The First Order Stark Effect In Hydrogen For n = 3 Johar M. Ashfaque University of Liverpool
May 11, 2014
Johar M. Ashfaque
String Phenomenology
Introduction
I will briefly mention the main result that was covered in my undergraduate dissertation titled “Time-Independent Perturbation Theory In Quantum Mechanics�, namely the first order Stark effect in hydrogen.
Johar M. Ashfaque
String Phenomenology
What is the Stark Effect? The splitting of the spectral lines of an atom in the presence of an external electric field is known as the Stark effect. Table shown below, corresponding to the Stark effect in the hydrogen atom provides a summary in the case of n = 2, 3, and 4: n
l
m
Matrix
Total Elements
2 3 4
0, 1 0, 1, 2 0, 1, 2, 3
0, ±1 0, ±1, ±2 0, ±1, ±2, ±3
4×4 9×9 16×16
16 81 256
Off-Diagonal Elements By Symmetry 6 36 120
Table: Stark effect corresponding to the cases where n = 2, 3, and 4.
Johar M. Ashfaque
String Phenomenology
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 off-diagonal elements. In our case, we need to compute only 36 off-diagonal elements. Johar M. Ashfaque
String Phenomenology
The Case: n = 3 Contd.
Definition The perturbation produced by the electric field ξ = (0, 0, ξ) is Ĥ 0 = eξz = eξr cos(θ) where e is the electron charge. It is clear, that the eξz has odd parity since eξr cos(π − θ) = −eξr cos(θ).
Johar M. Ashfaque
String Phenomenology
The Case: n = 3 Contd. Denote the 9 states as ψ300 , ψ310 , ψ320 , ψ311 , ψ321 , ψ31−1 , ψ32−1 , ψ322 , ψ32−2 . Definition A general matrix element is defined as D
E
nlm Ĥ 0 n0 l 0 m0
ZZZ =
eξ
=
eξ
∗ r 2 dr sin(θ)dθdφψnlm r cos(θ)ψn0 l 0 m0
∞
Z
r 3 drRnl (r )Rn0 l 0 (r ) ZZ 0 ∗ 0 (θ, φ) sin(θ) cos(θ)dθdφYlm (θ, φ)Ylm Z ∞ Z π eξ r 3 drRnl (r )Rn0 l 0 (r ) sin(θ) cos(θ)dθ 0
=
0
0 2π
Z
dφYl
m∗
m0
(θ, φ)Yl 0 (θ, φ).
0
Johar M. Ashfaque
String Phenomenology
The Case: n = 3 Contd. Definition The integral Z
∞
r 3 drRnl (r )Rn0 l 0 (r )
0
is the radial part. Definition The integral Z π
Z sin(θ) cos(θ)dθ
0
0
2π
0
dφYlm ∗ (θ, φ)Ylm 0 (θ, φ)
is the angular part. The integral over all space of any odd function is zero. Johar M. Ashfaque
String Phenomenology
The Case: n = 3 Contd.
The integral, as n = 3 is constant, gives Z 0
∞
r 3 drRnl (r )Rn0 l 0 (r ) =
Z 0
∞
( 0, if l = l 0 , r 3 drR3l (r )R3l 0 (r ) = K , if l 6= l 0
where K is a constant to be determined.
Johar M. Ashfaque
String Phenomenology
The Radial Wave Function Plots Of The Hydrogen Atom Maple plots of the radial wave function |rR(r )|2 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 |rR(r )|2 , for the hydrogen atom for n = 3 and l = 0, 1, 2. Johar M. Ashfaque
String Phenomenology
The Probability Density Plots Of The Hydrogen Atom Maple plots of the probability density |ψ|2 in polar coordinates, for the normalized wave-functions of the hydrogen atom for n = 3 with Z = 1 are presented:
ψ310
ψ31±1
ψ320
ψ32±1
Figure 10: Maple plots of the sections through the probability density |ψ|2 in polar coordinates, for the normalized wave-functions of the hydrogen atom, illustrating the relative probability of finding the electron at a given distance r /a. Johar M. Ashfaque
String Phenomenology