“atom” very light Like the rotating molecule, the e ect

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“atom” very light Like the rotating molecule, the e ect
The hydrogen atom: same Hamiltonian as a diatomic molecule, with one “atom” very light
me e-
mP
m m
m = e P ≈ me
me+mP
r
+
Like the rotating molecule, the effective potential has the original potential (Coulomb in this case)
in it, and also the ‘centrifugal potential’ when angular momentum of the electron increases and\
tugs at the ‘bond’ to the proton.
veff(r)
rotation part
coulomb part
The lowest energy solution looks similar to the harmonic oscillator gaussian and the particle in a box
sine wave, but has a ‘cusp’ at r=0 because the potential goes to -∞ there:
x
R10(r)
EnℓM=En=E1s could depend on all three
quantum numbers, but
depends only on n !
The energy Enℓm depends only on n. This is an accident of symmetry for hydrogen atom, for other
atoms the energy depends on n and on ℓ (just like it depended on ℓ for the rotating molecule)
E
n=∞
n=3
n=2
n=2
IP = Ionization Potential
ℓ = 0(s)
ℓ = 1(p)
ℓ = 2(d)
R20•Y00
R21•Y1-1
R21•Y11 R21•Y10
R10(r) • Y00(θ,ℓ) = Ψ1s
m
n
ℓ
ℓ e
or
Ψ(r,θ,ϕ) =
R32 Y20
=dz2
z
x
z
R21 • (Y1-1 - Y11)
iR21 • (Y1-1 + Y11)
R21
y
y
R20
x
z
y
kre– kr/2
r
(2 - kr) e– kr/2
r
R10
x
1/e
e– kr
a0=k-1
note: |ψ|2 = probability per unit
volume is high, but
P = r2drsinθdθdℓ|ψ|2 0 as r 0
r

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