pauli atom

7/21/2019 Pauli Atom

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Recap: Last few days (1)

• We showed how quantum mechanics describes identical particlesIt does not distinguish between them

• We introduced a new operator the e!change operator "1#

where "1# f(1#) $ f(#1)

• We showed how this operator can be used to describe wa%efunctions

that are either even or odd under the exchange of coordinates

describing particles

• &or identical particles we saw '"1# $*+ ,on -egenerate .& of

are also .& of "1#



Recap: Last few days (#)

• In cases where they were degenerate solutions we could find alinear combination of wa%efunctions which were also solutions:

ψ / $ 'ψ n(!1) ψ m(!#) / ψ m(!1)ψ n(!#)

which is an even function

ψ − $ 'ψ n(!1) ψ m(!#) − ψ m(!1)ψ n(!#)

which is an odd function • We described a more complete amiltonian for 1 electron atom by

introducing concepts such as spin spin orbit coupling+

• We introduced spin .& 0(σ) $ α(σ)β(σ)+ We said the fullwa%efunction is gi%en by

#

1

#

1

( ) terms smaller r V r

Ze

m

pr H SO +++++

1

#)(

#

*

#

++−= σ πε

σ

( ) ( )σ φ S r n



Recap: Last few days (2)

• We saw that the quantum numbers (n 3 m3) determined the shape ofa wa%efunction (as well as the energy)+



4he two electron atom

( )

terms smaller r V r V r

e

r

Ze

r

Ze

m

p

m

pr r H H

SOSO +++

−+−+==

)()(

1

##

)(#1

##11

1#

#

#

#

1

##

#

#

1##11

σ σ πε

σ σ

)#1()( ##11 H r r H =σ σ

orb P 1#

spin P 1# spinorb

P P P 1#1#1# =

Electron 1

Electron 2

Nucleus

1r #r

1#1# r r r −=In short we write

Recall the e!change operator "1#

interchanges all

coordinates (both orbital and spin of electrons 1 and #)

We can separate the e!change operator in two parts+ If

is the e!change operator of the orbital coordinates and if

is the e!change operator of the spin coordinates then



&rom the form of the amiltonian we see that it is symmetric under

interchange of all coordinates of electrons 1 and #+

4herefore

( )[ ] *#11# = H P

4his implies that the .& of (1#) are either e%en or odd functions underthe e!change of coordinates+

( )

terms smaller r V r V

r

e

r

Ze

r

Ze

m

p

m

pr r H H

SOSO +++

−+−+==

)()(

1

##)(#1

##11

1#

#

#

#

1

##

#

#

1##11

σ σ

πε σ σ



( ) )()(1

## #*1*

#

#

1

#

*

#

#

#

1#1* r H r H

r Ze

r Ze

m p

m pr r H += +−+= πε

4he simplified amiltonian now based only on spatial coordinates loo5s li5e

)()()()()()( #

*

##*1

*

11* r E r r H and r E r r H mmmnnn φ φ φ φ ==

4he solution is obtained from the solutions to the indi%idual parts:

( )#1* r r H )()( #1 r r mn φ φ

***

mnnm E E E +=

4he .& of are gi%en by the product

and the .6 is

1#

#

r

e

To calculate the wavefunctions for the two electron atom,

We can simplify amiltonian as follows:

,eglect the spin orbit and smaller terms

,eglect the (large7) term for the moment.



*

nm E

( ) ( ) ( ) ( )#1##11 r r and r r nmmn φ φ φ φ φ φ ==

Recall there is a case of degeneracy for a specific set of integers n≠m where

two distinct .& ha%e energy .

4hese need not be .& of "1# but we can pic5 a linear combination which is an

.& of "1#

+

( ) ( ) ( ) ( ) ##1#11#11# φ φ φ φ φ φ === r r r r P P nmmn

orborb

1#1# φ φ =orb P

( )#1#

1φ φ φ +=+ ( )#1

#

1φ φ φ −=−

−−++ −== φ φ φ φ orborb P and P 1#1#

Lets consider

Li5ewise

0o φ1 and φ

# are not .& of "

1#orb

8ut linear combinations of φ1 and φ# are .& of "1#orb

and these are described byand

where both states are degenerate and ha%e energy .n/.

m+



4he spin description

4o get the full wa%efunction we must multiply the orbital part by the appropriate spin

function+

4he one electron spin functions are α and β+ 9s we ha%e not considered the effect ofspin on the energy the energy of the state is not affected by which spin state theelectron is in+

0o in the two electron atom we multiply the orbital state by the product of the spinstates such as α(σ1) β(σ#) which indicates that that electron 1 is in the spin upstate and electron # is in spin down state

4here are four possible linearly independent spin functionsα(σ1) α(σ#) ↑↑α(σ1) β(σ#) ↑↓β(σ1) α(σ#) ↓↑β(σ1) β(σ#) ↓↓



9nd as these ha%e the same energy we can pic5 linear combination of

the abo%e spin functions+

4he following linear combinations are particularly useful:

4he four spin states are orthonormal

( ) ( ) ( )

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

( ) ( ) ( ) ↓↓=

↓↑+↑↓+=

↑↑=

−

+

+

+

#1#1

1

#1#1#1

*

#1#1

1

#

1

σ β σ β σ σ

σ α σ β σ β σ α σ σ

σ α σ α σ σ

S

S

S

( ) ( ) ( ) ( ) ( )( ) ↓↑−↑↓−=− #1#1#1

*

#

1 σ α σ β σ β σ α σ σ S

0ince these are e%en under interchange

of spin coordinates they are called 0/

functions

e%en .&s of "1#spin

&urther analysis shows that these three

states ha%e a total 0pin 0$1+

0ince this is odd under interchange

of spin coordinates this is called a 0

function+ 9n odd .& of "1#spin

&urther analysis shows that this state

has a total 0pin 0$*+



The full wavefunction is a product of orbital and spin functions.

4he twoelectron wa%efunction based on single electron n m states is

φ/ ; spin function (one of four abo%e)

φ− ; spin function (one of four abo%e)

-eeper analysis of these functions shows that:

• 0/ functions are .& of total spin and .6 of total spin is 1< while the = componentsof total spin (1 * 1) in units of < for states respecti%ely+

• 0 is an .& of total spin with .6 $ *> the = component is =ero

0/ functions are spin triplets

0− is a spin singlet

We can now consider thefollowing full wa%efunctions:

Which are appro!imate .& of

the original amiltonian (1#)+

( ) ( )

( ) ( )

( ) ( )

( ) ( )#1

*

#1

#1#1

#1

*

#1

#1#1

1*1

1*1

σ σ φ

σ σ φ

σ σ φ

σ σ φ

−−

+−

−+

++

−=

−=

S r r

iS r r

S r r

iS r r

i

i



We saw that (1#) commutes with "1# ?the operator which e!changes

the full coordinates (orbital @ spin) of electrons 1 and #+

4he abo%e eight wa%efunctions are .& of "1#+

Where φ/0/ and φ:0: are e%en under "1# while φ/0: and φ:0/ are odd

under "1#+

( ) ( )

( ) ( )

( ) ( )

( ) ( )#1

*

#1

#1#1

#1

*

#1

#1#1

1*1

1*1

σ σ φ

σ σ φ

σ σ φ

σ σ φ

−−

+−

−+

++

−=

−=

S r r

iS r r

S r r

iS r r

i

i

( ) ( ) ( ) ( ) ±±±±±±±± ±=±±== S S S P P S P spinorb φ φ φ φ 1#1#1#



"auli .!clusion "rinciple

8ased on e!perimental e%idence "auli ad%anced the principle that thewa%efunctions for a system of electrons must be odd under the

e!change of all coordinates (orbital and spin) of any two electrons+

Aathematically we ha%e seen that electronic wa%efunctions can beeither e%en or odd>

Interpreting the "auli "rinciple suggests that nature only requires odd

functions to describe electrons+



4he "auli "rinciple is an established principle of quantum physics+

It says that the wavefunction for a system of identical particles issymmetrical under the interchange of all coordinates of any twoparticles if they are of integral spin ( particles, photons,….BOSOS!, while the wavefunction is anti"symmetrical underthe interchange of all coordinates of any two identical particles if

the particles are of half integral spins (electrons, protons,neutrons, mesons, ….#$%&'OS!

4he name 8osons and &ermions refer to the fact that the particles obeydifferent statistical rules and these rules were wor5ed out by 8ose

and .instein for integral spin particles and by &ermi and -irac forhalfintegral spin particles+



4he "auli "rinciple and two electron atom+

Returning to the two electron atom we see the "auli "rinciple restricts

the wa%efunctions to anti?symmetrical (odd) ones only+

4he only %alid wa%efunctions are therefore φ/0: and φ:0/i+

9nd since the energy of φ/ is abo%e that of φ: the energy le%els are:

φ+S- ↑↓ anti parallel spins (S=0)

φ-S+i. ↑↑ parallel spins (S=1)

2 H’12

4he Boulomb repulsion between the electrons allied with the "auli "rinciple

splits the two electron states into spin singlet (0$*) and spin triplets (0$1)+



,o two fermions can occupy the same state

Bonsider two electrons where the two spin ?orbit states are the same+ 4hat is the sameorbital quantum numbers apply (n 3 m3) describe the two states+

Bonsider the odd functions:

φ/0:

φ0/i:

Bonsider the e%en orbital function is described by

If the two electrons ha%e the same spin states eg+ 0pin up then the product of thespin wa%efunctions is the .%en function α(σ1) α(σ#)$0/

1 then the full spin orbital

is the e%en state

owe%er by the "auli "rinciple such an e%en spin orbital wa%efunction is notallowed+

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) *#

1*

#

1 #1#1#1#1#1

* =−=−=− σ β σ β σ β σ β σ α σ α σ α σ α σ σ or S

( ) *#

1#1 =−=− φ φ φ

1

++S φ

( ) ( )#mn1mn r r

φ φ φ =+



Bonsequence of "auli "rinciple

4hus we cannot ha%e two electrons in the same orbital and spin states+

We can e!press this as follows:

In the same system no two electrons can ha%e the same quantum

number numbers (orbital and spin)

4his is sometimes used as a simplified %ersion of the "auli "rinciple+

pauli atom

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