Operators - are performed on functions -are performed on vector functions and have directional qualities as well. These are referred to as vector operators.

Operatorsˆ ( ) ( )Of x g x

23 6d

x xdx

ˆ ( ) ( )Of x of x

ˆ ( ) ( )O f r g r

) ) ))d d d

dx dy dz

f(r f(r f(rf(r i j k

2

2

2sin( ) sin( )

dn x n n x

d x

- are performed on functions

-are performed on vector functions and have directional qualities as well. These are referred to as vector operators.

- can obey the Eigen equation, and thus have eigen values and eigen functions.- In general we are concerned with the function that obey this equation.

Classical Mechanics-Position

( )x t

3 2( ) 2 1x t t t

Example

( ) ( ) ( ) ( )t x t y t z t r i j k

2( ) ( 1) 0t t t r i j k

Example

Notice that we are using a function of time to describe the position not somefixed value.

This function tells you the position at any point in time.

Classical Mechanics-Position-3D

ˆ ˆ( ) ( ) ( )x t t x t x r i r i

ˆ ˆ( ) ( ) ( )y t t y t y r j r j

ˆ ˆ( ) ( ) ( )z t t z t z r k r k

ˆ( ) [ ( )]

ˆ ˆ ˆ ( )

( ) ( ) ( )

( ) ( ) ( )

t t

x y z t

t t t

x t y t z t

r r r

i j k r

i r i j r j k r k

i j k

Note that the operator is applied to the position function and the result is the quantity associated with the operator.

Ie. The x operator give you the x component of r(t), this is know as a projection operator.

The vector operator r can be constructed from the projector operators.

Classical Mechanics-Position-3D

2( ) ( 1) 0t t t r i j k

2 2ˆ ( ) ( 1) 0 ( 1) ( )x t t t t x t r i i j k

Example

2ˆ ( ) ( 1) 0 ( )y t t t t y t r j i j k

2ˆ ( ) ( 1) 0 0 ( )z t t t z t r k i j k

2

2

ˆ ˆ ˆ ˆ( ) [ ( )] ( 1) 0

ˆ ˆ ˆ( ) ( ) ( )

( 1) 0

t t x y z t t

x t y t z t

t t

r r r i j k i j k

r i r j r k

i j k

Classical Mechanics-Velocity-1D

( )( )

dx tv t

dt

ˆd

vdt

3 2

2

ˆ( ) ( ) ( )

2 1

6 2

dv t v x t x t

dtd

t tdt

t t

ˆ( ) ( ) ( )d

v t v x t x tdt

Example

Classical Mechanics-Velocity-3D

( ) ( ) ( )( )

dx t dy t dz tt

dt dt dt v i j k

ˆd

vdt

ˆ( ) ( ) ( )d

t v t tdt

v r r

2

2

ˆ( ) ( ) ( 1) 0

( 1) 0

(2 1 0 )

dt v t t t

dtd d dt t

dt dt dtt

v r i j k

i j k

i j k

Example

Classical Mechanics-Velocity-3D

( ) ( ) ( )ˆ( ) [ ( )] ( ) ( ) ( )x y z

dx t dy t dz tt t v t v t v t

dt dt dt v v r i j k i j k

ˆ ˆ ˆ( ) ( )

( ) ( ) ( )ˆ ˆ( )

ˆ ( ) ( ) ( )

( ) ( ) ( )

x

x y z

x y z

v t x v t

d dx t dy t dz tx t xdt dt dt dt

x v t v t v t

v t v t v t

r r

r i j k

i j k

i i j k

ˆ( )x x

dv t v

dt i

ˆyd

vdt

j ˆzd

vdt

k

Classical Mechanics-Velocity-3D

ˆ ˆ ˆ ˆ( ) [ ( )] ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

x y z

x y z

t t v v v t

d t d t d t

dt dt dt

dx t dy t dz t

dt dt dtv t v t v t

v v r i j k r

r r ri i j j k k

i j k

i j k

ˆ ˆ ˆ ˆx y zv v v v i j k

The corresponding vector operator to velocity can be reconstructed from the projector operators of the components:

Classical Mechanics-Velocity-3D

Example

2ˆ ( ) ( 1) 0 (2 1 0 ) 2 ( )x x

dv t t t t t v t

dt r i i j k i i j k

2ˆ ( ) ( 1) 0 (2 1 0 ) 1 ( )y y

dv t t t t v t

dt r j i j k j i j k

2( ) ( 1) 0t t t r i j k

2ˆ ( ) ( 1) 0 (2 1 0 ) 0 ( )z z

dv t t t t v t

dt r k i j k k i j k

2ˆ ˆ ˆ ˆ( ) [ ( )] ( 1) 0

ˆ ˆ ˆ[ ( )] [ ( )] [ ( )]

( ) ( ) ( )

2 0

x y z

x y z

x y z

t t v v v t t

v t v t v t

v t v t v t

t t

v v r i j k i j k

r i r j r k

i j k

i j k

Classical Mechanics-Acceleration-1D

2

2

( )( )

d x ta t

dt

2

2ˆ

da

dt

2

2

23 2

2

ˆ( ) ( ) ( )

2 1

12 2

da t a x t x t

dt

dt t

dtt

2

2ˆ( ) ( ) ( )

da t a x t x t

dt

Example

Classical Mechanics-Acceleration-3D

2 2 2

2 2 2

( ) ( ) ( )( )

d x t d y t d y tt

dt dt dt a i j k

2

2ˆ

da

dt

2

2ˆ( ) ( ) ( )

dt a t t

dt a r r

2

22

ˆ ( ) ( 1) 0

(2 0 0 ) ( )

da t t t

dtt

r i j k

i j k a

Example

Classical Mechanics-Force-1D2

2

( )( )

d x tF t m

dt

2

2ˆ dF m

dt

2

2

23 2

2

ˆ( ) ( ) ( )

2 1

12 2

dF t F x t m x t

dt

dm t tdtmt m

2

2ˆ( ) ( ) ( )

dF t F x t m x t

dt

Example

Classical Mechanics-Force-3D2 2 2

2 2 2

( ) ( ) ( )( )

d x t d y t d y tt m

dt dt dt

F i j k

2

2ˆ dF m

dt

2

2ˆ( ) ( ) ( )

dt F t t

dt F r r

2

22

ˆ( ) ( ) ( 1) 0

2 0 0

dt F t m t t

dtm

F r i j k

i j k

Example

Impulse and Momentum2

2

( )( ) ( )

d x tF t m ma t

d t

ˆˆ ˆ( ) ( ) ( )p x t F x t dt mv x t

2

2

( )( ) ( )

d x tF t dt m dt mdv t

d t MomentumImpulse

2

2

( )( )

d x tF t dt F t m dt m v F t m v

d t

( ) ( ) ( )p t F t dt mv t In general

For a constant force

Momentum-1D( )

( ) ( )dx t

p t mv t mdt

ˆ ˆd

p mv mdt

( )ˆ ˆ( ) ( ) ( )

dx tp t p x t mv x t m

dt

3 2

2

( )ˆ ˆ( ) ( ) ( )

2 1

6 2

dx tp t p x t mv x t m

dtd

m t tdt

mt mt

Example

Momentum-3D

( ) ( ) ( ) ( )( ) ( )

d t dx t dy t dz tt m t m m m m

dt dt dt dt

rp v i j k

ˆ ˆd

p mv mdt

( )ˆ ˆ( ) ( ) ( )

d tt p t mv t m

dt

rp r r

2

ˆ ˆ( ) ( ) ( )

( 1) 0

2 0

t p t mv t

dm t tdtmt m

p r r

i j k

i j k

Example

Impulse-1D

( ) ( )p t F t dt( )

( )dV t

F tdx

ˆ ˆˆd

p Fdt Vdtdx

( )( )

dV tp t dt

dx

ˆˆ( ) ( ( )) ( ( ))

ˆ( ( ))

p t p x t F x t dt

dV x tdt

dx

Force can be thought of asa change in potential energy with change in position

Impulse-1D

2

3 22

3 2

2

ˆˆ( ) ( ) ( ) 2 1

2 1

6 2

dp t p x t F x t dt m t t dt

dtd

m t tdt

mt mt

2

00

( ) 2ˆ ( ) 2 ( ) / 2 cos( ) sin( )

( )

d kx t kxp x t dt kx t dt kx t dt t

dx t

Examples

20

1 ( ( ))( ( )) ( ) ( ( )) ( ) cos( )

2 ( )

dV x tV x t kx t F x t where x t x t

dx t

i) In terms of the Force operator:

ii) In terms of the Potential operator:

Impulse-3D

( ) ( )t t dtp F ( )t V t F

ˆˆ Vdt p

( ) ( )t V t dt p

ˆ ˆˆ( ) ( ( )) ( ( )) ( ( ))t t t dt V t dt p p r F r r

Angular Momentum

( ) ( )t t L r p

( ) ( ) ( ) ( ) ( ) ( )x y zx t y t z t p t p t p t L

( ) ( ) ( )

( ) ( ) ( )x y z

x t y t z t

p x p t p t

i j k

L

Angular Momentum

x y zL L L L i j k

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x z y

y x z

z x y

L y t p t z t p t

L z t p t x t p t

L y t p t x t p t

Angular Momentum

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

x z y

y x z

z y x

L t y t p t z t p t

L t z t p t x t p t

L t x t p t y t p t

ˆˆˆ ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆˆ ˆ ˆˆ ˆˆ ˆ

ˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆ

x z y

y x z

z y x

dz dyL yp zp m y z

dt dt

dx dzL zp xp m z x

dt dt

dy dxL xp yp m x y

dt dt

ˆ ˆ ˆ ˆx y zL L L L i j k

Kinetic Energy22 2( ) ( ) ( )

( )2 2 2

mv t m dx t p tK t

dt m

( ) ( ) ( ) ( ) ( ) ( )( )

2 2 2

m d t d t m t t t tK t

dt dt m

r r v v p p

2 22 ˆˆ ˆ

2 2

m d pK mv

dt m

ˆ ˆ ˆ ˆˆ ˆˆ2 2 2

m d d mK

dt dt m

r r v v p p

22 2ˆ ˆ[ ( )] ( ) [ ( )]ˆ ( )

2 2 2

mv x t m dx t p x tK x t

dt m

Potential Energy

( ) ( )V d r F r r( ) ( )V t F t dx

( )( )

dV tF t

dx ( ) ( )V F r r

Hooks Law

( ) ( )F t kx t

2( ) ( )2

kV t x t

Coulombs Law2

2( )

4 ( )

zeF t

x t

2

( )4 ( )

zeV t

x t

ˆ ˆ( ( )) ( ( )) ( )V x t F x t dx t ˆ ˆV Fdx

ˆ ˆ( ( )) ( ( ))V t t d r F r r

ˆ ˆV d F r

Conservation of Energy

( ) ( )E K t V t Total energy remains constant, as long as V is not an explicit function of time. (i.e V(x(t)))

ˆ ˆ ˆ ˆ( ) ( ) ( ) 0dE d d

K x t V x t K V x tdt dt dt

ˆ ˆ ˆ ˆ( ) ( ) ( ) 0dE d d

K t V t K V tdt dt dt

r r r

2 ˆ ˆˆˆ ˆ ˆ ˆ ˆ( ) ( )2 2

pH K V V x V

m m

p pr

Hamiltonian

Conservation of Energy-Hook’s Law

ˆ ˆ ( ) 0 ?dE d

K V x tdt dt

2( ) ( )

2

kfor V x x t

2 2ˆ ( ) ( )

2 2

p x t k x tdE d

dt dt m

22

2

2

1 ( ) ( )2 ( )

2 2

( ) ( ) ( )2 ( )

2

d dx t k dx tm x t

m dt dt dt

m dx t d x t dx tkx t

dt dt dt

21 1ˆ ˆ( ) ( ) ( )

2 2

d dp x t p x t k x t

m dt dt

2

2

( ) ( )( )

( )( ) ( ) 0

dx t d x tm kx t

dt dt

dx tma t F t

dt

Since: Newtons Law F – ma = 0

1( ) ( ) 2 ( ) ( )

2 2

d d d k dm x t m x t x t x t

m dt dt dt dt