Download Outline of Section 6

Hydrogen-like atoms
Roadmap for solution of Hydrogen-like atoms
• Start from 3D TISE for electron in Coulomb potential of nucleus
• Separate variables to give 1D radial problem and angular problem
Solution of angular part already known
in terms of spherical harmonics.
• Simplify 1D radial problem using substitutions and atomic units
• Solve radial problem
Extract asymptotic solution at large r
Use Frobenius method
Find eigenvalues by requiring normalizable solutions
Reminder: SE in three
dimensions
H-atom is our first example of the 3D Schrödinger equation
Wavefunction and potential energy are now
functions of three spatial coordinates:
Kinetic energy involves three
components of momentum
Interpretation of
wavefunction:
V  x   V  r   V  x, y , z 
px2  p y2  pz2
px2
p2


2m
2m
2m
2
2
2
 2
2
2
2 
2


 


2m x 2
2m
2m  x 2 y 2 z 2 
d r  r, t 
3
  x     r     x, y , z 
  r, t 
2
2
probability of finding particle in a probability density at r
volume element centred on r
(probability per unit volume)
Time-independent Schrödinger equation
Ĥ  r   r   E  r 

2
2m
 2  r   V  r   r   E  r 
The Hamiltonian for a hydrogenic atom
In a hydrogenic atom or ion with nuclear
charge +Ze there is the Coulomb
attraction between electron and nucleus.
This has spherical symmetry – potential
only depends on r. This is known as a
CENTRAL POTENTIAL
The Hamiltonian operator is
-e
Ze2
V (r )  
4 0 r
r
+Ze
2
2
Ze
Hˆ  
2 
2me
4 0 r
NB: for greater accuracy we should use the reduced mass not the electron mass
here. This accounts for the relative motion of the electron and the nucleus (since
the nucleus does not remain precisely fixed):
me   
me mN
; me  electron mass, mN  nuclear mass
me  mN
Hamiltonian for hydrogenic atoms
The natural coordinate system is spherical polars. In
this case the Laplacian operator becomes (see 2B72):
2
1


L


2  2  r 2   2 2
r r  r 
r
Lˆ  
2
2
 1  
 
1 2 
 sin    sin     sin 2   2 




So the Hamiltonian is
2
2
2
2
ˆ
Ze


L
Ze
 2 
Hˆ  r   
2 


r

2
2
2me
4 0 r
2me r r  r  2me r
4 0 r
2
TISE for H-like atom is
(m = me from now on)
Ĥ  r   r   E  r 
  2  (r )  Lˆ2 (r ) Ze 2


 (r )  E (r )
r

2
2
2mr r 
r  2mr
4 0 r
2
Angular momentum and the H atom
2
ˆ2


L
Ze


2
Hˆ  r   

r

2
2
2mr r  r  2mr
4 0 r
2
Reminder:
d Q
i
Q
 H , Q 
Ehrenfest’s theorem
dt


t
CONCLUSIONS
• The angular momentum about any axis and the total
angular momentum commute with the Hamiltonian
• They are therefore both conserved quantities
So we will look for a solution of the form
• We can have simultaneous eigenfunctions of these
operators and the Hamiltonian
• We can have well-defined values of these quantities and
the energy at the same time
2
[H , L ]  0 
[H , Lz ]  0 
d L
2
dt
d Lz
dt
0
0
The angular wavefunction
This suggests we look for
separated solutions of the form
 (r)   (r   )  R(r )Ylm (   )
The angular part are the eigenfunctions of the total angular momentum operator
L2. These are the spherical harmonics, so we already know the corresponding
eigenvalues and eigenfunctions (see §5):
LˆzYlm  ,    m Ylm  ,  
1
Y00 ( ,  ) 
4
Lˆ2Ylm  ,    l  l  1 2Ylm  ,  
Y11 ( ,  )  
Eigenvalues of Lˆ2 are l (l  1) 2 , with l  0,1, 2,
Eigenvalues of Lˆz are m , with  l  m  l
l = principal angular momentum quantum number.
3
sin  exp(i )
8
Y10 ( ,  ) 
3
cos 
4
Y11 ( ,  ) 
3
sin  exp(i )
8
m = magnetic quantum number (2l+1 possible values).
Note: this argument works for any spherically-symmetric
potential V(r), not just the Coulomb potential.
The radial equation
Substitute separated solution into the time-independent Schrödinger equation
 (r , ,  )  R(r )Ylm ( ,  )
Get radial equation
  2   Lˆ2
Ze 2


 (r )  E (r )
r

2mr 2 r  r  2mr 2 4 0 r
2
d  2 dR  l (l  1)

r

2
2mr dr  dr 
2mr 2
2
2
Ze2
R
R  ER
4 0 r
Note that this depends on l but not on m: R(r) and E therefore involve
the magnitude of the angular momentum but not its orientation.
The radial equation (2)
Define a new radial
function χ(r) by:
R(r ) 
Get radial equation for χ(r)
 (r )
d  2 dR  l (l  1)

r

2
2mr dr  dr 
2mr 2
2
r
d 2   l (l  1)


2
2
2m dr
2
mr

2
2
2
Ze2
R
R  ER
4 0 r
Ze2 

   E
4 0 r 
The effective potential
New radial equation looks like the
1D Schrödinger equation with an
effective potential
l (l  1)
Veff (r ) 
2mr 2
2
d 2   l (l  1)


2
2
2m dr
2
mr

2
Ze2

4 0 r
2
Ze2 

   E
4 0 r 
d 2

 Veff (r )  E
2
2m dr
2
1D TISE
V(r)
r
d 2

 V ( x)  E
2
2m dx
2
l (l  1)
2mr 2
2
is known as the centrifugal
barrier potential
The centrifugal barrier
Where does the centrifugal barrier come from?
CLASSICAL ARGUMENT
l (l  1)
Vcb (r ) 
2mr 2
2
Fixed l corresponds to fixed angular momentum for the electron. L  mv r
so as r becomes small,v must increase in order to maintain L. This causes an
increase in the apparent outward force (the ‘centrifugal’ force).
For circular motion
mv 2
L2
F
 3
r
mr
L  mv r
dV
L2
F 
V 
dr
2mr 2
Alternatively, we can say that the energy required to supply the extra
angular speed must come from the radial motion so this decreases as
if a corresponding outward force was being applied.
Roadmap for solution of radial equation
d 2   l (l  1)


2
2
2m dr
2
mr

2
2
Ze2 

   E
4 0 r 
 nlm (r)  R(r )Ylm (   )
 (r )
R(r ) 
r
• Simplify equation using atomic units
• Solve equation in the asymptotic limit (large r)
Gives a decaying exponential solution
 (r ) 
 exp( r )
r 
• Define new radial function by factoring out asymptotic solution
 (r )  F (r ) exp( r )
• Solve equation for F(r) using the series (Frobenius) method
F (r )   a p r p  s
p
• Find that solution is not normalizable unless series terminates.
This only happens if the eigenvalues have certain special values.
Hence we find the eigenvalues and eigenstates of the H-atom.
Atomic units
There are a lot of physical constants in these expressions. It makes atomic
problems simpler to adopt a system of units in which as many as possible of
these constants are one. In atomic units we set:
Planck constant  1 (dimensions [ ML2T 1 ])
Electron mass me  1 (dimensions [ M ])
Constant apearing in Coulomb's law
e2
4 0
 1 (dimensions [ ML3T 2 ])
It follows that:
 4 
Unit of length =  2 0 
 5.29177 1011 m=Bohr radius, a0
 e  me
2
2
 e2  me
18
Unit of energy = 
 2  4.35974 10 J  27.21159eV=Hartree, Eh
 4 0 
In these units the radial equation becomes
d 2   l (l  1)


2
2
2m dr
 2mr
2
2
Ze2 
1 d 2   l (l  1) Z 


    E
   E  
2
2
4 0 r 
2 dr
r
 2r
Asymptotic solution of
radial equation (large r)
Consider the radial equation at very large
distances from the nucleus, when both terms
in the effective potential can be neglected.
We are looking for bound states of the atom
where the electron does not have enough
energy to escape to infinity (i.e. E < 0):
Put E  
This gives
1 d 2   l (l  1) Z 


    E
2
2
2 dr
r
 2r
2
2
d 2
2


   (r )  exp( r )
2
dr
For normalizable solutions we must take the decaying solution
Inspired by this, rewrite the solution in terms
of yet another unknown function, F(r):
 (r )  F (r ) exp( r )
Differential equation for F
Derive equation for F
1 d 2   l (l  1) Z 


    E
2
2
2 dr
r
 2r
 (r )  F (r ) exp( r )
Differential equation for F:
 d2
d l (l  1) 2Z 
 dr 2  2 dr  r 2  r  F  r   0


Series solution (1)
Look for a power-series solution (Frobenius method). The point r = 0 is a
regular singular point of the equation so at least one well-behaved series
solution should exist (see 2B72).
Substitute
F (r )   a p r p  s
p

a r
p 0
p
p  s 2
 d2
d l (l  1) 2Z 
 dr 2  2 dr  r 2  r  F  r   0



 p  s  p  s  1  l l  1   a p r p  s 1 2  p  s   2Z   0
p 0
Series solution (2)
The indicial equation that fixes s comes from equating coefficients of
the lowest power of r which is s - 2

 apr
p 0
p  s 2

 p  s  p  s  1  l l  1   a p r p  s 1 2  p  s   2Z   0
p 0
s ( s  1)  l (l  1)  0
 s 2  s  l (l  1)  0
  s  l  s  (l  1)   0
s  l , or s  l  1
We need the regular solution that will be
well-behaved as r→0, so take
s  l 1
Series solution (3)
General recursion relation comes from equating coefficients of r to the power p+l

a r
p 0
p
p l 1

 p  l  1 p  l   l  l  1   a p r p l 2  p  l  1  2Z   0
p 0
a p 1
ap

2   p  l  1  Z 
 p  l  2  p  l  1  l  l  1
Series solution (4)
For p→∞ we find:
a p 1
ap
a p 1
2


p 
p
ap

2   p  l  1  Z 
 p  l  2  p  l  1  l  l  1
(remember a p is coefficient of r pl 1 in the expansion)
Compare with:

(2 r ) n
(2 ) n
n
exp(2 r )  
  bn r with bn 
.
n!
n!
n 0
n 0

Coefficient of r p l 1 would be bp l 1.
Coefft. of r p l  2 bp l  2 (2 ) p l  2 ( p  l  1)!
2
2






Coefft. of r p l 1 bp l 1 (2 ) p l 1 ( p  l  2)! ( p  l  2) p 
p
So, our series behaves for large p just like exp(2κr).
Series solution (5)
So, if the series continues to arbitrarily
large p, the overall solution becomes
 (r )  F (r ) exp( r )
 (r ) exp(2 r )  exp( r )  exp( r )
(not normalizable)
To prevent this the series must terminate after a
finite number of terms. This only happens if
Z

 ( p  l  1) for some integer p  0,1, 2
 n where n is a positive integer  l : n  l  1, l  2
So finally the energy is
2
Z2
E
 2
2
2n
with n > l
n is known as the principal quantum number.
It defines the “shell structure” of the atom.
Summary of solution so far
 nlm (r )  Rnl (r )Ylm (   )
 nl (r )

Ylm (   )
Each solution of the time-independent
Schrödinger equation is defined by
three quantum numbers n,l,m
r
Fnl (r )e Zr / n

Ylm (   )
r
The radial solution depends on
n and l but not m
Fnl (r )  r
l 1

a r
p 0
p
p
The energy only depends on the
principal quantum number n
which is bigger than l
a p 1
ap

2 Z  p  l  1  n 
n  p  l  2  p  l  1  l  l  1 
Z2
En   2 , n  1, 2,3
2n
n  l 
Example
What is the radial wavefunction for n = 2 and l = 0?
The hydrogen energy spectrum
1
En   2
2n
In Hartrees  Eh  27.2eV
In eV ground state energy
= -13.6eV
= - ionisation energy
This simple formula agrees with
observed spectral line frequencies
to within 6 parts in ten thousand
n > l so
n 1
n2
l 0
l  0,1
n3
l  0,1, 2
Traditional spectroscopic nomenclature:
l = 0: s states (from “sharp” spectral lines)
l = 1: p states (“principal”)
l = 2: d states (“diffuse”)
l = 3: f states (“fine”)
…and so on alphabetically (g,h,i… etc)
The energy spectrum: degeneracy
For each value of n = 1,2,3… we have a
definite energy:
For each value of n, we can have n
possible values of the total angular
momentum quantum number l:
Z2
En   2 (in atomic units)
2n
l = 0, 1, 2,…, n-1
For each value of l and n we can have 2l+1
values of the magnetic quantum number m:
The total number of states (statistical weight)
associated with a given energy En is therefore:
(This neglects electron spin. See Section 7.)
m  l , (l  1),
0,
(l  1), l
n 1
2
(2
l

1)

n
.

l 0
The fact that the energy is independent of m is a feature of all spherically
symmetric systems and hence of all atoms. The independence on l is a
special feature of the Coulomb potential, and hence just of hydrogenic
atoms. This is known as accidental degeneracy.
The radial wavefunctions
Rnl(r) depends on n and l but not on m
Z
R10 (r )  2  
 a0 
3/ 2
exp(  Zr / a0 )
1  Z 
R21 (r ) 


3  2a0 
 Z 
R20 (r )  2 

2
a
 0
3/ 2
3/ 2
 Zr 
  Zr 
exp
 


a
2
a
 0
 0

  Zr 
Zr 
1

exp




2
a
2
a
0 

 0
 Z 
R32 (r ) 


27 10  3a0 
4
4 2 Z 
R31 (r ) 


9  3a0 
 Z 
R30 (r )  2 

 3a0 
3/ 2
3/ 2
3/ 2
2
 Zr 
  Zr 
exp
 


a
3
a
 0
 0 

  Zr 
Zr   Zr 
1

exp

 


6
a
a
3
a
0
0
0

 


 2 Zr 2Z 2 r 2 
  Zr 
1


exp



2 
 3a0 27 a0 
 3a0 
For atomic units set a0 = 1
 nlm (r )  Rnl (r )Ylm (   )
Rnl (r ) 
 nl (r )
r
Fnl (r )e  Zr / n

r
The radial wavefunctions (2)
Z
R10 (r )  2  
 a0 
3/ 2
exp(  Zr / a0 )
1  Z 
R21 (r ) 


3  2a0 
 Z 
R20 (r )  2 

 2a0 
3/ 2
3/ 2
 Zr 
  Zr 
exp
 


a
2
a
 0
 0

  Zr 
Zr 
1

exp




 2a0 
 2a0 
 Z 
R32 (r ) 


27 10  3a0 
4
R31 (r ) 
4 2 Z 


9  3a0 
 Z 
R30 (r )  2 

 3a0 
3/ 2
3/ 2
3/ 2
2
 Zr 
  Zr 
exp
 


 a0 
 3a0 

  Zr 
Zr   Zr 
1

exp

 


 6a0   a0 
 3a0 
 2 Zr 2Z 2 r 2 
  Zr 
1


exp



2 
 3a0 27 a0 
 3a0 
Only s states (l = 0) are finite at the origin.
Radial functions have (n-l-1) zeros
(excluding r = 0).
Full wavefunctions are:
 nlm (r)  Rnl (r )Ylm (   )
Normalization chosen so that:


0
drr 2 Rnl2  r   1
Asymptotic solution
n1
Rnl (r ) 


r
exp(Zr / n)
r 
Solution near r = 0
l
Rnl (r ) 


r
r 0
Radial probability density
Total probability density
|  nlm (r ) |  R (r ) Ylm (   )
2
|  nlm (r ) | d r
2
3
2
nl
2
(Rnl is real)
= probability of finding particle in a
volume element centred on (r, θ, φ)
Integrate over all angles using normalization of spherical harmonics
d   sin  d d
d 3r  r 2 drd 
Radial probability density
r R (r )dr 
2
2
nl
 
(solid angle element)
R (r ) Ylm (   ) r 2 drd 
2
nl
2
,
r 2 Rnl2 (r )dr
= probability of finding the particle in a
spherical shell centred on r, i.e. at any angle
r 2 Rnl2 (r )dr   nl2 (r )dr
so this is analogous to the 1D case
Angular probability density
Solid angle probability density
| Ylm (   ) |
2
= probability density of finding
particle in a solid angle element
| Ylm (   ) |2 d  
| Ylm (   ) |2 sin  d d
d   sin  d d
= probability of finding particle between
θ and θ + dθ and φ and φ + dφ
Total probability density
|  nlm (r ) | d r   r R (r )dr  . | Ylm (   ) | sin  d d 
2
3
2
2
nl
2
= (Radial probability) x (Angular probability)
Radial probability density
Radial probability density
r 2 Rnl2 (r )
Orbital
n l
<r> (au)
1s
1 0
1.5
2s
2 0
6.0
2p
2 1
5.0
3s
3 0
13.5
*
r   d 3r  nlm
(r )r nlm (r )

  dr r Rnl  r 
3
0
2
Comparison with Bohr model
Bohr model
Angular momentum (about any axis)
assumed to be quantized in units of
Planck’s constant:
Lz  n , n  1, 2,3,
Electron otherwise moves
according to classical
mechanics and has a single
well-defined orbit with radius
n 2 a0
rn 
, a0  Bohr radius
Z
Energy quantized and
determined solely by angular
momentum:
Z2
En   2 Eh , Eh  Hartree
2n
Quantum mechanics
Angular momentum (about any axis)
shown to be quantized in units of
Planck’s constant:
Lz  m , m  l ,
,l
Electron wavefunction spread over
all radii. Can show that the
quantum mechanical expectation
value of 1/r satisfies
1  Z  1 , a  Bohr radius
r
rn 0
n 2 a0
Energy quantized, but
determined solely by principal
quantum number n, not by
angular momentum:
Z2
En   2 Eh , Eh  Hartree
2n
The remaining approximations
These results are not exact because we have made several approximations.
• We have neglected the motion of the nucleus. To fix this we should replace me
by the reduced mass μ. This improves agreement with experiment by an order of
magnitude (simple formula gives spectral lines to within 4 parts in 100 thousand!)
me   
me m p
me  m p
and E (n)  me

1
1
 E ( n) 
E ( n)   
1 m / m
1  me / m p
e
p

 Z2
 2 Eh
 2n
• We have used a non-relativistic treatment of the electron and in particular have
neglected its spin (see §7). Including these effects give rise to
“fine structure”
(from the interaction of the electron’s orbital motion with its spin)
“hyperfine structure”
(from the interaction of the electron’s spin with the nuclear spin)
• We have neglected the fact that the EM field between the nucleus and the
electron is itself a quantum object. This leads to “quantum electrodynamic”
(QED) corrections, and in particular to a small “Lamb shift” of the energy levels.
Summary
Energy levels in au
Z2
En   2 , n  1, 2,3
2n
n  l 
In Hartrees  Eh  27.2eV
Ground state energy = -1/2 au = -13.6eV = - ionisation energy
Statistical weight
n 1
g    2l  1  n 2
l 0
Wavefunction
Fnl (r )  r
 nlm (r )  Rnl (r )Ylm (   )
Fnl (r )e  Zr / n

Ylm (   )
r
Radial probability density
r 2 Rnl2 (r )
a p 1
ap

l 1

p
a
r
 p
p 0
2 Z  p  l  1  n 
n  p  l  2  p  l  1  l  l  1 