Andre Schneider P622
|
|
- Jesse Fletcher
- 5 years ago
- Views:
Transcription
1 Andre Schneder P6 Probem Set #0 March, 00 Srednc 7. Suppose that we have a theory wth Negectng the hgher order terms, show that Souton Knowng β(α and γ m (α we can wrte β(α =b α O(α 3 (. γ m (α =c α O(α (. m(µ = Keepng ony the hghest order terms n α n α(µ m(µ. (.3 α(µ dm = ( αc O(α m (.4 dα = ( α b O(α. (.5 dα we obtan Thus, or dα α = b α = b n µ n µ = (b α. (.6 dm = c αm dm m = c αd(α n m = c ( n. (.7 b b α n m = c b n (α. (.8 If the mts of the ntegra are from m(µ and m(µ on the RHS and α(µ and α(µ on the LHS n m(µ n m(µ = c b (n α(µ n α(µ (.9 whch mpes that m(µ = c/b α(µ m(µ. (.0 α(µ
2 Srednc 8. Consder ϕ 4 theory L = ϕ µ ϕ µ ϕ mm ϕ 4 λλ µ ϕ 4, (. n d =4 dmensons. Compute the beta functon to O(λ, the anomaous dmenson of m to O(λ, and the anomaous dmenson of ϕ to O(λ. Souton Rewrtng the Lagrangan as L = µ ϕ 0 µ ϕ 0 m 0 ϕ 0 4 λ 0ϕ 4 0, (. where the new feds and coupngs are reated to the od ones by ϕ 0 = ϕ / ϕ, (.3 m 0 = m / ϕ m, (.4 λ 0 = λ λ ϕ, (.5 and snce the s now ony cance the nfntes they can be wrtten as the foowng sums ϕ = m = λ = n= n= n= a n (λ n, (.6 b n (λ n, (.7 c n (λ n. (.8 From the resuts to Π( and V 4 (,, 3, 4 to O(λ (probems 4.5 and 6. we have that Thus, chosng the s to cance ony the nfntes we obtan A =0, (.9 B = λ (4π ( µ n, (.0 m C = 3λ (4π 3 ( µ n. (. m a (λ =O(λ, (. b (λ = λ (4π O(λ, (.3 c (λ = 3λ (4π O(λ. (.4 Frst we consder the nvarance of λ 0 wth respect to µ. We start defnng G(λ, n( λ ϕ = G n (λ n (.5 n= we have that n λ 0 = G(λ, n λ n µ. (.6
3 Dferentatng wth respect to n µ and requrng that λ 0 s ndependent of t mpes Thus, where the beta functon s gven by 0= d n λ 0 G(λ, dλ = λ dλ λ = ( λg (λ λg (λ dλ λ.... (.7 dλ = λ β(λ (.8 β(λ =λ dg dλ (λ = λ d dλ (c (λ a (λ O(λ 3 = 3λ (4π O(λ3. (.9 In the equaton above G (λ was obtaned expandng n( λ ϕ to frst order n λ and comparng wth n= G n(λ/ n. Now we consder the nvarance of m 0 wth respect to µ. Defnng we have that M(λ, n( / m / ϕ = n= M n (λ n (.0 n m 0 = M(λ, n m. (. Dferentatng wth respect to n µ and requrng that λ 0 s ndependent of t mpes 0= d n m 0 M(λ, dλ = λ dm m ( M = (λ M (λ dλ... dm m. (. Thus, usng resuts (.8 and (.9, we obtan the anomaous dmenson of m γ m (λ ( dm M m = (λ M (λ... (λ β(λ = λ dm (λ O(λ dλ = λ d dλ (b (λ a (λ O(λ = λ (4π O(λ. (.3 In the equaton above M (λ was obtaned expandng n(m / ϕ / to frst order n λ and comparng wth n= M n(λ/ n. To obtan the anomaous dmenson of ϕ we consder the propagator n the MS normazaton scheme, ( = d 4 xe x 0 Tϕ(xϕ(0 0 (.4 3
4 and the bare propagator 0 ( = d 4 xe x 0 Tϕ 0 (xϕ 0 (0 0, (.5 whch shoud be ndependent of µ. The two are reated by Tang the ogartm and dferentatng wth respect to n µ, we obtan 0= d n 0 ( = d n ϕ ( 0 ( = ϕ (. (.6 ( n µ dλ λ dm (. (.7 m The anomaous dmenson γ ϕ (λ of ϕ s the frst RHS n the equaton above, that s, whch s of order O(λ. γ ϕ (λ n ϕ = a (λ a (λ a (λ... (.8 3 Srednc 5. Derve the fermon-oop correcton to the scaar propagator by worng through ( ( ( (η, η, J exp g d 4 x 0 (η, η, J (3. J(x η α (x η α (x and show that t has a reatve mnus sgn reatve to the case of the scaar oop. Fgure : Feynman dagram for the fermon-oop correcton to the scaar propagator. Souton The fermon-oop correcton for a scaar propagator (see fgure ( subjected to the nteracton L Yu = gϕψγψ, (3. where γ can be γ 5 or the dentty I matrx, can be obtaned notng that 0 (η, η, J = exp d 4 xd 4 yη(xs(x yη(y exp d 4 xd 4 yj(x (x yj(y. (3.3 where S(x y = (x y = d 4 p ( /p me p(x y (π 4 p m, (3.4 ɛ d 4 e (x y (π 4 M ɛ (3.5 4
5 are the free fed propagators. Thus, expandng both 0 (η, η, J and exp constrant (0, 0, 0 = we have that (η, η, J = P f =0 P f! V =0 g V! ( ( d 4 x J(x L Yu ( η α (x Pf d 4 yd 4 zη(ys(y zη(z V s=0 J(x, η, α(x ( γ P s! η α (x η α (x wth the normazaton V Ps d 4 vd 4 wj(v (v wj(w (3.6 we obtan the partton functon to a orders. However, snce we are ony nterested on the process descrbed n fgure ( we ony need to tae nto connected dagrams that have two externa scaars and two vertces, that s, V =, P s = and P f =. Thus, to obtan the one fermon oop correcton (FL to the propagator we have to evauate FL (η, η, J = (g!!! = g 3» d 4 x» J(x η γ «d 4 x α(x η α (x J(x η β (x γ» d 4 yd 4 zη(ys(y zη(z d 4 y d 4 z η(y S(y z η(z» d 4 vd 4 wj(v (v wj(w» d 4 v d 4 w J(v (v w J(w d 4 xd 4 x J(x J(x d 4 vd 4 wj(v (v wj(w» η γ α(x η α (x η β (x γ η β (x d 4 yd 4 zη(ys(y zη(z! η β (x d 4 v d 4 w J(v (v w J(w d 4 y d 4 z η(y S(y z η(z. (3.7 Now, snce the functona dervatves wth /η(x and /η(x must act on dfferent ntegras so that we have a cosed fermon oop, the ast equaty becomes FL (η, η, J = g d 4 xd 4 x γs(x xγs(x x 6 J(x J(x» d 4 vd 4 wj(v (v wj(w d 4 v d 4 w J(v (v w J(w. (3.8 The mnus sgn came from the odd number of antcommutaton reatons between the feds η, η and ther functona dervatves. The factor of two came from the two possbe ways the fermon functona dervatves can act on the two ntegras. Note that factor of /6 reduces to / when we consder that the functona dervatves /J(x and /J(x have to act on dfferent ntegras so that we have a cosed fermon oop. Thus the one fermon-oop correcton s Π FL = g d 4 xd 4 x γs(x xγs(x x. (3.9 Foowng the same scheme we have for the oop correcton of a scaar oop " «# SL (η, η, J = (g d 4 3» 4 x d 4 vd 4 wj(vs(v wj(w! J(x 4!» = g 5 d 4 xd 4 x J(x J(x d 4 vd 4 wj(v (v wj(w d 4 v d 4 w J(v (v w J(w» J(x J(x J(x J(x d 4 v d 4 w J(v (v w J(w d 4 v d 4 w J(v (v w J(w = g d 4 xd 4 x (x x (x x 6 J(x J(x» d 4 vd 4 wj(v (v wj(w d 4 v d 4 w J(v (v w J(w. (3.0 From the frst to the second equaty a factor of 4! arses from the number of ways we can rearrange the four v and w ntegras. From the second to the thrd equaty a factor of arses from the two possbe ways the functona dervatves can act on the doube and trpe prme ntegras. As n the 5
6 fermon oop case the factor of /6 reduces to / when we consder that the functona dervatves /J(x and /J(x have to act on dfferent ntegras, that s Π SL = g d 4 xd 4 x (x x (x x. (3. whch s the same as that of the fermon oop correcton Π FL except for a factor of. 4 Srednc 5.3 Consder mang ϕ a scaar rather than a pseudoscaar, so that the Yuawa nteracton s L Yu = gϕψψ. In ths case renormazabty requres us to add a term L ϕ 3 = 6 κκϕ 3, as we as a near term n ϕ to cance the tadpoes. Fnd the one-oop contrbutons to the renormazng factors for ths theory n the MS scheme. Souton Lagrangan The Lagrangan for ths theory s L = L 0 L L 0 = Ψ/ Ψ mψψ µ ϕ µ ϕ M ϕ (4. L = g gϕψψ 4 λλϕ 4 6 κκϕ 3 Y Yϕ L ct (4. L ct = ( Ψ Ψ/ Ψ ( m mψψ ( ϕ µ ϕ µ ϕ ( M M ϕ (4.3 The Scaar Propagator To obtan the one oop contrbutons to ths theory we start wth the correctons to the scaar propagator shown n fgure (. Fgure : Feynman dagrams for the one-oop correctons to the scaar propagator. 6
7 The contrbuton of the frst dagram n fgure ( s gven by Π Ψoop ( = ( (g ( d 4 (π 4 Tr S(/ / S(/ (4.4 where S(/p = /p m p m ɛ. (4.5 Snce the denomnator of the propagator s just a number the trace nsde the ntegra can be evauated notng that Tr( / / m( / m = 4( m The denomnators can be combned usng Feynman s formua ( m m = supressg the ɛs, where q = x and D = x( x m. Changng the ntegraton varabes from to q = x we have that where Π Ψoop ( = 4g 0 dx 4N. (4.6 0 dx (q D (4.7 d 4 q N (π 4 (q D, (4.8 N = q x( x m ( xq = q D ( xq. (4.9 Movng to d =4 spacetme dmensons and mang g g µ / we obtan Π Ψoop ( = 4g µ 0 dx dx d 4 q N (π 4 (q D = 4g µ d 4 q q D 0 (π 4 (q D = g 6 4π dxd 0 D n(d/µ = g 6m 4π dxd D n(d/µ 0 = g 6m 4π fnte (4.0 where µ =4πe γ µ and n the ast ne ony the dvergent terms are expct snce we are nterested n the MS scheme. The contrbuton of the second dagram, from probem 4.5 of the prevous homewor set, s Π ϕ 4 = λm 6π fnte (4. where the fnte terms were not wrten expctey snce we are worng n the MS scheme. The thrd dagram correcton s Π ϕ 3( = (κ ( d 4 ( ( (π 4 (4. where (p = p M ɛ. (4.3 7
8 A smar ntegra was obtaned n probem 6.. Usng the resut from there Π ϕ 3( = The ast dagram gves the contrbuton of the counterterms, that s, κ 6π fnte. (4.4 Π ct ( = ( ϕ ( M M. (4.5 Snce we want the sum of a one-oop contrbutons to the scaar propagator to be fnte we must have whch mpes that The Fermon Propagator g 6m 4π λm 6π κ 6π ( ϕ ( M M = 0 (4.6 g ϕ = 4π (4.7 M = ( λ 6π g 6m 4π M κ 6π M. (4.8 Now we move on to the one-oop correctons to the fermon propagator. The Feynman dagrams for ths case can be seen on fgure (3. Fgure 3: Feynman dagrams for the one-oop correctons to the scaar propagator. The contrbuton of the frst dagram n fgure (3 w be the same as the one for the theory wth pseudoscaar ϕ: ( Σ oop (/p = (g d 4 S( (π 4 /p / ( Σ oop (/p = g 6π ( /p m fnte. (4.9 The dveregent terms cance f addng Σ oop (/p wth eads to a fnte resut. Thus, Σ ct (/p = ( Ψ /p ( m m (4.0 g Ψ = 6π (4. m = g 8π. (4. 8
9 p p p p p p Fgure 4: Feynman dagrams for the one-oop correctons to the fermon scaar vertex. The Fermon-Scaar Vertex The Feynman dagrams for the one-oop correcton to a fermon-scaar vertex can be seen on fgure (4. The contrbuton of the frst dagram s ( V Y (p,p = (g 3 = g 3 = g 3 3 d 4 S( (π 4 /p / S(/p / ( d 4 (/p / m( /p / m (π 4 ((p m ((p m ( M df 3 d 4 q (π 4 where, wth the transformaton q = x p x p, N (q D 3 (4.3 N =(/q x /p ( x /p m( /q x /p ( x /p m (4.4 D = x ( x p x ( x p x x p p (x x m x 3 M. (4.5 Rewrtng N = q Ñ (near n q, where Ñ =( x /p ( x /p m(x /p ( x /p m. (4.6 Snce the terms that are near n q ntegrate to zero and ony the frst term s dvergent, the contrbuton of V Y,oop (p,p to the propagator n the MS scheme s (see secton 5 n Srednc V Y (p,p= The contrbuton of the second dagram s nu snce V Y (p,p= ( 3 6 κ(g = 6 κg s fnte (the argest term s of order Thus, we obtan The Three-Scaar Vertex g3 8π fnte. (4.7 d 4 (π 4 S(/ (( (( d 4 (π 4 S(/ (( (( (4.8 d 4 (π 4 5 and ths ntegra converges. g = g 8π. (4.9 The Feynman dagrams for the one-oop correcton to a three-scaar vertex can be seen on fgure (5. 9
10 3 3 3 Fgure 5: Feynman dagrams for the one-oop correctons to the three-scaar vertex. Not a possbe permutatons were drawn. The contrbuton of the frst dagram to the one-oop correcton of the three scaar vertex s nu snce ( 3 V 3 (,, 3 = (κ 3 d 4 (π ( 4 (( (( (4.30 s fnte. The second dagram contrbuton s ( 3 V 3 (,, 3 = ( (g 3 = g 3 df 3 d 4 q (π 4 where, from secton 6 n Srednc, d 4 (π S(/ S(/ 4 / S(/ / N d 4 q D 3 q N (π 4 q D 3 d 4 (π S(/ S(/ 4 / 3 S(/ / (4.3 q = x x, (4.3 q = x x 3 3, (4.33 N =3mTr/q/q = mq, (4.34 N =3mTr/q/q = mq, (4.35 D = x 3 x x 3x x x 3 m, (4.36 D = x x x x 3 3 x x 3 m, (4.37 f we eep ony the dvergent terms. Snce ony terms proportona to q 3 and q can dverge and Tr/a/b/c = 0 the ony term that s eft s proportona to 3mTr/q/q = mq. Thus, The thrd dagram contrbuton s V 3 (,, 3 = ( (κ( λ = 3λκ 6π usng resut (3.9 from Srednc. Addng a contrbutons we obtan V 3 (,, 3 = mg3 8π = 3mg3 π. (4.38 d 4 (π ( 4 (( (( (( (4.39 κ = 3λ 6π 3mg3 κπ (4.40 0
11 Fgure 6: Feynman dagrams for the one-oop correctons to the four-scaar vertex. Not a possbe permutatons were drawn. The Four-Scaar Vertex The Feynman dagrams for the one-oop correcton to the four-scaar vertex can be seen on fgure (6. The contrbuton of the frst dagram s V 4 (,, 3, 4 = g 4 d 4 (π 4 Tr S(/ S(/ / S(/ / S(/ / / 3 ( 5 permutatons of, 3 and 4 = 3g4 π (4.4 usng the resut from secton 5 n Srednc. The resut s the same as n the case of a pseudoscaar ϕ. Ths happens because the ony dvergent part n the ntegra above s the one proportona to ( n the numerator and the four γ 5 s from the pseudoscaar nteracton w not change ths factor. The contrbutons of the second and thrd dagrams are V 4 (,, 3, 4 =κ 4 V 4 (,, 3, 4 = κ λ d 4 ( (π 4 (( (( (( 3 ( permutatons of, and 3, (4.4 d 4 ( (π 4 (( (( 3 4 ( 5 permutatons of, 3 and 4. (4.43 Both are fnte and, therefore, do not contrbute to the vertex correcton n the MS scheme. The contrbuton of the fourth dagram s V 4 (,, 3, 4 = λ = 3λ 6π usng the resut from secton 3 n Srednc. Thus, we have 5 Srednc 5. d 4 ( (π 4 (( ( permutatons of, 3 and 4 (4.44 (4.45 λ = 3g4 λπ 3λ 6π. (4.46 Compute the one-oop contrbutons to the anomaous dmenson of m, M, Ψ and ϕ. Souton Lagrangan The Lagrangan for the Yuawa theory L = Ψ Ψ/ Ψ m mψψ ϕ µ ϕ µ ϕ M M ϕ g g µ / ϕψγ 5 Ψ 4 λλ µ ϕ 4 (5.
12 can be rewrtten as where L = Ψ 0 / Ψ 0 mψ 0 Ψ 0 µ ϕ 0 µ ϕ 0 M 0 ϕ 0 g 0ϕ 0 Ψ 0 γ 5 Ψ 0 4 λ 0ϕ 4 0 (5. Choosng the s to cance the nfntes we have ( a n (λ Ψ = n = g 6π fnte n= ( b n (λ g m = n, = 8π fnte ϕ = M = g = λ = n= n= n= n= n= Ψ 0 = / Ψ Ψ, (5.3 m 0 = m Ψ m, (5.4 ϕ 0 = ϕ / ϕ, (5.5 M 0 = / M / ϕ M, (5.6 g 0 = g ϕ / Ψ µ/ g, (5.7 λ 0 = λ ϕ λ. (5.8 c n (λ n g, = 4π d n (λ n, = e n (λ n = g 8π f n (λ n = whch mpes that, to frst order, Ψ 0 = g 3π m 0 = g 6π ϕ 0 = g 8π ( λ M 0 = 3π g 8π g 0 = 5g 6π λ 0 = ( fnte ( λ 6π g m π M ( fnte ( 3λ 6π 3g π λ ( fnte ( fnte (5.9 (5.0 (5. (5. (5.3 (5.4 Ψ, (5.5 m, (5.6 ϕ, (5.7 ( m M M, (5.8 µ / g, (5.9 ( 3λ 6π g π 3g λπ µ λ. (5.0 To obtan the anomaous dmensons we mae use of the foowng resuts (secton 5 Srednc: dg = g β g(g, λ (5. dλ = λ β λ(g, λ (5.
13 Anomaous dmenson of m The anomaous dmenson of m s found from where s the anomaous dmenson of m. Anomaous dmenson of M 0= d n m 0 = d = g 8π The anomaous dmenson of M s found from 0= d n M 0 = d ( = 3π ( λ 3π g 8π dλ g 4π g 6π dg n m γ m (5.3 γ m dm m = g 6π (5.4 ( m M n M ( m M dg g π m M (γ M γ m γ M (5.5 where, eepng ony the O( 0 terms, we obtan γ M M dm = λ 3π g 8π ( m M (5.6 s the anomaous dmenson of M. Anomaous dmenson of Ψ The anomaous dmenson of the fed Ψ s found from s the anomaous dmenson of Ψ. γ Ψ = d n Ψ = d = g 6π g 3π dg = g 3π (5.7 3
14 Anomaous dmenson of ϕ The anomaous dmenson of ϕ s found from 0= d n ϕ = d g 8π = g dg 4π = g 8π (5.8 s the anomaous dmenson of ϕ. 4
Srednicki Chapter 51
Srednici Chapter 51 QFT Probems & Soutions A. George September 7, 13 Srednici 51.1. Derive the fermion-oop correction to the scaar proagator by woring through equation 5., and show that it has an extra
More informationQuantum Field Theory Homework 5
Quantum Feld Theory Homework 5 Erc Cotner February 19, 15 1) Renormalzaton n φ 4 Theory We take the φ 4 theory n D = 4 spacetme: L = 1 µφ µ φ 1 m φ λ 4! φ4 We wsh to fnd all the dvergent (connected, 1PI
More informationSrednicki Chapter 14
Srednc Chapter 4 QFT Problems & Solutons A. George September, Srednc 4.. Derve a generalzaton of Feynman s formula, = Γ ( α ) A αa... A αn n Γ(α df xα n ) (n )! ( x A ) Hnt: start wth: Γ(α) A α = dt t
More informationProblem 10.1: One-loop structure of QED
Problem 10.1: One-loo structure of QED In Secton 10.1 we argued form general rncles that the hoton one-ont and three-ont functons vansh, whle the four-ont functon s fnte. (a Verfy drectly that the one-loo
More informationNote 2. Ling fong Li. 1 Klein Gordon Equation Probablity interpretation Solutions to Klein-Gordon Equation... 2
Note 2 Lng fong L Contents Ken Gordon Equaton. Probabty nterpretaton......................................2 Soutons to Ken-Gordon Equaton............................... 2 2 Drac Equaton 3 2. Probabty nterpretaton.....................................
More informationDifferentiating Gaussian Processes
Dfferentatng Gaussan Processes Andrew McHutchon Aprl 17, 013 1 Frst Order Dervatve of the Posteror Mean The posteror mean of a GP s gven by, f = x, X KX, X 1 y x, X α 1 Only the x, X term depends on the
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationSupplementary material: Margin based PU Learning. Matrix Concentration Inequalities
Supplementary materal: Margn based PU Learnng We gve the complete proofs of Theorem and n Secton We frst ntroduce the well-known concentraton nequalty, so the covarance estmator can be bounded Then we
More informationModelli Clamfim Equazione del Calore Lezione ottobre 2014
CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g
More informationHomework & Solution. Contributors. Prof. Lee, Hyun Min. Particle Physics Winter School. Park, Ye
Homework & Soluton Prof. Lee, Hyun Mn Contrbutors Park, Ye J(yej.park@yonse.ac.kr) Lee, Sung Mook(smlngsm0919@gmal.com) Cheong, Dhong Yeon(dhongyeoncheong@gmal.com) Ban, Ka Young(ban94gy@yonse.ac.kr) Ro,
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationCausal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
More informationHW #6, due Oct Toy Dirac Model, Wick s theorem, LSZ reduction formula. Consider the following quantum mechanics Lagrangian,
HW #6, due Oct 5. Toy Drac Model, Wck s theorem, LSZ reducton formula. Consder the followng quantum mechancs Lagrangan, L ψ(σ 3 t m)ψ, () where σ 3 s a Paul matrx, and ψ s defned by ψ ψ σ 3. ψ s a twocomponent
More information19 Quantum electrodynamics
Modern Quantum Feld Theory 77 9 Quantum electrodynamcs 9. Gaugng the theory Consderng the Drac Lagrangan L D = @/ m we observed the presence of a U( symmetry! e, assocated wth the Noether current j µ =
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationProf. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model
EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several
More informationBernoulli Numbers and Polynomials
Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationLECTURE 21 Mohr s Method for Calculation of General Displacements. 1 The Reciprocal Theorem
V. DEMENKO MECHANICS OF MATERIALS 05 LECTURE Mohr s Method for Cacuaton of Genera Dspacements The Recproca Theorem The recproca theorem s one of the genera theorems of strength of materas. It foows drect
More information2 Finite difference basics
Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one- The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T
More informationAPPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14
APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce
More information1. relation between exp. function and IUF
Dualty Dualty n consumer theory II. relaton between exp. functon and IUF - straghtforward: have m( p, u mn'd value of expendture requred to attan a gven level of utlty, gven a prce vector; u ( p, M max'd
More informationQuantum Field Theory III
Quantum Feld Theory III Prof. Erck Wenberg February 16, 011 1 Lecture 9 Last tme we showed that f we just look at weak nteractons and currents, strong nteracton has very good SU() SU() chral symmetry,
More informationQuantum Runge-Lenz Vector and the Hydrogen Atom, the hidden SO(4) symmetry
Quantum Runge-Lenz ector and the Hydrogen Atom, the hdden SO(4) symmetry Pasca Szrftgser and Edgardo S. Cheb-Terrab () Laboratore PhLAM, UMR CNRS 85, Unversté Le, F-59655, France () Mapesoft Let's consder
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationResearch on Complex Networks Control Based on Fuzzy Integral Sliding Theory
Advanced Scence and Technoogy Letters Vo.83 (ISA 205), pp.60-65 http://dx.do.org/0.4257/ast.205.83.2 Research on Compex etworks Contro Based on Fuzzy Integra Sdng Theory Dongsheng Yang, Bngqng L, 2, He
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More information8.323 Relativistic Quantum Field Theory I
MI OpenCourseWare http://ocw.mt.edu 8.323 Relatvstc Quantum Feld heory I Sprng 2008 For nformaton about ctng these materals or our erms of Use, vst: http://ocw.mt.edu/terms. MASSACHUSES INSIUE OF ECHNOLOGY
More informationψ ij has the eigenvalue
Moller Plesset Perturbaton Theory In Moller-Plesset (MP) perturbaton theory one taes the unperturbed Hamltonan for an atom or molecule as the sum of the one partcle Foc operators H F() where the egenfunctons
More information22.51 Quantum Theory of Radiation Interactions
.51 Quantum Theory of Radaton Interactons Fna Exam - Soutons Tuesday December 15, 009 Probem 1 Harmonc oscator 0 ponts Consder an harmonc oscator descrbed by the Hamtonan H = ω(nˆ + ). Cacuate the evouton
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationClassical Field Theory
Classcal Feld Theory Before we embark on quantzng an nteractng theory, we wll take a dverson nto classcal feld theory and classcal perturbaton theory and see how far we can get. The reader s expected to
More informationMARKOV CHAIN AND HIDDEN MARKOV MODEL
MARKOV CHAIN AND HIDDEN MARKOV MODEL JIAN ZHANG JIANZHAN@STAT.PURDUE.EDU Markov chan and hdden Markov mode are probaby the smpest modes whch can be used to mode sequenta data,.e. data sampes whch are not
More informationV.C The Niemeijer van Leeuwen Cumulant Approximation
V.C The Nemejer van Leeuwen Cumulant Approxmaton Unfortunately, the decmaton procedure cannot be performed exactly n hgher dmensons. For example, the square lattce can be dvded nto two sublattces. For
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( )
More informationGroup Analysis of Ordinary Differential Equations of the Order n>2
Symmetry n Nonlnear Mathematcal Physcs 997, V., 64 7. Group Analyss of Ordnary Dfferental Equatons of the Order n> L.M. BERKOVICH and S.Y. POPOV Samara State Unversty, 4430, Samara, Russa E-mal: berk@nfo.ssu.samara.ru
More informationStatistical Mechanics and Combinatorics : Lecture III
Statstcal Mechancs and Combnatorcs : Lecture III Dmer Model Dmer defntons Defnton A dmer coverng (perfect matchng) of a fnte graph s a set of edges whch covers every vertex exactly once, e every vertex
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More information1 Renormalization of Yukawa theory
Quantum eld Theory-II Problem Set n. 5 - Solutons UZH and ETH, S-2016 Prof. G. Isdor Assstants: A. Greljo, D. Marzocca, J. Shapro Due: 15-04-2016 http://www.physk.uzh.ch/lectures/qft/ndex2.html 1 Renormalzaton
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More informationLecture 10: Euler s Equations for Multivariable
Lecture 0: Euler s Equatons for Multvarable Problems Let s say we re tryng to mnmze an ntegral of the form: {,,,,,, ; } J f y y y y y y d We can start by wrtng each of the y s as we dd before: y (, ) (
More informationFeynman parameter integrals
Feynman parameter ntegrals We often deal wth products of many propagator factors n loop ntegrals. The trck s to combne many propagators nto a sngle fracton so that the four-momentum ntegraton can be done
More informationMAE140 - Linear Circuits - Fall 13 Midterm, October 31
Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationπ e ax2 dx = x 2 e ax2 dx or x 3 e ax2 dx = 1 x 4 e ax2 dx = 3 π 8a 5/2 (a) We are considering the Maxwell velocity distribution function: 2πτ/m
Homework Solutons Problem In solvng ths problem, we wll need to calculate some moments of the Gaussan dstrbuton. The brute-force method s to ntegrate by parts but there s a nce trck. The followng ntegrals
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationSolution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another well-known teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More information3. Stress-strain relationships of a composite layer
OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman.. tress-stran reatonshps of a composte ayer Introducton
More informationAssociative Memories
Assocatve Memores We consder now modes for unsupervsed earnng probems, caed auto-assocaton probems. Assocaton s the task of mappng patterns to patterns. In an assocatve memory the stmuus of an ncompete
More informationTextbook Problem 4.2: The theory in question has two scalar fields Φ(x) and φ(x) and the Lagrangian. 2 Φ ( µφ) 2 m2
PHY 396 K. Solutons for problem set #11. Textbook Problem 4.2: The theory n queston has two scalar felds Φx) and φx) and the Lagrangan L 1 2 µφ) 2 M2 2 Φ2 + 1 2 µφ) 2 m2 2 φ2 µφφ 2, S.1) where the frst
More informationThe Expectation-Maximization Algorithm
The Expectaton-Maxmaton Algorthm Charles Elan elan@cs.ucsd.edu November 16, 2007 Ths chapter explans the EM algorthm at multple levels of generalty. Secton 1 gves the standard hgh-level verson of the algorthm.
More informationSolutions Homework 4 March 5, 2018
1 Solutons Homework 4 March 5, 018 Soluton to Exercse 5.1.8: Let a IR be a translaton and c > 0 be a re-scalng. ˆb1 (cx + a) cx n + a (cx 1 + a) c x n x 1 cˆb 1 (x), whch shows ˆb 1 s locaton nvarant and
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationMulti-dimensional Central Limit Argument
Mult-dmensonal Central Lmt Argument Outlne t as Consder d random proceses t, t,. Defne the sum process t t t t () t (); t () t are d to (), t () t 0 () t tme () t () t t t As, ( t) becomes a Gaussan random
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More information( ) r! t. Equation (1.1) is the result of the following two definitions. First, the bracket is by definition a scalar product.
Chapter. Quantum Mechancs Notes: Most of the matera presented n ths chapter s taken from Cohen-Tannoudj, Du, and Laoë, Chap. 3, and from Bunker and Jensen 5), Chap... The Postuates of Quantum Mechancs..
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy
More informationHidden Markov Models & The Multivariate Gaussian (10/26/04)
CS281A/Stat241A: Statstcal Learnng Theory Hdden Markov Models & The Multvarate Gaussan (10/26/04) Lecturer: Mchael I. Jordan Scrbes: Jonathan W. Hu 1 Hdden Markov Models As a bref revew, hdden Markov models
More informationCalculus of Variations Basics
Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y
More informationBoundary Value Problems. Lecture Objectives. Ch. 27
Boundar Vaue Probes Ch. 7 Lecture Obectves o understand the dfference between an nta vaue and boundar vaue ODE o be abe to understand when and how to app the shootng ethod and FD ethod. o understand what
More informationGraph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationarxiv: v1 [math.ho] 18 May 2008
Recurrence Formulas for Fbonacc Sums Adlson J. V. Brandão, João L. Martns 2 arxv:0805.2707v [math.ho] 8 May 2008 Abstract. In ths artcle we present a new recurrence formula for a fnte sum nvolvng the Fbonacc
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,
More informationLecture 14: Forces and Stresses
The Nuts and Bolts of Frst-Prncples Smulaton Lecture 14: Forces and Stresses Durham, 6th-13th December 2001 CASTEP Developers Group wth support from the ESF ψ k Network Overvew of Lecture Why bother? Theoretcal
More informationMAE140 - Linear Circuits - Fall 10 Midterm, October 28
M140 - Lnear rcuts - Fall 10 Mdterm, October 28 nstructons () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationThe Dirac Equation for a One-electron atom. In this section we will derive the Dirac equation for a one-electron atom.
The Drac Equaton for a One-electron atom In ths secton we wll derve the Drac equaton for a one-electron atom. Accordng to Ensten the energy of a artcle wth rest mass m movng wth a velocty V s gven by E
More informationk p theory for bulk semiconductors
p theory for bu seconductors The attce perodc ndependent partce wave equaton s gven by p + V r + V p + δ H rψ ( r ) = εψ ( r ) (A) 4c In Eq. (A) V ( r ) s the effectve attce perodc potenta caused by the
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationLowest-Order e + e l + l Processes in Quantum Electrodynamics. Sanha Cheong
Lowest-Order e + e + Processes n Quantum Eectrodynamcs Sanha Cheong Introducton In ths short paper, we w demonstrate some o the smpest cacuatons n quantum eectrodynamcs (QED), eadng to the owest-order
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationLecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES
COMPUTATIONAL FLUID DYNAMICS: FDM: Appromaton of Second Order Dervatves Lecture APPROXIMATION OF SECOMD ORDER DERIVATIVES. APPROXIMATION OF SECOND ORDER DERIVATIVES Second order dervatves appear n dffusve
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More information[WAVES] 1. Waves and wave forces. Definition of waves
1. Waves and forces Defnton of s In the smuatons on ong-crested s are consdered. The drecton of these s (μ) s defned as sketched beow n the goba co-ordnate sstem: North West East South The eevaton can
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationThis model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:
1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors
More information