Post on 02-Jun-2018
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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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Coordinate axes and vectors
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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Coordinate axes and vectors
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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Coordinate axes and vectors
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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Coordinate axes and vectors
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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Coordinate axes and vectors
A vector could be
the distanceand
direction
you need to walk to getfrom A to BA
B
r
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Coordinate axes and vectors
A vector could be
A forcehow hard you are
pushing
andwhat direction you arepushing
F
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Coordinate axes and vectors
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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Coordinate axes and vectors
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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Coordinate axes and vectors
Then we can write
G=Gx i+Gyj+Gzk
x
y
z
G
Gx
Gy
Gz
Gxi
Gyj
Gzk
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Coordinate axes and vectors
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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Vector cross product
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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Vector cross product
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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