The First Order Stark Effect In Hydrogen

1. The First Order Stark Eect
In Hydrogen For n = 3
Johar M. Ashfaque
University of Liverpool
May 11, 2014
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2. Introduction
I will brie
y mention the main result that was covered in my
undergraduate dissertation titled Time-Independent Perturbation
Theory In Quantum Mechanics, namely the

3. rst order Stark eect
in hydrogen.
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4. What is the Stark Eect?
The splitting of the spectral lines of an atom in the presence of an
external electric

5. eld is known as the Stark eect. Table shown
below, corresponding to the Stark eect in the hydrogen atom
provides a summary in the case of n = 2; 3; and 4:
n l m Matrix Total Elements O-Diagonal
Elements By
Symmetry
2 0, 1 0, 1 44 16 6
3 0, 1, 2 0, 1, 2 99 81 36
4 0, 1, 2, 3 0, 1, 2, 3 1616 256 120
Table: Stark eect corresponding to the cases where n = 2; 3; and 4.
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6. The Case: n = 3
For n = 3, the degeneracy of the energy level of the hydrogen
atom is 9. We can see this degeneracy more clearly in the form of
nlm representation as
300, 310, 320, 311, 321, 31-1, 32-1, 322, 32-2,
where l = 0; 1; 2 and m = 0;1;2. We conclude that we are
dealing with a 9 9 matrix and set out to compute the elements
of this matrix.
Due to symmetry, we only need to compute
n2(n2 1)
2
o-diagonal elements. In our case, we need to compute only 36
o-diagonal elements.
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7. The Case: n = 3 Contd.
De

8. nition
The perturbation produced by the electric

9. eld = (0; 0; ) is
^ H0 = ez = er cos()
where e is the electron charge.
It is clear, that the ez has odd parity since
er cos( ) = er cos():
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10. The Case: n = 3 Contd.
Denote the 9 states as 300, 310, 320, 311, 321, 311, 321,
322, 322.
De

11. nition
A general matrix element is de

12. ned as
D
nlm

15. ^ H0

18. n0l 0m0
E
= e
ZZZ
r 2dr sin()dd
nlmr cos() n0l 0m0
= e
Z 1
0
r 3drRnl (r )Rn0l 0 (r )
ZZ
sin() cos()ddY m
l
(; )Y m0
l 0 (; )
= e
Z 1
0
r 3drRnl (r )Rn0l 0 (r )
Z
0
sin() cos()d
Z 2
0
dY m
l
(; )Y m0
l 0 (; ):
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19. The Case: n = 3 Contd.
De

20. nition
The integral Z 1
0
r 3drRnl (r )Rn0l 0(r )
is the radial part.
De

21. nition
The integral
Z
0
sin() cos()d
Z 2
0
dY m
l
(; )Y m0
l 0 (; )
is the angular part.
The integral over all space of any odd function is zero.
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22. The Case: n = 3 Contd.
The integral, as n = 3 is constant, gives
Z 1
0
r 3drRnl (r )Rn0l 0(r ) =
Z 1
0
r 3drR3l (r )R3l 0(r ) =
(
0; if l = l 0,
K; if l6= l 0
where K is a constant to be determined.
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23. The Radial Wave Function Plots Of The Hydrogen Atom
Maple plots of the radial wave function jrR(r )j2 for the hydrogen
atom for n = 3 and l = 0; 1; 2 are presented:
R30 R31 R32
Figure 9: Maple plots of the radial wave function
jrR(r )j2, for the hydrogen atom for n = 3 and l =
0; 1; 2.
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24. The Probability Density Plots Of The Hydrogen Atom
Maple plots of the probability density j j2 in polar coordinates, for
the normalized wave-functions of the hydrogen atom for n = 3
with Z = 1 are presented:
310 311 320 321
Figure 10: Maple plots of the sections through the prob-
ability density j j2 in polar coordinates, for the normal-
ized wave-functions of the hydrogen atom, illustrating
the relative probability of

25. nding the electron at a given
distance r=a.
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