HYDROGEN ATOM AND HYDROGEN-LIKE IONS
Hydrogen is the simplest of all the atoms with only one electron surrounding the nucleus.
Ions such as He+ and Li2+ are hydrogen-like since they also have only a single electron.
In each case the mass of the electron is much less the nuclear mass, therefore, we will
assume a stationary nucleus exerting an attractive force that binds the electron. This is the
Coulomb force with corresponding potential energy Uc(r)
(1)
U c (r)  
Z e2
4 o r
The Coulomb force between the nucleus and electron is an example of a central force
where the attractive force on the electron is directed towards the nucleus. This is a three
dimensional problem and it best to use spherical coordinates (r, , ) centered on the
nucleus (Fig. 1).
Z
electron
r
z = r cos

nucleus
Y

r sin
x = r sin cos
y = r sin sin
X
Fig. 1. Spherical coordinates of the electron (r, , ) centered on the nucleus. The
distance between the electron and nucleus is r,  is the angle between the Z axis
and the radius vector and  is the angle between the X axis and the projection of the
radius vector onto the XY plane.  ranges from 0 to 2 and the azimuthal angle 
from 0 to .
The Schrodinger equation in spherical coordinates can be expressed as
(1)
1   2 
r
r 2 r  r
1  


 2
 sin 
r
 r sin  r 
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1  2



 2 2
2
 r sin  
2
2
 E  U 
0
1
where  is the reduced mass of the system. The time independent wavefunction  in
spherical coordinates where the variables have been separated can be taken as
(2)
 ( r,  ,  )  R( r )( ) ( )
In applying the principle of separation of variables to the three dimensional Schrodinger
equation (Eq. 1) in spherical coordinates leads to the differential equation for  ( )
(3)
d 2( )
 ml 2 ( )
2
d
azimuthal equation
This is called the azimuthal equation. The solution of the azimuthal equation (Eq. 3) is
(4)
( )  exp  iml  
The function  ( ) has a period of 2 if ml is an integer. Since all physical quantities are
derived from the wavefunction, the wavefunction must be singled valued for , 2, … .
The differential equation for ( ) is called the angular equation
(5)
1 d
sin  d
( )  
ml 2 

sin


l
(
l

1)

( )


d  
sin 2  

angular equation
The solutions ( ) for the angular equation are polynomials in cos known as the
associated Legendre polynomials Pl ml (cos  ) . The products ( ) ( ) describing the
angular dependence of the wavefunction are known as the spherical harmonics Yl ml ( , ) .
The functions ( ) are polynomials in sin and cos of order l.
Finally, to complete the process, the radial equation becomes
(6)
2
1 d  2 dR  2m 
l (l  1) 
r

E

U

R0
c


2
2 
r dr  dr 
2m r 2 

radial equation
The associated Laguerre functions are the solutions of the radial equation are in
polynomials in r).
The physically acceptable wavefunctions that are solutions of the three differential
equations (Eqs. 3, 5 and 6) and that satisfy the boundary conditions imposed on the
system depend upon three quantum numbers n, l, ml where
principal quantum number n = 1, 2, 3, …
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2
orbital quantum number
l = 0, 1, 2, 3, … , (n-1)
magnetic quantum number ml = 0, 1, 2, 3, … , l
The set of three quantum numbers n, l, ml determine the state of the electron.
The differential equation in  (Eq. 3) and in  (Eq. 5) are independent of the potential
energy function Uc(r). The total energy E and the potential energy Uc(r) appear only in
the radial differential equation (Eq. 6). Therefore, it is only the radial equation (Eq. 6)
containing the potential energy term Uc(r) that determines the allowed values for the total
energy E.
Physically acceptable solutions of the radial equation (Eq. 6) for hydrogen atom and
hydrogen-like ions can only be found if the energy E is quantized and has the form
(7)
En  
Z 2 me4
 4  o 
2
2
2
1
13.6 Z 2

eV
n2
n2
where the principal quantum number is n = 1, 2, 3, … and n > l. The negative sign
indicates that the electron is bound to the nucleus. If the energy were to become positive,
then the electron would no longer be a bound particle and the total energy would no
longer be quantized. The quantized energy of the electron is a result of it being bound to a
finite region. For hydrogen-like species, the total energy depends only on the principal
quantum number n, this is not the case for more complex atoms. The ground state is
specified by the unique set of quantum n = 1, l = 0, ml = 0. For the first excited state there
are four independent wavefunctions with quantum numbers: n = 2, l = 0, ml = 0; n = 2, l =
1, ml = -1; n = 2, l = 1, ml = 0; n = 2, l = 1, ml = 1. This means that the first excited state is
four-fold degenerate.
We can define a pseudo-wavefunction g(r) = r R(r) which leads to a one dimensional
Schrodinger equation
(8)
 2  d 2 g (r)

 U eff ( r ) g ( r )  E g ( r )

2
 2m  dr
where the effective potential energy has two contributions due to the Coulomb interaction
Uc and the angular motion of the electron
(9)
U eff  U c  U l
Uc  
Ze2
 4  o  r
Ul 
l (l  1)
2mr 2
In many applications in atomic physics it is important to know the behaviour of the
wavefunctions since measureable quantities can be obtained by calculating various
expectation values. From the wavefunction of a given state (n l ml) we can calculate the
probability of finding an electron from the corresponding probability density function
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*
*
(10) nlml *nlml   nlm
eiEnt /  nlml eiEnt /   nlm
 nlml  Rnl* *lml *ml Rnl lml ml
l
l
The probability of finding the electron does not depend upon the azimuthal angle  since
(11) *ml ( )ml ( )  eiml eiml  1
The three dimensional behavior of the probability density is completely dependent on the
product of the radial probability density Pnl ( r )  Rnl* ( r ) Rnl ( r ) and a directionally
dependent modulation factor *lml ( )lml ( ) . The probability of finding the electron at
any position within a spherical shell of radius r and thickness dr is
(12) P( r )dr  Rnl* ( r ) Rnl ( r )4 r 2 dr  4 g nl* ( r ) g nl dr
where the factor 4 rdr gives the differential volume enclosed between the spheres.
The modulation factor *lml ( ) lml ( ) can be conveniently shown as a polar diagram
with the origin at the center of the nucleus (r = 0). A line drawn from the origin to a point
on the curve is equal to the value of *lml ( )lml ( ) where the angle  is measured
between the line and the Z axis. The full angular dependence is shown by visualizing the
three-dimensional surface obtained by rotating the curve on the polar diagram through the
2 range of the azimuthal angle .
MATLAB: Scripting, Investigations, Questions
pot_hydrogen.m
The m-script pot.hydrogen.m evaluates the potential energy functions (Eq. 9). Fig. 2
shows the potential energy functions for the state l = 1. A shallow potential energy well is
formed which binds the electron to the nucleus. For radial positions near r = 0, the
effective potential energy is positive and has a negative slope. Thus, the force between
the electron and the nucleus is repulsive. Only when l = 0 is there an appreciable
probability of finding the electron very close to the nucleus. For any state with l > 0 there
is zero probability of finding the electron a the center of the nucleus.
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L=1
20
Uc
UL
15
Ueff
potential energy U (eV)
10
5
0
-5
-10
-15
-20
0.5
1
1.5
radial position r (m)
2
2.5
x 10
-9
Fig.2. The effective potential energy for a single electron bound to the nucleus in
the state l = 1.. The effective potential has two contributions, the coulomb
interaction action and the potential associated with the angular momentum of the
electron.
a_legendre.m
There is a Matlab function called legendre.m which evaluates the associated Legendre
polynomials. You need to specify the angle and the order of the polynomial l. The
function returns a set of rows for each of the allowed values of ml and set of columns for
each angle. The following m-script illustrates how to call the legendre.m script for
angles ranging from 0 to 2 rad with l = 3. The rows correspond to ml = 0, 1, 2, 3. The
vector A gives the value for the associated Legendre polynomial l = 3 and ml = 2 which
has been normalized to a maximum of 1 for each angle :
close all; clear all; clc
num = 5;
theta_min = 0;
theta_max = 2*pi;
theta = linspace(theta_min,theta_max,num)
L = 3;
m_L = 2;
% angle theta (rad)
% orbital quantum number
% magnetic quantum number
PLm = legendre(L,cos(theta));
A = PLm(m_L+1,:)/max(abs(PLm(m_L+1,:)));
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% number of angles
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% associated Legendre polynomials
% associated Legendre polynomial
%
L, m_L normalized to 1.
5
Sample results:
theta =
0
1.5708
3.1416
4.7124
6.2832
PLm =
1.0000 -0.0000
0
1.5000
0
0.0000
0
-15.000
A=
-1.0000
0
0
0
0.0000
1.0000
1.5000
0
-0.0000 0
-15.000 0
0.0022
0.0993
-0.0000
-0.9934
Z axis
Z axis
P(l,m l ) 2 = P(3, 0) 2
P(l,m l ) = P(3, 0)
(a)
(b)
Fig. 3. Polar diagrams (a) the associated Legendre polynomial and (b) directional
dependence of the one electron probability function for the state l = 3 and ml = 0.
The m-script a_legendre.m was used to create the polar plots.
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Exploration
Use the m-script a_legendre.m to view the polar plots of the associated Legendre
polynomials and corresponding probability density functions for a wide range of l and ml
values.
Investigations and questions
References
The probability of finding an electron in a and the probability of finding the electron
must have physically acceptable properties because
m-scripts
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