atomic hidrogen 1

8/17/2019 Atomic Hidrogen 1

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Phy107 Fall 2006 1

From Last Time…

• Light waves are particles

and matter particles are waves!

• Electromagnetic radiation (e.g. light)

made up of photon particles

• Matter particles show wavelike properties like

interference


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Phy107 Fall 2006 2

• Light: Is quantized. Has energy and momentum:

• Electromagnetic radiation(light) has a dual nature.

It exhibits both wave and article characteristics

• The photoelectric effectshows the particle characteristics of light

• Interference and diffraction

Photon: particle and wave

E = hf = hc

λ =1240 eV − nm

λ

p = E

c= hf

c= h

λ

λ = h

p


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Phy107 Fall 2006 3

Wavelengths of massive objects

• de"roglie wavelength #

λ = h p

• #mv for a nonrelativistic

(v $$c) particle with mass.

λ = h

mv

λ = h p= hc

2 mc2 E kinetic

!ame

constant as

be"ore

#inetic energy

rest energy


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Phy107 Fall 2006 4

Wavelength of eV electrons

• %or an electron&

λ =1240 eV − nm

2 × 0.511 MeV

1

E kinetic

=1.23 eV

1/ 2 − nm

E kinetic

• ' e electron& λ#'.* nm• '+ e electron λ#+.*, nm• '++ e electron λ#+.' nm

constant

#inetic energy rest energy


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Phy107 Fall 2006 5

Wavelength of 100 eV objects

• %or an electron&

λ =1240 eV − nm

2 ×m0 MeV

1

100eV

=88 eV

1/ 2 − nm

m0 MeV

• '++ e electron& λ#+.' nm• '++ e proton λ#+.++, nm # ., pm• Electron .-'' Me& roton ,/+ Me

constant

#inetic energy rest energy


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Phy107 Fall 2006 6

Wave reflection from crstal

• If electron are waves the0 can interfere• Interference of waves reflecting from different

atomic la0ers in the cr0stal.

• 1ifference in path length 2 spacing etween atoms

side view

$e"lection "rom

to lane$e"lection

"rom next

lane


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Phy107 Fall 2006 8

Particle interference• 7sed this interference idea to to learn aout the structure of matter

• '++ e electrons4 λ # +.'nm8 9r0stals also the atom

• '+ :e electrons4

8 Inside the nucleus& *. fermi& '+;


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Phy107 Fall 2006 9

Let%s st&d electron waves

• =ere is a wave4

>where is the electron?8 @ave eAtends infinitel0 far in B x and %x direction

x

λ = h

p


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Phy107 Fall 2006 10

'nalog with so&nd

• Cound wave also has the same characteristics

• "ut we can often locate sound waves

8 E.g. echoes ounce from walls. 9an make a sound pulse

• EAample48 =and clap4 duration 2 +.+' seconds

8 Cpeed of sound # */+ mDs

8 Cpatial eAtent of sound pulse # *./ meters.

8 *./ meter long hand clap travels past 0ou at */+ mDs


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Phy107 Fall 2006 11

(eat fre)&enc: spatial locali*ation

• @hat does a sound particleF look like?

8 Gne eAample is a eat freuenc0F etween two notes

8 Two sound waves of almost same wavelength added.

QuickTime™ and a TIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and a TIFF (LZW) decompressor

are needed to see this picture.

&onstructiveinter"erence

Large

amplitude

&onstructiveinter"erence

Large

amplitude

'estructiveinter"erence

Cmall

amplitude


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Phy107 Fall 2006 12

+a,ing a particle o&t of waves

//+ =6 B

/*, =6

//+ =6 B

/*, =6 B

/*H =6

//+ =6 B

/*, =6 B

/*H =6 B

/*3 =6 B

/*< =6


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Phy107 Fall 2006 13

-patial e$tent

of locali*ed so&nd wave

∀ ∆ x spatial spread of wave packetF• Cpatial eAtent decreases as the spread in

included wavelengths increases.

-8

-4

0

4

8

-15 -10 -5 0 5 10 15J

∆ x


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Phy107 Fall 2006 14

-ame occ&rs for a matter wave

• 9onstruct a locali6ed particle 0 adding together

waves with slightl0 different wavelengths.

• Cince de "roglie sa0s λ # h D & each of these

components has slightl0 different momentum.8 @e sa0 that there is some uncertaint0F in the momentum

or the energ0

• nd still donFt know eAact location of the particle!

8 @ave still is spread over ∆ x (uncertaint0F in position)8 9an reduce ∆ x ut at the cost of increasing the spread in

wavelength (giving a spread in momentum).


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Phy107 Fall 2006 15

.nterpreting

• %or sound& we would Just sa0 that the sound pulse is

centered at some position& ut has a spread.

• 9anFt do that for a uantum;mechanical particle.

• Man0 measurements indicate that the electron is

indeed a point particle.• Interpretation is that the magnitude of electron wave;

pulseF at some point in space determines the

probabilit of finding the electron at that point.

-8

-4

0

4

8

-15 -10 -5 0 5 10 15J


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Phy107 Fall 2006 16

/eisenberg ncertaint Principle

• 7sing∆ x # position uncertaint0∆ momentum uncertaint0

• =eisenerg showed that the product

( ∆ x ) • ( ∆ ) is alwa0s greater than ( h D /π )

Gften write this as

where is pronounced h;arF

lanckFs

constant

∆ x( ) ∆ p( )~ h /2

h=

h

2π


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Phy107 Fall 2006 17

Thin,ing abo&t &ncertaint

%or a classical particle& #mv & so an

uncertaint0 in momentum corresponds to an

uncertaint0 in velocit0.

∆ x( ) ∆ p( )~ h /2

∆ x( ) ∆v( )~ h /2m

This sa0s that the uncertaint0 is small for massive oJects&

ut ecomes important for ver0 light oJects& such as

electrons.

Large& massive oJects donFt show effects of uantum

mechanics.


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Phy107 Fall 2006 18

ncertaint principle )&estion

Cuppose an electron is inside a oA ' nm in width.There is some uncertaint0 in the momentum of

the electron. @e then suee6e the oA to make

it +.- nm. @hat happens to the momentum?

. Momentum ecomes more uncertain

". Momentum ecomes less uncertain

9. Momentum uncertaint0 unchanged


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Phy107 Fall 2006 19

sing )&ant&m mechanics

• Kuantum mechanics makes astonishingl0

accurate predictions of the ph0sical world

• 9an appl0 to atoms& molecules& solids.

• n earl0 success was in understanding

8 Ctructure of atoms

8 Interaction of electromagnetic radiation with atoms


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Phy107 Fall 2006 20

Planetar model of atom

• ositive charge is concentrated inthe center of the atom ( nucleus )

• tom has 6ero net charge4

8 ositive charge in nucleus cancelsnegative electron charges.

• Electrons orit the nucleus like

planets orit the sun

• (ttractive) 9oulom force pla0srole of gravit0

nucleus

electrons


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Phy107 Fall 2006 21

!ifference between atoms

• 5o net charge to atom8 numer of oriting negative electrons same as

numer of positive protons in nucleus

8 1ifferent elements have different numer of

oriting electrons

• =0drogen4 ' electron

• =elium4 electrons

• 9opper4 , electrons• 7ranium4 , electrons!

• Grgani6ed into periodic tale of elements


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Phy107 Fall 2006 22

Elements in same

column have similar

chemical properties


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Phy107 Fall 2006 23

• 9ircular motion of oriting electronscauses them to emit electromagnetic radiationwith freuenc0 eual to orital freuenc0.

• Came mechanism 0 which radio waves are emitted

0 electrons in a radio transmitting antenna.• In an atom& the emitted electromagnetic wave

carries awa0 energ0 from the electron.

8 Electron predicted to continuall0 lose energ0.8 The electron would eventuall0 spiral into the nucleus

8 However most atoms are stable!

Planetar model and radiation


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Phy107 Fall 2006 24

'toms and photons

• EAperimentall0& atoms do emit electromagneticradiation& ut not Just an0 radiation!

• In fact& each atom has its own fingerprintF of

different light freuencies that it emits.

=0drogen

Mercur0

@avelength (nm)

/++ nm -++ nm


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Phy107 Fall 2006 25

/drogen emission spectr&m

• =0drogen is simplest atom

8 Gne electron oriting aroundone proton.

• The (almer -eries ofemission lines empiricall0

given 0

n # *& λ #


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Phy107 Fall 2006 26

/drogen emission

• This sa0s h0drogen emits onl0photons of a particular wavelength& freuenc0

• hoton energ0 # h" &

so this means a particular energ0.

• 9onservation of energ04

8 Energ0 carried awa0 0 photon is lost 0 theoriting electron.

h h h d


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Phy107 Fall 2006 27

The (ohr hdrogen atom

• etained planetar0F picture4 oneelectron orits around one proton

• Gnl0 certain orits are stale

• adiation emitted onl0 whenelectron Jumps from one staleorit to another.

• =ere& the emitted photon has anenerg0 of E initial%E final

Ctale orit

Ctale orit '

E initial

E "inal

hoton


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Phy107 Fall 2006 28

nerg levels

• Instead of drawing orits& we can Just indicate the energ0

an electron would have if it were in that orit.Nero energ0

n#'

n#

n#*n#/

E 1= −

13.6

12

eV

E 2= −

13.6

22

eV

E 3= −

13.6

32

eV

E n e r g 0

a A i s

Energ0 uanti6ed!


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Phy107 Fall 2006 29

mitting and absorbing light

hoton is emitted when electrondrops from one uantumstate to another

Nero energ0

n#'

n#

n#*n#/

E 1 = −13.6

12 eV

E 2= −

13.6

22

eV

E 3 = −

13.6

32

eV

n#'

n#

n#*n#/

E 1 = −13.612 eV

E 2= −

13.6

22

eV

E 3= −

13.6

32

eV

soring a photon of correct

energ0 makes electron Jump to

higher uantum state.

hoton

asored

h" #E *%E +

hoton

emitted

h" #E *%E +


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Phy107 Fall 2006 30

Photon emission )&estion

n electron can Jump etween the allowed uantum states

(energ0 levels) in a h0drogen atom. The lowest threeenerg0 levels of an electron in a h0drogen atom are;'*.< e& ;*./ e& ;'.- e.

These are part of the seuence E n %'*.


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Phy107 Fall 2006 31

• Each orit has a specific energ0

E n=-13.6/n2

• hoton emitted when electron

Jumps from high energ0 to low

energ0 orit.E i – E f = h f

• hoton asorption induces

electron Jump fromlow to high energ0 orit.

E f – E i = h f

• grees with eAperiment!

nerg conservation for (ohr atom


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Phy107 Fall 2006 32

$ample: the (almer series

• ll transitions terminate atthe n# level

• Each energ0 level has

energ0 E n#;'*.< D n) e

• E.g. n#* to n# transition

8 Emitted photon has energ0

8 Emitted wavelength

E photon = −13.6

32

− −

13.6

22

=1.89 eV

E photon = hf = hc

λ , λ =

hc

E photon

=1240 eV − nm

1.89 eV = 656 nm


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Phy107 Fall 2006 33

9ompare the wavelength of a photon produced from

a transition from n#* to n#' with that of a photon

produced from a transition n# to n#'.

-pectral 2&estion

n=2

n=3

n=1

. λ31 < λ21

". λ31 = λ21

9. λ31 > λ21

E31 > E21 so λ31 < λ21


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Ph 107 Fall 2006

(&t wh3

• @h0 should onl0 certain orits e stale?• "ohr had a complicated argument ased on

Ocorrespondence principleP

8 That uantum mechanics must agree with classicalresults when appropriate (high energies& large si6es)

• "ut incorporating wave nature of electron gives

a natural understanding of these

uanti6ed oritsF

atomic hidrogen 1

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