Perturbation Theory Development

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Lecture 14 Perturbation Theory Development Pseudo
Example (H atom) Atomic Units
F. Grieman
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The problem!!
1e- H1 1e-
H2
Perturbation Theory Thus Far H Ho H
?m ?m(o) ?m(1) Em Em(o) Em(1)
If ?m ?m(o) then Em(1)
lt?m(o) H ?m(o) gt Formal Development of
Perturbation Theory Math Tools 1) ?m
expansion in complete orthonormal set of ?m(o)
like vectors ?m ?i aim?i(o) 2)
Power Series solution introduce ? 0 ? 1 (no
perturbation to full perturbation) H Ho
?H ?m ?m(o) ? ?m(1) ?2 ?m(2)
Em Em(o) ? Em(1) ?2 Em(2)
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Solve Sch. Eq. in general Substitute
Expansions H ?m
Em
?m (Ho ?H) (?m(o) ??m(1) )
(Em(o) ?Em(1) ) (?m(o) ??m(1) ) Multiply
out collect terms with same power of ? ?o
Ho ?m(o) Em(o) ?m(o) zero order Sch. Eq.
already solved!!! ?1 H ?m(o) Ho ?m(1)
Em(1) ?m(o) Em(o) ?m(1) ?2 ?m(1) ???.
Use expansion ?m(1) ?i aim?i(o) find
aim H ?m(o) Ho ?i aim?i(o) Em(1) ?m(o)
Em(o) ?i aim?i(o) Lets Find Em(1) Multiply
on left by ?m(o), integrate then use
orthogonality lt?m(o) H ?m(o)gt ?i aimlt?m(o)
Ho ?i(o)gt Em(1) lt?m(o) ?m(o)gt Em(o) ?i
aimlt ?m(o) ?i(o) gt lt?m(o) H ?m(o)gt
ammlt?m(o) Ho ?m(o)gt Em(1) (1)
Em(o) ammlt ?m(o) ?m(o) gt lt?m(o)
H ?m(o)gt amm Em(o)
Em(1)
Em(o) amm
lt?m(o) H ?m(o)gt Em(1)
?
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Lets Find ?m(1) the first correction to the
wave function H ?m(o) Ho ?i aim?i(o)
Em(1) ?m(o) Em(o) ?i aim?i(o) what we had
before Now instead multiply on left by ?k(o),
integrate then use orthogonality lt?k(o) H
?m(o)gt ?i aimlt?k(o) Ho ?i(o)gt Em(1) lt?k(o)
?m(o)gt Em(o) ?i aimlt ?k(o) ?i(o) gt lt?k(o)
H ?m(o)gt akmlt?k(o) Ho ?k(o)gt Em(1)
(0) Em(o) akmlt ?k(o)
?k(o) gt lt?k(o) H ?m(o)gt akm
Ek(o) 0
Em(o) akm Solve for
lt?k(o) H ?m(o)gt Hkm
? interaction coeffiecient akm
_______________________________
______________________
Em(o) -
Ek(o) Em(o) - Ek(o) ? energy
denominator
' i ? m Him So, ?m(1) ?i aim?i(o)
?i' ________________ ?i(o)
Em(o) - Ei(o)


Him Then, ?m ?m(o) ?m(1) ?m(o)
?i' ________________ ?i(o)

Em(o) - Ei(o)
?
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Him ?m ?m(o)
?i' ________________ ?i(o)
Em(o) - Ei(o)
Him aim
________________
Em(o) - Ei(o)
interaction mixes zero order states
?1(o) ?2(o) ?3(o)
?1(o) H11 H11 H11
matrix rep. ?2(o) H11 H11
H11 ?3(o) H11 H11
H11 . .
. . . .
. . .
. . .
Energy Denominator The closer in
energy, the stronger the interaction
actual
zero order
Real Molecule Harmonic Oscillator
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Atomic Units a natural set of units mass
me length ao h2/(mee2) charge e
angular momentum h linear momentum h/ao
Energy mee4/h2 e2/ao 1 Hartree -2
E1s(H) -27.2 eV e.g. E1s(H) -½H
-13.6 eV H atom Hamiltonian H(SI)
(-h2/2me) ?2(r,?,?) Ze2/r (r(meters))
r(m) aor(a.u.) dr aodr d/dr
dr/drd/dr 1/aod/dr ? d2/dr2 (1/ao2) d2/dr2
chain rule Then ?2(SI) (1/ao2) ?2(a.u.)
Substituting H(SI)
(-h2/2me) (1/ao2) ?2(a.u.) Ze2/aor
(e2/ao) -½ ?2(a.u.) Z/r(a.u.)
Energy in Hartrees
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H atom energy En -(meZ2e4)/(2h2) (1/n2)
-Z2/2n2 Hartrees
He atom Hamiltonian in atomic units H
(-½ ?12 Z/rN1) (-½ ?22 Z/rN2)
1/r12