The First Order Stark Effect In Hydrogen For $n=3$
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The First Order Stark Effect In Hydrogen For n =3 Johar M. Ashfaque University of Liverpool May 11, 2014 Johar M. Ashfaque String Phenomenology
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Transcript of The First Order Stark Effect In Hydrogen For $n=3$
The First Order Stark EffectIn 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 myundergraduate dissertation titled “Time-Independent PerturbationTheory In Quantum Mechanics”, namely the first order Stark effectin hydrogen.
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What is the Stark Effect?
The splitting of the spectral lines of an atom in the presence of anexternal electric field is known as the Stark effect. Table shownbelow, corresponding to the Stark effect in the hydrogen atomprovides a summary in the case of n = 2, 3, and 4:
n l m Matrix Total Elements Off-DiagonalElements By
Symmetry2 0, 1 0, ±1 4×4 16 63 0, 1, 2 0, ±1, ±2 9×9 81 364 0, 1, 2, 3 0, ±1, ±2, ±3 16×16 256 120
Table: Stark effect corresponding to the cases where n = 2, 3, and 4.
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The Case: n = 3
For n = 3, the degeneracy of the energy level of the hydrogenatom is 9. We can see this degeneracy more clearly in the form ofnlm 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 aredealing with a 9× 9 matrix and set out to compute the elementsof 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 36off-diagonal elements.
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The Case: n = 3 Contd.
Definition
The perturbation produced by the electric field ξ = (0, 0, ξ) is
H ′ = 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).
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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⟨nlm
∣∣∣ H ′ ∣∣∣ n′l ′m′⟩ = eξ
∫∫∫r 2dr sin(θ)dθdφψ∗nlmr cos(θ)ψn′ l′m′
= eξ
∫ ∞0
r 3drRnl(r)Rn′ l′(r)∫∫sin(θ) cos(θ)dθdφY m
l∗(θ, φ)Y m′
l′ (θ, φ)
= eξ
∫ ∞0
r 3drRnl(r)Rn′ l′(r)
∫ π
0
sin(θ) cos(θ)dθ∫ 2π
0
dφY ml∗(θ, φ)Y m′
l′ (θ, φ).
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The Case: n = 3 Contd.
Definition
The integral ∫ ∞0
r3drRnl(r)Rn′l ′(r)
is the radial part.
Definition
The integral∫ π
0sin(θ) cos(θ)dθ
∫ 2π
0dφYm
l∗(θ, φ)Ym′
l ′ (θ, φ)
is the angular part.
The integral over all space of any odd function is zero.
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The Case: n = 3 Contd.
The integral, as n = 3 is constant, gives∫ ∞0
r3drRnl(r)Rn′l ′(r) =
∫ ∞0
r3drR3l(r)R3l ′(r) =
{0, if l = l ′,
K , if l 6= l ′
where K is a constant to be determined.
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The Radial Wave Function Plots Of The Hydrogen Atom
Maple plots of the radial wave function |rR(r)|2 for the hydrogenatom 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.
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The Probability Density Plots Of The Hydrogen Atom
Maple plots of the probability density |ψ|2 in polar coordinates, forthe normalized wave-functions of the hydrogen atom for n = 3with Z = 1 are presented:
ψ310 ψ31±1 ψ320 ψ32±1
Figure 10: Maple plots of the sections through the prob-
ability density |ψ|2 in polar coordinates, for the normal-
ized wave-functions of the hydrogen atom, illustrating
the relative probability of finding the electron at a given
distance r/a.
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