Quantum Model of the Hydrogen Atom_Clase Martes_15!10!2013

8/13/2019 Quantum Model of the Hydrogen Atom_Clase Martes_15!10!2013

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Quantum Model of the

Hydrogen Atom


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Simple Harmonic Oscillator

To explain blackbody radiation Planckpostulated that the energy of a simpleharmonic oscillator is quantized

In his model vibrating charges act as simpleharmonic oscillators and emit EM radiation

The quantization of energy of harmonic

oscillators is predicted by QM


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Simple Harmonic Oscillator (SHO)

Lets write down the Schrdinger Equation for SHO

For SHO the potential energy is

Time independent Schrdinger Equation for SHO

in 1D

mk

xmkxxU

22)(

222

xEx

xm

x

x

m

22

22

2

22


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Simple Harmonic Oscillator

Plancks expression for

energy of SHO

Energy of SHO obtained

from the solution of theSchrdinger equation

Thus, the Planck formulaarises from the Schrdinger

equation naturally n = 0 is the ground state

with energy h

2;2

,...3,2,1,0

2

1

2

1

h

n

hnnE

nhE

Term htells usthatquantum SHO always

oscillates. These are calledzero point vibrations


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Quantum Model of the

Hydrogen Atom

Potential Energy

Time-independent Schrdinger Equation

Schrdinger Equation in so-called sphericalor polar coordinates

2222

here,)( zyxrr

ekrU e

E

r

ek

rrrr

rrm e

2

2

2

222

2

2

2

sin

1

sin

11

2

Ezyx

ek

zyxm e

222

2

2

2

2

2

2

22

2


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The wavefunction is the function of three variablesnow and can be written as

We had one variable (quantum box or simpleharmonic oscillator)one quantum number

Here we can assume that both wavefunction andenergy should in general depend on threequantum numbers, corresponding to eachcoordinate

Wavefunction of the

Hydrogen Atom

)()()(,, gfrRr


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Wavefunction of the

Hydrogen Atom

The three quantum numbers, corresponding toeach coordinate are

nis the principal quantum number; it correspondsto coordinate r

lis the orbital quantum number; associated withthe coordinate , and determines the magnitude of

the electrons angular momentum, L. mlis the magnetic quantum number; associated

with the coordinate , and it determines theorientation of Lin the magnetic field


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The Hydrogen Atom

Thus we can write in general:

Quantum numbersdetermine the quantum state Often we say that the electron is in the state (n, l, ml)

The energy of the particle in a quantum statedepends on all quantum numbers

),,(),,( ,, rr lmln

lmln

EE ,,


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The Hydrogen Atom

An electron in a hydrogen atom only has

physically reasonable solutions when E, l and mlhave the values given indicated below:

lm

nlnln

a

ek

h

meE

n

EE

rr

l

emln

mln

l

l

...,,2,1,0

)i.e.(1...,,2,1,0...,3,2,1

28,

),,(),,(

0

2

20

4

020

,,

,,


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The Hydrogen Atom

The electron energy in thehydrogen atom depends only onthe principal quantum number, n

Same result as in Bohrs model.

2

0

,, n

EE

lmln


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Hydrogen Atom:

From Bohr to Schrdinger

The electron orbital (angular) momentum is

L is thus quantized This is more general than the Bohrs Postulate 3, since it

allows for orbits with angular momentum of zero (!!)

For large values of l(l >> 1):

)1( llL

lllLl

)1(


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The Hydrogen Atom:


1. Remember:Electrons simply

exists within the atom, and we canonly know the probability offinding the electron at a certaincoordinate

2. Bohrs orbits correspond to the

coordinate, where the probability offinding the electron is the largest1. Moreover, Bohrs theory was limited to

the states with highest angularmomentum, l = n 1

The wavefunction gives physical meaning ofthe orbits:


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The Hydrogen Atom:


There are no orbits!!! We describe the electron through

the quantum states

The energy of the electron is

constant when electron in a givenquantum state

Thus, no energy can be takenfrom electron by radiation

Why doesnt the accelerating electron emit EMWwhen in states corresponding to Bohrs Orbits?


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The Hydrogen Atom:


Since the energy of electron in a hydrogen doesnot depend on different land ml(in general forother atoms it does), It is naturally to assume

that photons are only emitted or absorbed withenergies corresponding to the differencebetween various energy sates

220

20,,

11

...,3,2,1,

nk

EEEh

nn

EEE

knkn

nmln l


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The Hydrogen Atom

Since, the energy depends only on the principal

quantum number, the energy of an electron indifferent quantum states- with the same n, butdifferent land/or mlis the same

The different sates having the same energy arecalled Degenerate States

The number of such states (having the sameenergy or energy level) is referred to as

degeneracy of the energy level In general, the degeneracy arises from the

symmetry in a system


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Degeneracy: The Hydrogen Atom

Example:n = 3 there are 9 states (describedby different wavefunctions) that have thesame energy

2,1,02,1,03 lmln

232322321132320311131310300,,,,,,,,

2,1,0;2,1,0;3

lmln

eV51.1

9

6.13

3

2

03

EE


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Degeneracy

Degeneracy can be lifted if additional

interactions (forces) applied to a system Consider a hydrogen atom in the magnetic field,

B, applied along z-direction. The energy ofelectron is then

This energy now depends on ml: we saydegeneracy over mlwas lifted

How many energy levels with differentdegeneracy are there for n = 3?

l

e

nlm m

m

Be

n

EE

l 220


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More Features of the Atomic

Wavefunctions

Electrons in the different quantum states areunlikely to be found in the same spatial regions

Electrons in the lower angular momentum states

are more likely to be found closer to the nucleusthan those in states with higher angularmomentum

In multi-electron atomsthe degeneracy over

angular momentum is liftedand the sates withthe same nbut different l have slightly differentenergies


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Space Quantization

ldetermines the magnitude of the angular

momentum, L:

ml

determines orientation of Lin space when atomis in a magnetic field

Taking the magnetic field along z-direction, we can showthat

Thus, not only is the angular momentum quantized, butalso its component in some direction, usually taken tobe the z-direction

)1( llLl

lz

mL


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Space Quantization

Therefore, L, in an atom, cannot have anyarbitrary orientation (in magnetic field) withrespect to z-direction, but rather have onlydiscrete orientations:Space Quantization

However, as long as we do not have apreferred direction (e.g. defined bymagnetic field) space quantization ismeaningless

lz

mL


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Space Quantization: Example, l = 2


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Space Quantization: Zeeman Effect

11

2

0;0;1

Em

m

BeEE

mln

l

e

total

l

1

1

0

2

1,0;1,0;2

2

e

total

l

m

BeEE

mln


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Space Quantization:

Stern-Gerlach Experiment

Silver atoms studied inthe non-uniformmagnetic field with thegradient along z-

direction (dB/dz) Although the atoms are

neutral, they possess amagnetic dipole, and

the inhomogeneousmagnetic field acts onthis dipole with force


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The atoms are deflected as they travel through the magnet The stronger the force, the greater the deflection, and thus

the father away from the center of screen the atoms wouldland as detected on the screen

mdz

dB

m

e

dz

Bmm

ed

dz

dU

Fe

e

z

2

)2(

Bmm

eU

m

m

eL

m

e

BBBU

e

ez

e

zzz

2So,

22

:atomaninorbitinelelctronanFor

:zBBfieldmagneticain

dipolemagneticaofUenergy,potentialMagnetic


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Only two lines were observed on the screenon which the atoms land: one above and onebelow of the center !! Moreover, there were no central line and the distance

from the center to each positions was the same

http://mutuslab.cs.uwindsor.ca/schurko/nmrcourse/animations/stern-gerlach/sgpeng.htmhttp://mutuslab.cs.uwindsor.ca/schurko/nmrcourse/animations/stern-gerlach/sgpeng.htmhttp://mutuslab.cs.uwindsor.ca/schurko/nmrcourse/animations/stern-gerlach/sgpeng.htmhttp://mutuslab.cs.uwindsor.ca/schurko/nmrcourse/animations/stern-gerlach/sgpeng.htm


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Stern-Gerlach experiment indeed showeddiscrete nature of the momentum, confirmingspace quantization

However, the result contradict quantitative

prediction of the Schrdinger The experiment by Phipps and Taylor with

hydrogen atoms, where ml= 0 (and thus Lz= 0)showed the two emission lines in the spectrum,

suggesting that the effect is NOT due to OrbitalMomentum

Thus, there may be a magnetic dipole other thanthe one associated with orbital momentum!


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The Spin

The Spin is a [quantum] property of the electron.

Spin quantum number is always there

Other quantum numbers can change depending onthe specifics of the potential the electron is in.

For instance: In hydrogen, in addition to the spin, an electron is

characterized by three quantum numbers n, l, ml In a 1D Infinite potential well, in addition to the spin, an

electron is characterized by only one quantum number n

The spin is intrinsic to the electron!


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The Hydrogen Atom Wavefunction:

Revisited

Now we need to add another quantumnumber, associated with the electron spin

2

1

...,,2,1,0

)i.e.(1...,,2,1,0

...,3,2,1),,(),,( ,,,

s

l

mmln

m

lm

nlnl

nrr sl

G S f


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The Ground State of the

Hydrogen Atom

For the hydrogen ground state we have

Reminder: The hydrogen atom is unusual since

states with the same lhave the same energy.

21or

21;0;0;1

),,(),,( ,,,

sl

mmln

mmln

rrsl


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The Ground State of Atoms other

than Hydrogen

As soon as an atom contains more than oneelectron, the states with different lno longer havethe same energy, and degeneracy over orbital

quantum number is lifted What is the ground sate of a multi-electron system?

Perhaps in the lowest energy state all electrons in theatom have the same four quantum numbers

NO!!!


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Paulis Exclusion Principle

NO TWO ELECTRONS IN A SYSTEM CAN BEIN THE SAME QUANTUMS STATE

In other words, no two electrons can have thesame values of the quantum numbers

For atoms these are:

This allows us to understand the Periodic Table The electron configuration of any atom must

satisfy the Paulis exclusion principle

sl mmln ;;;


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Paulis Exclusion Principle: Atoms

Definitions:

a shell is the set of states with the same n(thusdifferent l, mland ms)

a sub-shell the set of states with the same n and

l(thus different mland ms) Using the rules for the atomic quantum

numbers, we determine that the number of

electrons in a sub-shell is 2(2l+1)

a shell is 2n2


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Additional Rules for Determining

the Ground State-Configuration

1. The total energy of the electron increases with increasing n(energy less negative)

2. Within a given shell (a given n), the l = 0states alwayshave

the lowest energy3. The energy of the sub-shells generally increases with l :

El=0< El=1< El=2< El=3

since electron with lowest value of lcan be closer tonucleus and does not feel shielding of (+)nuclear charge by () electron cloud

El i C fi i f h


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Electronic Configuration of the

Atomic Ground State: Examples

We shall use letters instead of numbers to denote the orbitalquantum number:

l = 0s; l = 1p;

l = 2d; l = 3f We shall use superscript above the letter to indicate the

number of electrons in a given sub-shell

Hydrogen (1): 1s1

Lithium (3): 1s22s1

Nitrogen (7): 1s22s2p3

Argon (18): 1s22s2p63s2p6


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Potassium (19): 1s22s2p63s2p64s1

Calcium (20): 1s22s2p63s2p64s2

Scandium (21): 1s22s2p63s2p6 d14s2 Note that 3d sub-shell starts to fill in after 4s

sub-shell is filled up

The same is true for 4d vs 5s sub-shells


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Nickel (28): 1s22s2p63s2p6d84s2

Copper (29): 1s22s2p63s2p6d104s1

Silver (47): 1s22s2p63s2p6d84s2p6d105s1 Note that there is no nd9 state for either Cu, Ag,

or Au (they loose one s-electron from the nextshell)


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Bees This should be a learning experience. So lets discuss your

project before the due date After ~ 600 seconds some bees have reached the boundary

and escaped

Up to this time would expect that the variance of the averagethat variance begins to approach the diffusion relation derivedby Einstein,

D is the diffusion coefficient, is the mean free path, and isthe mean free time

Could you describe one characteristic of the bees motion thatyou choose. For example, you could consider the timedependence of the number of bees escaping per unit time or

2

22

3

1,inces,

3

1

0case,thisinsince,6)()var(

vDvvD

xDtxxxx

Quantum Model of the Hydrogen Atom_Clase Martes_15!10!2013

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