Hydrogenlike atoms

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Hydrogen-like atoms Angular Wavefunctions Solving
Radial Schroedinger Equation One Electron Wave
Functions
F. Grieman
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Last time H atom Wave Function ?(r,?,f)
R(r) T(?) F(f) R(r)Y(?,f) Separate Sch. Eq.
angular part just what we had before L2
l(l1)h2 l 0, 1, 2, Lz mh ml 0, 1,
2, l Yl,ml (?,f) Tl, ml(?) Fml (f) Fml
(f) (2p)-1/2 eimlf Using L we found Tl,l
N sinl ? Yl,l Nl,l sinl ? (2p)-1/2 eilf Using
L- we can find the rest For example Y1,1
(3/8p)1/2 sin ? (2p)-1/2 eif
Y1,0 (3/4p)1/2 cos ?
Y1,-1 (3/8p)1/2 sin ? (2p)-1/2
e-if Simple math functions - Lets look at them

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Separation of variables via product function
?(r,?,f) R(r) Y(?,f)
Last time
Other side of equation
Radial Kinetic Angular Kinetic
Potential Energy
Energy Energy Physical
Interpretation
Veff effective
potential
energy
Moment of Inertia I mer2
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For l ? 0 Veff gives bound state. ?? l 0
(s) not bound ?? But l 0 is bound!!!! Consider
P. in a B. Kinetic Energy En n2h2
8ma2 (E T cannot
0) As a?, E? ? T? In 1 e- atom E T V as
r?, V?, but!!! T? Lowest energy is balance
of energies
Veff from e2/r l(l1)?2/r2
Potential Well in Veff traps electron
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Solving Radial Equation via Raising and Lowering
Operators (omitted in lecture website
handout)
En ? f(l,ml) ? gn n2 degeneracy
for n 1, 2, 3,
l 0, 1, 2, n-1 ml 0, 1, 2,
l Example n 3 gn 9 l
0 , 1 , 2
ml 0 -1, 0, 1
-2, -1, 0, 1, 2 ? 9 states

with same E3
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Wave functions Raising Lowering Operator
Derivation Omitted
Rn, l Associated Laguerre Polynomials (we
will look at them later) Complete wave functions
called Orbitals ?(r, ?, ? ) Rn,l (r)Yl,ml (?,
?) (Associated Laguerre Polynomial
Spherical harmonic) n principal quantum
number l angular momentum quantum
number 0, 1, 2, 3, 4
s, p, d, f, g (from spectroscopy -
sharp, principal, diffuse, fundamental ml
magnetic quantum number (z-axis defined by
magnetic field) Wave function symbol
?n,l,ml ?n, l label, ml or direction Examples
?1s, ?2s, ?2po, ?2pz, ?2px or simply 1s,
2s, 2po, 2pz, 2px
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Normalization 1 lt??gt ltRRgt lt??gt
lt??gt Independently
normalized
1 1 1 d?
r2dr sin?d?
d? limits 0 ?8
0?? 0?2? Probabilities P(e) very
different than Bohr We will look at R, RR,
r2RR, Y, And then Pd? RRr2dr YYsin? d?d?
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(Lec 13 start)
Fall off as -Zr
Fall off more slowly As Zr/2

Fall off even more slowly
Nodes l angular nodes n-l-1 radial nodes n-1
total nodes
radial node
Compare position and amplitudes
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Radial Probability Density
Radial nodes
Radial node
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Radial Distribution Function
?02? ?0?RRYYr2sin?drd?d? r2RRdr
d? r2dr as r ? 0, d? ? 0 rmost probable
rmp maximum dr2R2/dr 0 ltrgt f(l) Note
2s local maximum inside 2p
maximum But!!! lt1/rgt ? f(l,m) Z/n2ao ? ltVgt ?
f(l,m) For 1 e- atom E ? f(l,m)
Can determine lt r gt, lt r2 gt, lt 1/r gt lt r gt
n2ao/Z1 ½1- l(l1)/n2 f(l) lt r gt
decreases as l increases decreases as Z
increases Orbital H1s H2s H2p He1s lt r
gt(ao) 3/2 6 5 3/4