QMGreensite_merged
16.1. THEKRONIG-PENNYMODEL 255 providingthatweimposetheperiodicityrequirementthatx+Ldenotes thesame pointasx.The“periodicdelta-f ...
256 CHAPTER16. LIVEWIRESANDDEADSTARS Butalso ψE(x) = TaT−aψE = λEλ′EψE = ψE(x) (16.14) Thismeansthat λ′E =(λE)−^1. Insert thatfa ...
16.1. THEKRONIG-PENNYMODEL 257 ButaccordingtoBloch’stheorem,intheregion−a<x< 0 (regionII), ψII(x) = e−iKaψI(x+a) = e−iKa[A ...
258 CHAPTER16. LIVEWIRESANDDEADSTARS level,butatthispointwemustinvoketheExclusionPrinciple:Therecanbenomore thanoneelectroninany ...
16.2. THEFREEELECTRONGAS 259 While,attheboundaries,weimposethe”box”conditions ψ(0,y,z) = ψ(L,y,z)= 0 ψ(x, 0 ,z) = ψ(x,L,z)= 0 ψ( ...
260 CHAPTER16. LIVEWIRESANDDEADSTARS Accordingto(16.31), R= ( 2 mL^2 ̄h^2 π^2 EF ) 1 / 2 (16.34) sothenumberofelectrons,intermso ...
16.2. THEFREEELECTRONGAS 261 Thepressurewithwhichanyobjectresistscompressionisgivenby p=− dET dV (16.42) whereET istheenergyofth ...
262 CHAPTER16. LIVEWIRESANDDEADSTARS Thentheinwardgravitationalpressureis pG = − dEG dV = − 1 5 ( 4 π 3 ) 1 / 3 GM^2 V−^4 /^3 (1 ...
16.2. THEFREEELECTRONGAS 263 theequationalsopredictednegativeenergystates E=− √ p^2 c^2 +m^2 c^4 (16.51) Thishadtheuncomfortable ...
264 CHAPTER16. LIVEWIRESANDDEADSTARS ...
Chapter 17 Time-Independent Perturbation Theory ConsideraHamiltonianwhichhasthisform: H=H 0 +alittlebitextrapotential (17.1) whe ...
266 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY Thistime H 0 =− ̄h^2 2 m ∇^2 − e^2 r (17.9) istheusualHydrogenatomHamiltonian, ...
267 or 0 = (H 0 φ(0)n −En(0)φ(0)n )+λ(H 0 φ(1)n +Vφ(0)n −E(0)n φ(1)n −En(1)φ^0 n)+... +λN(H 0 φ(nN)+Vφ(nN−1)− ∑N j=0 En(j)φ(nN−j ...
268 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY andthisconditioneliminatestheambiguitymentionedabove. Ofcourse,afterhaving com ...
269 andmakinguseoftheconstraint(17.23),wehave 0 =−〈φ(0)n |V|φ(nN−1)〉+En(N) (17.30) Puttingeverythingtogether,wehavetheiterativee ...
270 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY andthesecond-ordercorrectiontotheenergy, E^2 n = 〈φ(0)n |V|φ^1 n〉 = 〈φ(0)n |V ...
17.2. EXAMPLE-THEANHARMONICOSCILLATOR 271 17.2 Example - The Anharmonic Oscillator ConsidertheHamiltonian H = − ̄h^2 2 m d dx^2 ...
272 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY Then,tofirstorderinλ En=h ̄ω(n+ 1 2 )+ 3 λ ( ̄h 2 mω ) 2 [1+ 2 n(n+1)] (17.49) ...
17.3. PERTURBATIONTHEORYINMATRIXNOTATION 273 isveryeasy. Theeigenvectorsare %φ(0) 1 = 1 0 0 . . , %φ(0 ...
274 CHAPTER17. TIME-INDEPENDENTPERTURBATIONTHEORY Now supposeweusethe states{φn}tospan theHilbertspace, instead oftheset {φ(0)n ...
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