Degenerate Perturbation Theory

HTML
DOWNLOAD

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Degenerate Perturbation Theory"

ÊÊΔιομήδης Σπανός
8 χρόνια πριν
Προβολές:

1 R.G. Griffi BioNMR School page 1 Degeerate Perturbatio Theory 1.1 Geeral Whe cosiderig the CROSS EFFECT it is ecessary to deal with degeerate eergy levels ad therefore degeerate perturbatio theory. The basic ideas are outlied below. 1.2 Degeerate Perturbatio Theory Whe two or more states a ad b have idetical eergies the the eergy deomiator ( ) vaishes ad the coefficiet c m C m ( 2) ad Ε Ε Ε m ( ) = m H N ( 2 & E ) Ε o o N = m H N 2 Ε m Ε o o Ε m diverge. The topic of how to deal with these situatios, which are relatively commo, is degeerate perturbatio theory ad is cosidered here Twofold degeeracy This is the simplest case to cosider two fold degeeracy, which yields H ψ = E ψ H ψ b = E ψ b ψ a ψ b = The eergies are idetical, E, ad the wavefuctios are ormalized ad orthogoal. A liear combiatio of ψ a ad ψ b is a eigefuctio of the uperturbed Hamiltoia. ψ = α ψ α + β ψ b = α a + β b with eergy E. I may cases, a small perturbatio will lift the degeeracy as λ goes 1

2 R.G. Griffi BioNMR School page 2 λ might be a electric, E, or magetic field, B, whose stregth tues λ to some ew value. As λ decreases the upper state reduces to oe choice of a liear combiatio of a or b while the lower state evolves to a orthogoal liear combiatio Time idepedet Schroediger Equatio We wat to solve the TISE with H = H + λ H ad E = E O + λe 1 + λ 2 E 2 + i i i H ψ = E ψ ψ =ψ O + λψ 1 + λ 2 ψ 2 + i i i Isertig ad collectig terms ( ) + i i i = E ψ + λ ( E 1 ψ + E ψ 1 ) + i i i H ψ + λ H ψ + H ψ 1 H ψ = E ψ cacels leavig for the λ 1 terms. H ψ 1 + H 1 ψ = E ψ 1 + E 1 ψ Takig the ier product with ψ α ψ a H ψ 1 + ψ a H ψ = E ψ a ψ 1 + E 1 ψ a ψ Because H is Hermitia H ψ a ψ 1 = E ψ a ψ 1 Isertig the liear combiatio of states Or usig other otatio where ψ a H αψ a + β ψ b α ψ a H ψ W ij = ψ i + β ψ = E 1 ψ a αψ a + β ψ b = α E 1 αw aa + βw ab = α E 1 (1) H ψ j ( ij = a, b, i i i) Similarly, the ier product with ψ b yields α + β = β E 1 ψ b H αψ a + βψ b = E 1 ψ b αψ + βψ b

3 R.G. Griffi BioNMR School page 3 α ψ b H ψ a + β ψ b = E 1 ψ b βψ b α + β = β E 1 (2) The W ij 's are i priciple kow quatities, that is they are matrix elemets of H 1 with the uperturbed wavefuctios ψ a ad ψ b. To obtai a useful form. 1) Multiple (2) by W ab ad 2) use (1) to elimiate β W ab the substitute λ + = β E 1 ( ) ( ) = α W ab = W ab βe 1 α W ab E 1 α W aa =αe 1 ( ) β W ab =α E 1 W aa Assumig α ad some algebra yields α W ab E 1 W aa ( )( E 1 ) = ad the quadratic ( E 1 ) 2 E 1 ( W aa + ) + ( W aa ) = ad some more algebra (see the last page of the otes) E 1 ± 2 W + W ± aa bb ( W W aa bb ) W ab which is the fudametal result of degeerate perturbatio theory. The two roots correspod to the two perturbed eergies. For the case where α = β = 1 the αw aa + βw ab = αe 1 ad W ab =

4 R.G. Griffi BioNMR School page 4 E 1 = which is part of the geeral formula above. Ad whe α = 1, β =, =, we fid E + 1 = W aa = ψ a ( ) E 1 ± = 1 2 W ab + ± W aa H ψ a E 1 = = ψ b So by choosig the correct good zero order (uperturbed) states the we ca use odegeerate perturbatio theory. We ca ofte do this usig the followig theorem Theorem: A= Hermitia operator that commutes with H ad H ψ α ad ψ b are eigefuctios of H they are also eigefuctios of A with values Aψ a = µψ a Aψ b = µψ b µ the W αb = ad hece ψ α ad ψ b o are good states to use i perturbatio theory. Proof: ψ a ψ a Α, = Αψ a = ( µ ) ψ a Sice µ the W ab must vaish. [ Α, H ] = [ Α, H ] ψ b ψ a ψ a = H,Α ψ b = ( µ )W ab = Bottom lie if you have a degeerate problem 1) fid a Hermitia operator that commutes with H ad H 2) choose your uperturbed states that are eigefuctios of H ad Α 3) Use 1 st order perturbatio theory If such a operator is ot available the resort to degeerate perturbatio theory Higher Order Degeeracy Rewritig the results above i matrix form twofold degeeracy we obtai W aa W ab α β = E1 α β E ( 1) is the characteristic eigevalues of the W matrix ad the eigevectors ψ ± = 1 ( 2 ψ ±ψ a b ) are the expaded eigevectors.

5 R.G. Griffi BioNMR School page 5 matrix For the case of fold degeeracy we search for the eigevalues of the W ij = ψ i H ψ j Fidig suitable wavefuctios from the uperturbed wavefuctios amouts to costrictig a basis i the degeerate subspace that diagoalizes ω! Cosider a set of orthoormal states ψ j the uperturbed Hamiltoia. Hψ j = E j ψ j We ow costruct the liear combiatio. that are degeerate eigefuctios of ψ i ψ j ψ = α j ψ j It, too, is a eigefuctio of Ĥ, the uperturbed Hamiltoia with the same eigevalues: j=1 = S ij H ψ = α j H ψ j = E α j ψ j = E ψ j=1 We wat to solve the TISE for the perturbed Hamiltoia H = H + λ H We do the usual ad expad Ε ad ψ i a power series E = E + λe 1 + λ 2 E ψ =ψ + λψ 1 + λ 2 ψ j=1 yieldig Isertig ito Hψ = Eψ ad collectig terms i like powers of λ, we obtai ( H + λ H )( ψ + λψ 1 + λ 2 ψ ) = ( E + λe 1 + λ 2 E ) ( ψ + λψ 1 b + λ 2 ψ ) H ψ + λ ( H ψ 1 + H ψ ) +... E ψ + λ ( E ψ 1 + E 1 ψ ) +... The zeroth order terms cacel, to first order we obtai H ψ 1 + H ψ = E ψ 1 + E 1 ψ Ier product with ψ j yields ψ j H ψ 1 + ψ j H ψ = E ψ j ψ However whe ψ j H ψ 1 + H ψ j ψ 1 = E ψ j ψ 1 the first terms cacel leavig

6 R.G. Griffi BioNMR School page 6 ψ j H 1 ψ = E 1 ψ j ψ Now usig ψ = ψ ad explorig the orthoormality of ψ 1 { } or defiig we obtai α =1 ψ j H ψ = E 1 α ψ j ψ W j = ψ j =1 H ψ = E 1 α j W j α = E 1 α =1 This is the geeralizatio from the 2-fold to the fold case. It is the eigevalue equatio for the matrix W (whose j th elemet i the ψ j basis is W j. E 1 is the eigevalue ad the eigevector (i the ψ j First order correctios to the eergy are the eigevalues of W. { } basis) is x j = α j { }

7 R.G. Griffi BioNMR School page 7 α + β = β E 1 *W ab αw ab = β W ab E 1 αw ab ( E 1 ) = αw aa =αe 1 β W αb = ( E 1 W aa )α αw ab α ( E 1 W aa )( E 1 ) = α ( E 1 ) 2 E 1 ( W aa + ) + ( W aa ) = ( ) ± ( W aa + ) 2 4( 1) ( W aa W ba ) 2 4( 1) ( W aa ) E 1 = W aa W + W aa bb ( 4W aa + 4W ab ( ) ± W aa W aa 1 2 ( ) ± W 2 aa 2W aa + W 2 bb 2 W + W aa bb ( ) + 4 W ab ( ) 2 2 W + W aa bb ( ) ± ( W aa ) W 2 ab

Σχετικά έγγραφα

SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6

SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES Readig: QM course packet Ch 5 up to 5. 1 ϕ (x) = E = π m( a) =1,,3,4,5 for xa (x) = πx si L L * = πx L si L.5 ϕ' -.5 z 1 (x) = L si