalent algebraic formula for the vector

cross product is

A short-hand way of writing this is

which is the same as the determinantnotation used with matrix algebra

z y x z yx y xy z z xa a a a ab bab b b b ja ib k

x

x y z

y za a

b b b

a ak

bi j


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dm qm background mathematics lecture 3

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