5.74 TIME-DEPENDENT QUANTUM MECHANICS

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1 p TIME-DEPENDENT QUANTUM MECHANICS The time evolutio of the state of a system is described by the time-depedet Schrödiger equatio (TDSE): i t ψ( r, t)= H ˆ ψ( r, t) Most of what you have previously covered is time-idepedet quatum mechaics, where we mea that H ˆ is assumed to be idepedet of time: H ˆ = H ˆ ( r ). We the assume a solutio of the form: ψ( r, t)= ϕ( r )T ( t) i 1 Tt () t Tt ()= H ˆ r ()ϕ r ϕ() r Here the left-had side is a fuctio of t oly, ad the right-had side is a fuctio of r oly. This ca oly be satisfied if both sides are equal to the same costat, E H ˆ r ϕ r ()ϕ r () Time-Idepedet Schrödiger Eq. = E H ˆ r ()ϕ()= r Eϕ() r H is operator correspodig to E Secod eq.: i 1 T Tt () t = E t + ie Tt ()= 0 Solutio: Tt ()= Aexp( iet / )= Aexp( iωt)

2 p. 2 So, for a set of eigevectors ϕ () r with correspodig eigevalues E, there are a set of correspodig eigesolutios to the TDSE. ψ ( r, t)= a ϕ r exp( iω t) ω = E / While the complete wavefuctio icludes time-depedet terms, the probability desity P = ψ * ( r, t) ψ( r, t)dr = ψ( r, t)ψ r, t is idepedet of time. Therefore, ϕ ( r ) are called statioary states. However, more geerally a system may be represeted as a liear combiatio of eigestates: ψ( r, t)= c ψ r, t = c e iω t ϕ ( r ) For such a case, the probability desity will oscillate with time: coherece. e.g., two eigestates ψ( r, t)= c 1 ϕ 1 e iω 1 t + c 2 ϕ 2 e iω 2 t pt ()= ψ * 2 ψ = c 1 ϕ 1 + c 2 ϕ c 1* c 2 ϕ 1* ϕ 2 e i ( ω 2 ω 1 t) + c 2* c 1 ϕ 2* ϕ 1 e +i ( ω 2 ω 1)t probability desity oscillates as cos( ω 2 ω 1 )t This is a simple example of coherece. Icludig mometum (a wavevector) of particle leads to a wavepacet.

3 p. 3 TIME EVOLUTION OPERATOR More geerally, we wat to uderstad how the wavefuctio evolves with time. The TDSE is liear i time. Sice the TDSE is determiistic, we will defie a operator that describes the dyamics of the system: For the time-idepedet Hamiltoia: ψ ( t) =U( t,t 0 )ψ ( t 0 ) t ψ( r, t)+ ih ψ ( r, t )= 0 (1) To solve this, we will defie a operator T = exp( iht / ), which is a fuctio of a operator. A fuctio of a operator is defied through its expasio i a Taylor series: T = exp[ iht ] = 1 iht = f( H) + 1 2! Multiplyig eq. 1 from the left by T 1 = exp( iht / )we have: iht 2 exp iht t ψ( r, t) = 0 itegratig t 0 t : exp iht ψ ( r, t ) exp iht 0 ψ ( r, t 0)= 0 ψ( r, t)= exp Ht t 0 ψ( r, t 0 )= Ut, ( t 0 )ψ r, t 0 For fuctios of a operator A : Give a set of eigevalues ad eigevectors of A, i.e., Aϕ = a ϕ, you ca show by expadig the fuctio as a polyomial that f ˆ A ()ϕ = f a ϕ

4 p. 4 ψ ( r, t) = e E ( t t 0 )/ ψ ( r, t 0 ) or Ut, ( t 0 )= e iω ( t t 0 ) ϕ ϕ ω = E This form is useful whe ϕ are characterized; we ll develop U( t, t 0 ) more later. Time-evolutio of a coupled two-level system (2LS) It is commo to reduce or map problem oto a 2LS. We the discard remaiig degrees of freedom, or icorporate them as a heat bath, H= H0 + Hbath. Let s discuss the time-evolutio of a 2LS with a time-idepedet Hamiltoia. Cosider a 2LS with two (uperturbed) eigestates ϕ a ad ϕ b with eigeeergies ε a ad ε b, which are the coupled through a iteractio V ab. V ε + ϕ a ε a H = a ε a a + b ε b b + a V ab b + b V ba a = ε a V ab V ba ε b ϕ b ε b 2 ε - * Sice the Hamiltoia is Hermetia, ( H ij = H ji ), we suggest V = V = Ve ab * φ i ba iφ εa Ve H = +φ i Ve εb If we defie the variables E = ε a + ε b 2 = ε a ε b 2

5 p. 5 The we ca solve for the eigevalues of the coupled systems: ε ± = E ± 2 + V 2 Because the expressios get messy, we do t choose to fid the eigevectors for the coupled system usig this expressio. Rather, we use a substitutio where we defie: ta 2θ = V iφ 1 ta2θe H= E I + +φ i ta 2θe 1 We ow fid that we ca express the eigevalues as ε ± = E ± sec2θ 2θ (0 < θ < π/2) V We ow wat to fid the eigestates of the Hamiltoia, ϕ ±, H ϕ ± = ε ± ϕ ± where e.g. ϕ + = c a ϕ a + c b ϕ b : ψ = cos θ e ϕ + si θ e ϕ + φ i /2 i φ/2 a ϕ = si θ e ϕ + cos θ e ϕ φ i /2 i φ/2 a b b Orthoormal complete + orthogoal: ϕa ϕ a + ϕb ϕ b = 1 Notice that ϕ + correspods to ϕ a perturbed by the V ab iteractio. ϕ correspods to ( for θ 0 ϕ+ ϕa ; ϕ ϕb ) ϕ b We ca schematically represet the eergies of these states:

6 p. 6 ε Ε ε + ε a ε - ε b These eigestates exhibit avoided crossig. The time-evolutio of this system is give by our time-evolutio operator. Ut, ( t 0 )= ϕ + e ω + ( t t ) 0 ϕ + + ϕ e iω ( t t ) 0 ϕ ω ± = ε ± Now ϕ a ad ϕ b are ot the eigestates preparig ϕ a will lead to time-evolutio! Let s prepare the system so that it is iitially i state ϕ a. ( t 0 = 0) ψ( 0) = ϕ a What is the probability that it is foud i state ϕ b at time t? P ba ()= t ϕ b ψ () t 2 2 = ϕ b Ut, ( t 0 )ϕ a To evaluate this, you eed to ow the trasformatio from the ϕ a,b to the ϕ ± basis, S a+ = ϕ a ϕ + This gives:

7 p. 7 where the Rabi Frequecy Ω R = V 2 V 2 P ba ()= t si 2 Ω V R t Ω R represets the frequecy at which probability amplitude oscillates betwee ϕ a ad ϕ b states. Pba () t 2 V V t = π/ωr t Notice for V 0 ϕ ± ϕ a,b (the statioary states), ad there is o time-depedece. For V >>, the Ω R = V ad P =1 after t = π 2Ω R = π 2V.

8 p. 8 TIME-INDEPENDENT HAMILTONIAN There are two types of values that we ofte calculate: Correlatio amplitude: Ct ()= βϕ() t measures the resemblace betwee the state of your system at time t ad a target state β. The probability amplitude Pt = Ct 2 for a set of eigestates ϕ () = β ψ () = β ψ C t t U t,t 0 = = * m m,,j * m ω i t ωjt c m j e j c c c e 0 Expectatio values: At () = ψ t ()Aψ( t) ψ () t = e ω t c ϕ = ψ ()= t e ω m t c * m ϕ m m c ϕ At () = c m* c e iω m t ϕ m Aϕ ω m = E E m m, ω ω m = c m* () t c ()A t m m,

9 p. 9 DENSITY MATRIX ψ t = c t we showed For a system described by a wavefuctio At = ψ ( t)aψ ( t) = c m* ()c t ()ma t We will ofte fid it useful to defie a desity operator, m ρ() t ψ( t) ψ ( t) = c t,m ()c * m () t m = ρ m () t m (by defiitio) ρ m are the desity matrix elemets. Substitutig, we see that,m = A m ρ m ( t) At,m = Tr Aρ() t [ ] Trace Properties: 1) cyclic ivariace Tr( ABC)= Tr( CAB) = Tr( BCA) 2) ivariat to uitary trasformatio Tr( S AS) = Tr( A) Pure vs. Mixed States Why would we eed the desity matrix? It helps for mixed states. 1) pure states: a system characterized by a wavefuctio (previous page) 2) mixed states: ot characterized by sigle wavefuctio > statistical mixtures esemble at thermal equilibrium > idepedetly prepared states > o phase relatioship betwee elemets of mixture

10 p. 10 For a esemble of systems with a probability p of occupyig quatum state ϕ p = 1, with () = ψ ψ A t p t A t () t p () t () t () = ρ() ρ ψ ψ A t Tr A t Properties: * 1) ρ is Hermetia ρ m = ρ m 2) Tr( ρ)=1 Normalizatio 3) Tr ( ρ 2 )= 1 for pure state < 1 for mixed state Let s loo at the desity matrix elemets for a mixture: ρ = ρ m = p ψ ψ m m where ϕ = c c : expasio coefficiet for eigestate of wavefuctio = = c c P c * m ( c ) m * coefficiets for eigestate averaged over mixture Diagoal elemets ( = m) ρ = p C 2 = c c * = p probability of fidig a system i mixture i state POPULATION ( 0)

11 p. 11 Off-Diagoal Elemets ( m) complex have phase factor describe the evolutio of coheret superpositios. COHERENCES For a arbitrary state χ, the expectatio value of the desity matrix: χ ρ χ gives the total probability of fidig a particle i the pure state χ withi the mixture. We will sometimes refer to the desity matrix at thermal equilibrium ρ 0 ( or ρ eq ), which is characterized by thermally distributed populatios i the quatum states e ρ = p = Z where Z is the partitio fuctio. More geerally, the desity matrix ca be defied as βe where Z = Tr(e βh ). For H = E, ρ= e Z βh ρ = m βh e m βe = e δ

12 p. 12 TIME-EVOLUTION OF DENSITY MATRIX Follows aturally from defiitio of ρ ad T.D.S.E. i i ψ = H ψ ψ = ψ H H = H t t ρ = ψ ψ = ψ ψ + ψ ψ t t t t i i = H ψ ψ + ψ ψ H ρ i = H, ρ t [ ] Liouville-Vo Neuma Eq. For a time-idepedet Hamiltoia: ρ m () t = ρ ( t) m = ψ( t) ψ ( t) m () ( t t ) ω i ψ t = U t,t ψ t = e ψ t () ρ t = e ψ t ψ t m e ω i t t0 +ω i m t t0 m 0 0 ( t t ) ( t ) ω i m 0 = e ρm 0 E ω = m E m Populatios: () t ( t ) ρ =ρ time-ivariat m 0 Cohereces: oscillate at eergy splittig ω m