Post on 06-Apr-2018
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
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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, *****)
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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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
+ ( , , )
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Ze
r
-e
(r,,)
x
y
z
Hamiltonian forhydrogenic atom
Structure of Hydrogenic Atoms
In
HeZ
ro
quantum mechanics
{= h2 2
2 4 }
H =p
2
2
+ V x y z( , , )
V x y z eZro
( , , ) = 4
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
{ ( , , ) ( , , ) =
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
}
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
]
Schrdinger forhydrogenic atom
eq.
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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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
Structure of Hydrogenic Atoms
Ze
r
-e
(r,,)
x
y
z
Schrdinger forhydrogenic atom
eq.
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
Commutationfor hydrogenic atom
relations
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
Commutationfor hydrogenic atom
relations
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
Commutationfor hydrogenic atom
relations
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
Radial equationfor hydrogenic atom
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
Radial equationfor hydrogenic atom
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
Radial equationfor hydrogenic atom
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms
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
Radial equationfor hydrogenic atom
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
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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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
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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( )
) )
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
Structure of Hydrogenic Atoms Solutions to radial equationfor hydrogenic atom
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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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
Structure of Hydrogenic Atoms Solutions to radial equationfor hydrogenic atom
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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Structure of Hydrogenic Atoms Solutions to radial equationfor hydrogenic atom
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
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Structure of Hydrogenic Atoms Solutions to radial equationfor hydrogenic atom
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
+
//
//
//
( )
( )
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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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
8/3/2019 Chem 373- Lecture 16: Introduction to the Hydrogen Atom
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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.