Understanding Quantum Physics: Particle-Wave Behaviors, Quantized Energies, Tunneling, Slides of Physics

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Some

topics

in

Quantum

Theory

Particle

behaviors

of

electromagnetic

waves

Wave

behaviors

of

particles

Quantized

energies

Webassign

hint:

N

d

N

m

d

grooves/cm

For

sin

If

you

have

not

already

done

so

please

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to

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concerning

your

intentions

regarding

Exam

The

material

you

have

learned

up

to

now

in

PHY

was

known

in

and

is

basically

still

true

.^

Some

details

(such

as

at

high

energy,

short

times,

etc.

have

been

modified

with

Einstein’s

theory

of

relativity,

and

with

the

development

of

quantum

theory.

Which

of

the

following

technologies

do

not

need

quantum

mechanics.

A.

X

‐ray

diffraction

B.

Neutron

diffraction

C.

Electron

microscope

D.

MRI

(Magnetic

Resonance

Imaging)

E.

Lasers

Which

of

the

following

technologies

do

not

need

quantum

mechanics.

A.

Scanning

tunneling

microscopy

B.

Atomic

force

microscopy

C.

Data

storage

devices

D.

Microwave

ovens

E.

LED

lighting

Quantum

physics

Electromagnetic

waves

sometimes

behave

like

particles

one

“photon”

has

a

quantum

of

energy

E=

hf

momentum

p=

h/

hf/c

Particles

sometimes

behave

like

waves

“wavelength”

of

particle

related

to

momentum:

=h/p

quantum

particles

can

“tunnel”

to

places

classically

“forbidden” 

Stationary

quantum

states

have

quantized

energies

Classical

physics

Wave

equation

for

electric

field

in

Maxwell’s

equations

(plane

wave

boundary

conditions):

Equation

for

particle

trajectory

r

(t)

in

conservative

potential

U(

r )

and

total

energy

E

^



^



ct

x k E t x x c

t^





 



^ 

sin ˆ

) , (

:

example

for

max

2 2

2

2 2

j

E

E

E

 

2

0

0

2

example

12 for

t

g

t

t

E

U

d dt

m

k

v

r

r r

r

Docsity.com

Mathematical

representation

of

particle

and

wave

behaviors.

Consider

a

superposition

of

periodic

waves

at

t=0:

^

i^

xi k

E

t x E

sin

(^

max

single

wave

(one

value

of

k

)

superposed

wave

(many

values

of

k

)

^

^

^

^

2

max

2

sin

(^

i^

xi k

E

x E



x^  

x^ 



k^

=^

10





k^

=^

1^ 



x^



k^



^2



x

smaller

more

particle

like

k

smaller

more

wave

like

Wave

equations

Electromagnetic

waves:

Matter

waves:

(Schrödinger

equation)

2 2 2

2 2

x

c

t^

^ 

E

E ^

^

^

 t x x U x m t x t

i^

,

) (

2

,^

2 2

2

   

  



 





  







Electromagnetic waves

Matter waves

Vector –

E

or

B

fields

Second order

t

dependence

Examples:

Scalar – probability amplitudeFirst order

t

dependence

Examples:

^



^

 t

kx

c E

t x

B

t

kx

E

t x

E

y z

ω

sin

ω

sin

max max

/

0

sin(

(^

iEt e

kx

t x



Comparison

of

different

wave

equations

0 2 0

0

/

/

(^30)

8

1

) , (^

0

0

a e

E

e

e a

t r^

t iE

a r

 







^



Wave

‐like

properties

of

particles

Louis

de

Broglie

suggested

that

a

wavelength

could

be

associated

with

a

particle’s

momentum

x i x h i h p  

   









 2

λ

“Wave”

equation

for

particles

Schrödinger

equation

^

^

^

 t x t h i t x x U x m

2 2

2

^

^

 

^

^

^

 , , ) ( 2

ions

wavefunct

state

Stationary

2 2

2

/ t x E t x x U x m

e

t^

iEt





r

r

^

^

 

^

^

^



^



or

sin

particle

free

Example

2 2

2 2

0

2 2

2

/





 

h m

E

h mE

k

k m

E

(kx)e

x,t

t x E t x x m

e

t

r

U

-iEt/

iEt







r

r

Example:

Suppose

we

want

to

create

a

beam

of

electrons

( m

=9.1x

‐^31

kg)

for

diffraction

with

=1x

‐^10

m.

What

is

the

energy

E

of

the

beam?

^

 ^

^

17

2

10

31

2

34

2 2

eV

J

m

kg

J

h m

E

- -^







Electrons

in

an

infinite

box:

 

 

 

π

sin

ψ

ψ

for

ψ

ψ

0

2 2

2

n

L

x n

x

L x x x m x E

m

n

E

n

2 2 2 

Electrons

in

a

finite

box:

finite

probability

of

electron

existing

outside

of

classical

region