Dm Qm Background Mathematics Lecture 5

8/10/2019 Dm Qm Background Mathematics Lecture 5

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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



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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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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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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Dm Qm Background Mathematics Lecture 5