dm qm background mathematics lecture 5
TRANSCRIPT
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Quantum Mechanics for
Scientists and Engineers
David Miller
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Background mathematics 5
Sum, factorial and product notations
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Summation notation
If we want to add a set of numbers a 1, a 2, a 3,and a 4 , we can write
or we can use summation notation
If the range of j is obvious
Here j is an index
1 2 3 4S a a a a
4
1 1,2,3,4 1,...,4 j j j
j j jS a a a
j
j
S aplural of index
indexesor
indices
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For a set of numbers spaced by a constant amount,an arithmetic progression or sequence
i.e., where the nth number ise.g., the numbers 4, 7, 10 , and 13
i.e., , , andwith, therefore and terms in the
progressiongives the series (sum of the terms)
1 411
12
m
j
a aa j d m
Example arithmetic series
1 1na a n d
1 2 3 4a a a a
1 4a 2 7a 3 10a 4 13a 3d 4m
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For a set of numbers spaced by a constant amount,an arithmetic progression or sequence
i.e., where the nth number ise.g., the numbers 4, 7, 10 , and 13
i.e., , , andwith, therefore and terms in theprogression
gives the series (sum of the terms)
111
12
mm
j
a aa j d m
Example arithmetic series
1 1na a n d
1 2 3 4a a a a
1 4a 2 7a 3 10a 4 13a 3d 4m
4 134 34
2
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Example geometric series
With a constant ratio between successive terms,a geometric progression or sequence
i.e., where the nth term ise.g., the numbers 3, 6, 12 , and 24
i.e., , , andwith, therefore and terms in theprogression
gives the series (sum of the terms)
11
nna a r
1 2 3 4a a a a
1 3a 2 6a 3 12a 4 24a 2r 4m
3 6 12 24 45
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Example geometric series
With a constant ratio between successive terms,a geometric progression or sequence
i.e., where the nth term ise.g., the numbers 3, 6, 12 , and 24
i.e., , , andwith, therefore and terms in theprogression
gives the series (sum of the terms)
11
nna a r
1 2 3 4a a a a
1 3a 2 6a 3 12a 4 24a 2r 4m
4
11
1 1
m j
j j j
a a r
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Example geometric series
With a constant ratio between successive terms,a geometric progression or sequence
i.e., where the nth term ise.g., the numbers 3, 6, 12 , and 24
i.e., , , andwith, therefore and terms in theprogression
gives the series (sum of the terms)
11
nna a r
1 2 3 4a a a a
1 3a 2 6a 3 12a 4 24a 2r 4m
4
11 1
1 1
11
mm j
j j j
r a a r a
r
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Factorial notation
Quite often, we need a convenient way ofwriting the product of successive integers
e.g.,We can write this as
called four factorialand using the exclamation point !
The notation is obvious for most other casesNote, though, that we choose
1 2 3 4
1 2 3 4 4!
0! 1
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Product notation
Generally, when we want to write the product
of various successive terms
by analogy with the summation notation
we can use the product notation
For example, for all integers
1 2 3 4a a a a
4
1 2 3 41
j j
a a a a a
1
!n
p
n p
1n
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Background mathematics 5
Power series
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Analytic functions and power series
For a very broad class of the functions in physics
we presume they are analytic(except possibly at some singularities)
i.e., at least for some range of values of the
argument x near some point xothe function can be arbitrarily wellapproximated by a power series
i.e., with xo = 0 for simplicity we have
which may be an infinite series 2 31 2 3o f x a a x a x a x
f xThe ellipsis means weomit writingsome terms
explicitly
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Analytic functions and power series
For a very broad class of the functions in physics
we presume they are analytic(except possibly at some singularities)
i.e., at least for some range of values of the
argument x near some point xothe function can be arbitrarily wellapproximated by a power series
i.e., generally
2 31 2 3o o o o f x a a x x a x x a x x
f x
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Maclaurin series
Continuing
so
and
and so on
2
2 3 2 32 2 3 2 2! 3!d f f x a a x a a xdx
2
220
0 2!d f
f adx
3
330
0 3!d f
f adx
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Maclaurin and Taylor series
Continuing gives the Maclaurin series
Repeating the same procedure around with
gives the Taylor series
2 2
20 0 0
01! 2! !
n n
n x df x d f x d f f x f
dx dx n dx
2 31 2 3o o o o f x a a x x a x x a x x
2 2
21! 2! !o o o
n no o o
o n x x x
x x x x x xdf d f d f f xdx dx n dx
f x
o x x
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Maclaurin and Taylor series allow approximations
for small ranges of the argument about a pointExamples for small x
Approximations to first order in x
Note lowest order dependence on x for cos x
is second order
Power series for approximations
1/ 1 1 x x
1 1 / 2 x x sin x xtan x x ln 1 x x
exp 1 x x
2cos 1 / 2 x x
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