Chem 373- Lecture 16: Introduction to the Hydrogen Atom

8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom

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Lecture 16: Introduction to the Hydrogen

AtomThe material in this lecture covers the following in Atkins.

Section 13.1 The structure of hydrogenic atoms

(a) The separation of internal motion

Lecture on-lineThe Hamiltonian of the hydrogen atom (PDF Format)

The Hamiltonian of the hydrogen atom (PowerPoint)

Handout for this lecture


2/23

Audio-visuals on-line

Hydrogen atom Hamiltonian and energy levels

(PowerPoint)(Good account

from the Wilson Group,****)

Hydrogen atom Hamiltonian and energy levels (PDF)(Good account from the

Wilson Group,****)

Slides from the text book (From the CD included in Atkins ,**)Interactive Hydrogen Orbital Plots (For Mac users only)

( Visualizes all the

angular and radial wavefunctions of the hydrogen atom, *****)


3/23

Ze

r

-e

(r,,)

x

y

z

Hamiltonian forhydrogenic atom

The hydrogenic atom consists of an electron ofcharge (-e) amd mass M moving around a nuclei

of charge Ze and mass M.e

=m

m

reduced mass

e

e

M

M

The

+

Structure of Hydrogenic Atoms

The

V x y z

Hamiltonian is given by

H =p

2

2

+ ( , , )


4/23

Ze

r

-e

(r,,)

x

y

z

Hamiltonian forhydrogenic atom


In

HeZ

ro

quantum mechanics

{= h2 2

2 4 }

H =p

2

2

+ V x y z( , , )

V x y z eZro

( , , ) = 4


5/23


{ ( , , ) ( , , ) =

h2

2

2 4

eZ

r x y z E x y zo }

r

(x,y,z) (r,, )

X

Y

Z

rcos

rsin

We shall solve this eqution

in spherical coordinates

x r

y r

z r

==

=

sin cos

sin sin

cos

Schrdinger forhydrogenic atom

eq.

{H eZro

= h2

22 4

}


6/23


With

eZ

rr E r

o{ ( , , ) ( , , ) =

h2 2

2 4 }

and

= + rr r r r

L22

2 2 222 1[ ]

h

We can write

[ ]Hr r r r

LZe

ro= + +

h2 2

2 22

2

2 1

2 4

Where

L22

2 2

2

2

1= + +[ cot

sin

]


eq.


7/23

We finally get the Schrdinger equation

{ [ ] } ( , , ) ( , , ) + + =h

2 2

2 22

2

2 1

2 4

r r r r

LZe

rr E r

o


Ze

r

-e

(r,,)

x

y

z


eq.


8/23


We

H

shall first show

[L2, ] = 0

Since L does notdepend on rand commuteswith itself

2

We H

r r r rL

Ze

ro

have

[L2

[L2, ]

, ( ) ]

=

+ + =h

2 2

2 22

2

2 1

2 4

[L L

2 2, ( ) ] [ , ] + + =h2 22 2

22

24

1

2

rr r

Zer

r

Lo

0 0

0

Since L does notdepend on r

2

Commutationfor hydrogenic atom

relations


9/23


where

L iz = h

We can also show that

[H,Lz ] = 0

We can writer hH r

r r r rL

Ze

ro( ) [ ]= + +

2 2

2 22

2

2 1

2 4


relations


10/23


We have[ , ( )]L H rz

r=

Thus

[H,Lz ] = 0

[ , ] [ , ]L

r r r

Ze

r L rLz

oz +

h h2 2

2

2

2 22 4

1

2

Since L does notdepend on r

z Since L commutes

with L

z2

0 0

= 0


relations


11/23


SinceL H L H L Lz z[ , ] [ , ] [ , ]= = =0 0 0

2 2; ; ;

We must be able to find commeneigenfunctions to all three operators

H = E

L2 2 = h (l +1)l ; l = 0,1,2

L m ; m = -l, - l+1, ....,lz = h


relations


12/23


We l m

l mim

have already shown that( , ) = Y ( , ) =

2l+1

4P

are eigenfunctions to L and L

l,m l|m|

2z

(( | !|

( | !|)(cos ) exp[ ]

+

We

L

must also have that

R(r)Y ( , (l+1)lR(r)Y ( , ; l = 0,1, 2

L R(r)Y ( , m R(r)Y ( , ; m = -l, - l+1, ...., l

as L and L do not depend on r

l,m l,m

z l,m l,m

2z

2 2

) )

) )

=

=

h

h

Radial equationfor hydrogenic atom


13/23


WeEshall now seek solutions of the formHR(r)Y ( , R(r)Y ( ,l,m l,m ) )=

or

r r r mrL Ze

r

E

o{ [ ] } )

)

+ + +

=

h2 22 2

2

22 1

2 4

R(r)Y ( ,

R(r)Y ( ,

l,m

l,m

or

r r rZe

r mrL

E

o{ [ ] } ) { } )

)

+ + +

=

h

2 2

2 22

22

41

2

R(r)Y ( , R(r)Y ( ,

R(r)Y ( ,

l,m l,m

l,m



14/23


Thus

r r r

Ze

r mrL

E

o

R(r)Y ( , R(r)Y ( ,

R(r)Y ( ,

l,m l,m

l,m

{ [ ] } ) { } )

)

+ + +

=

h2 2

2 22

2

2

4

1

2

Now

r r r

Ze

r

l l

mr

E

o

by working on Y ( , )

Y ( , R(r) Y ( , R(r)

R(r)Y ( ,

l,m

l,m l,m

l,m

){ [ ] } ) ){( )

}

)

+ + + +

=

h h2 22

222

2

4

1

2

Where we have used that L is an eigenfunction of Y ( , )2 l,m



15/23


Y ( , R(r) Y ( , R(r)

R(r)Y ( ,

l,m l,m

l,m

){ [ ] } ) ){( )

}

)

+ ++

=

h h2 2

2

2

22

2

4

1

2r r r

Ze

r

l l

mr

E

o

Multiplying next by 1/ Y ( , ) from the right on both sides

R(r) R(r) R(r)

l,m

{ [ ] } ) {( )

} + + + =h h

2 2

2

2

22

2

4

1

2r r r

Ze

r

l l

mrE

o

Combining terms

R(r) R(r) R(r) R(r) + + + + =h h2 22

222

24

12

{ ) { ( ) }r r r

Zer

l lmr

Eo



16/23


Combining terms

R(r) R(r)R(r) R(r) + + + + =

h h2 2

2

2

22

2

4

1

2

{ ) {

( )}

r r r

Ze

r

l l

mrE

o

Veff

Correolis orcentrifugal term

Coulomb attraction


We have

R(r) R(r)R(r) R(r) + + +

+=

h h2 2

2

2

22

2

4

1

2

{ ) {( )

}r r r

Ze

r

l l

mrE

o


17/23

We have

R(r) R(r)R(r) R(r) + + +

+=

h h2 2

2

2

22

2

4

1

2

{ ) {( )

}r r r

Ze

r

l l

mrE

o

(1) The solutions R(r) to our equation must depend onl and E

(2) It turns our that for R(r) to be normalized

=

=

R r R r r dr

E

e

nn

o

( ) ( )

, , ...

2

0

4

2 2 2

1

1 2 3

can only take certain values given by

E = -Z

32

2

2

h

Structure of Hydrogenic Atoms Solutions to radial equationfor hydrogenic atom


18/23

( )

) )

3 It is further required that l in the

total solution

(r, , R(r)Y ( ,

l < n -1

l,m =

satisfies

Thus

(r, , R(r) Y ( ,

n = 1, 2, 3 ...

l < n - 1

m = - l, - l+ 1, ....l - 1, l

nl l,m ) )=

We have

R(r) R(r)R(r) R(r) + + +

+=

h h2 2

2

2

22

2

4

1

2

{ ) {( )

}r r r

Ze

r

l l

mrE

o



19/23

Introducing

Zr

a m eo e

= =2 2

2and a

4o

oh

We find that solutions R to

R (r) R (r)

R (r) R (r)

nl

nl nl

nl nl

( )

{ ) {

( )

}

r

r r r

Ze

r

l l

mr Eo nl + + +

+

=

h h2 2

2

2

22

2

4

1

2

are

r N n L r enl nl

l

n ln

of the form

( ) ( ),/

= 2

Normalization Polynomial Exponential



20/23


n l

1 0

0

1

R

e

e

R r e

n l,

//

//

,

//

( )

( )

2Z

a

1

2 2

Z

a

1

4 6

Z

a

o

o

o

=

3 22

3 24

2 0

3 24

2 21

2

2

n l

1 0

0

1

R

e

e

R r e

n l,

//

//

,

//

( )

( )

2Z

a

1

2 2

Z

a

1

4 6

Z

a

o

o

o

=

3 22

3 24

2 0

3 24

2 21

2

2


21/23


n R

e

e

e

nll

0

1

3 6 2 1

9

3 4 13

3 2

3 22 6

3 26

3 22 6

19 3

Za

127 6

Za

1

81 30

Z

a

o

o

o

+

//

//

//

( )

( )


22/23

1. You are not expected to be ableto solve the Schrdinger equation for

the hydrogen - like atom

H n,l,m n,l,m ( , , ) ( , , ) r E rn =

However you should be aware that theHamiltonian can

be written in the form

where r is the distance between the

hydrogen- like

atom and the electron whereasis the reduced mass.

r h

Hr r r r

L

Ze

ro= + +

2 2

2 2 22

2 1

2 4

[ ]

You should also be aware of

the following commutation

relations

[H,L and [H,L2

z ] ]= =0 0

What you need to know about the hydrogen atom

from the previous lecture


23/23

2. You are not required to memorize

the exact form of the eigenfunctions.

You should recognizedthat they can be

written as a product

of a radial part R (r) and

the spherical harmonics

Y ( , )

[eigenfunctions of L and L ]

as

(r,

R (r)Y ( ,

nl

lm2

z

n,l,m

nl lm

, )

)

=

What you need to know about the hydrogen atom

from the previous lecture

with the corresponding

energies given by

E = -Z me

32 e n

2 4

2

o

2 2 2

h

It is important that youremember the possible

quantum numbers for l

and m with respect to

a given n.

Documents

Chem 373- Lecture 16: Introduction to the Hydrogen Atom