dm qm background mathematics lecture 4

8/10/2019 Dm Qm Background Mathematics Lecture 4

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Quantum Mechanics for

Scientists and Engineers

David Miller


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Background mathematics 4

First derivative


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First derivative

For some function

The (first) derivative is the

slope

gradient or

rate of changeof y as we change x

If for some small infinitesimalchange in x, called dx

y changes by some smallinfinitesimal amount dy

y

x

y x

the first derivative is written

The ratio notation on theright is Leibniz notation

dy

y xdx


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y

xx1

1x

d y

d x

First derivative

The derivative at some specific

point x1 can be written

The value of the derivative isthe slope of the tangent line

the dashed line in the figure

at that point

Equal in value to the tangent

1

1

x

dyy x

dx

x3

3x

d y

d x

xy

yx


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First derivative

Looking at the slope

of the orange lineas we reduce

the orange line slopebecomes closer to the slopeof the black tangent line

y

x

x

x1

12

xy x

1 2

xy x

1 1

2 2

x xy x y x

x

x


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First derivative

In the limit as becomes very small

i.e., in the limit as tends to zero

this ratio becomes the (first) derivative

x

x

0limx

1

1 1

0

2 2limxx

x xy x y x

dy

dx x


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Sign of first derivative

If y increases as we increase x

sloping up to the right

If ydecreases as we increase x

sloping down to the right

y

xx1

0d y

d x0

d y

d x

x3

0d yd x

0d y

d x


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Derivative of a power

1 0.5 0 0.5 1

0.5

1

2y x

1n ndx nx

dx

2dy xdx

1 0.5 0 0.5 1

2

2


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The derivative of a straight line

The straight line has a constant slope

is a constant

1 0.5 0 0.5 1

1

1 y x

1 0.5 0 0.5 1

0.5

1

1dy

dx


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The derivative does not depend on the height

All these lines have the same slope

1 0.5 0 0.5 1

1

1

y x

1 0.5 0 0.5 1

0.5

1

1dy

dx


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1 0.5 0 0.5 1

1

1




0.25y x

1 0.5 0 0.5 1

0.5

1

1dy

dx


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1 0.5 0 0.5 1

1

1




0.5y x

1 0.5 0 0.5 1

0.5

1

1dy

dx


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The derivative of a constant

y is not changing with x

is zero

1 0.5 0 0.5 1

1

1

0.5y

1 0.5 0 0.5 1

0.5

1

0dy

dx


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Derivative of an exponential

exp expd

x xdx

1 0.5 0 0.5 1

1

2

3

expy x

1 0.5 0 0.5 1

1

2

3

expdy xdx


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Derivatives of sine and cosine

sin cosd

x xdx

1

1

siny x

1

1

cosdy

x

dx


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Derivatives of sine and cosine

cos sind

x xdx

1

1

cosy x

1

1

sindy

x

dx


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Background mathematics 4

Second derivative


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Second derivative

The second derivative is

The derivative of the derivative

The rate of change of thederivative or slope

2

2

d y d dyy x

dx dx dx

1 0.5 0 0.5 1

0.5

1

x


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Second derivative

The slope at is, for small

And similarly at

So1 0.5 0 0.5 1

0.5

1

x

/ 2x x

/2

0

x

y y xdy

dx x

/ 2x

/2

0

x

y x ydydx x

2

2 0limx

d y d dy

dx dx dx

/2 /2x x

dy dy

dx dx

x


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Second derivative


And similarly at

So1 0.5 0 0.5 1

0.5

1

x

/ 2x x

/2

0

x

y y xdy

dx x

/ 2x

/2

0

x

y x ydydx x

2

2 0limx

d y d dy

dx dx dx

0 01 y x y y y xx x x


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Second derivative


And similarly at

So1 0.5 0 0.5 1

0.5

1

x

/ 2x x

/2

0

x

y y xdy

dx x

/ 2x

/2

0

x

y x ydydx x

2

2 0limx

d y d dy

dx dx dx

2

2 0y x y y x

x


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Sign of second derivative

Going from a positive first

derivativeTo a negative first derivative

Gives a negative secondderivative

Going from a negative firstderivative

To a positive first derivative

Gives a positive secondderivative

y

xx2 x3 x4x1

0d y

d x 2

2 0

d y

dx

0d y

d x

2

2 0d y

d x


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Any region where the first

derivative is decreasing withincreasing x

Has a negative secondderivative

Any region where the firstderivative is increasing withincreasing x

Has a positive secondderivative

y

xx

2

2 0

d y

d x

2

2 0d y

d x

x x x x x


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Points where the derivative is

neither increasing ordecreasing

i.e., second derivative ischanging sign

correspond to zero secondderivative

Known as

inflection points

y

xx

2

2 0d y

d x

2

2 0d y

d x

x


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Curvature

The second derivative can be

thought of as thecurvature

of a function

Large positive curvature


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Curvature



of a function

Small positive curvature


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Curvature



of a function

Large negative curvature


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Curvature



of a function

Small negative curvature


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Curvature

The value of the curvature does

not depend on the height ofthe function

All these curves have the samecurvature


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Linearity linear superposition

For two functions

and

The derivative of thesum is the sum of the

derivatives

Example

Split into

So

So

u x v x

d du dv

u x v xdx dx dx

lnf x x x

u x x lnv x x

1du

dx

1dv

dx x

11d u vf xdx x


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Linearity multiplying by a constant

For a function

The derivative ofa constant a times a

function

is

a times the derivative

Example

Split into

So

So

u x

d du

au adx dx

1/2

f x a x a x

a 1/2u x x

1/21 1

2 2

dux

dx x

2

du af x adx x


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Linearity

An operation or function is linear if

linear superposition or additivitycondition

and

multiplication by a constant (or formallyhomogeneity of degree one) condition

f x

f y z f y f z

f ax a f x


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Example of nonlinear operation

The function

does not represent a linear operation

But

So for this function

is not in general equal to

2

f x x

2 2 2 2f y z y z y z yz

2 2f y f z y z

f x y

f x f y


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Product rule

For two functions

and

The derivative of theproduct is

Example

Split into

So

So

u x v x

d dv du

uv u vdx dx dx

2

sinf x x x

2u x x sinv x x

2du

xdx

cosdv

xdx

2 cos 2 sind uvf x x x x xdx


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Quotient rule

For two functions

and

The derivative of theratio or quotient is

Example

Split into

So

So2

du dvv u

d u dx dx

dx v v

u x v x

3

21

x

f x x

3u x x 21v x x

23du

xdx

2dv

xdx

d udx v

2 2 3

22

1 3 2

1

x x x x

x


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Quotient rule

For two functions

and


Example

Split into

So

So2

du dvv u

d u dx dx

dx v v

u x v x

3

21

x

f x x

3u x x 21v x x

23du

xdx

2dv

xdx

d udx v

2 2 3

22

3 1 21

x x x x

x


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Quotient rule

For two functions

and


Example

Split into

So

So2

du dvv u

d u dx dx

dx v v

u x v x

3

21

x

f x x

3u x x 21v x x

23du

xdx

2dv

xdx

d udx v

2 2 4

22

3 1 21

x x xx


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Quotient rule

For two functions

and


Example

Split into

So

So2

du dvv u

d u dx dx

dx v v

u x v x

3

21

x

f x x

3u x x 21v x x

23du

xdx

2dv

xdx

d udx v

2 4

222

3 21 1

x xx x


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Chain rule

For two functions

and

The derivative of thefunction of a function

Can be split into aproduct

Example

Split into

d df dgf g x

dx dg dx

f y g x 2

2

1h x x

21g x x 2f y y


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Chain rule

For two functions

and



Example

Split into

So

So

d df dgf g x

dx dg dx

f y g x 2

2

1h x x

21g x x 2f g g

2df g gdg

2dg

xdx

2 22 1 2 4 1dh x x x xdx


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Chain rule

For two functions

and



Example

Split into

So

So

d df dgf g x

dx dg dx

f y g x exph x a x

g x a x expf g g

expdf g

gdg

dg

adx

exp expdh ax a a axdx


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Chain rule

For two functions

and



Example

Split into

So

So

d df dgf g x

dx dg dx

f y g x 2

exph x a x

2g x a x expf g g

expdf g

gdg

2dg

axdx

2 2exp 2 2 expdh ax ax ax axdx


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dm qm background mathematics lecture 4

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