Quantum Mechanics II Pof. Bois Altshule Apil 25, 2 Lectue 25 We have been dicussing the analytic popeties of the S-matix element. Remembe the adial wave function was u kl () = R kl () e ik iπl/2 S l (k)e ik+iπl/2] () whee S l (k) = e i2δ l(k) which is followed fom unitaity. The geneal popety of S l wee found last time, assuming k is a complex numbe S l (k) = S l ( k), S l (k) = S l (k ) ] Also the phase is an odd function in k: δ l (k) = δ l ( k). We also found last time that thee ae two kinds of singulaities. The fist lies in the uppe halfplane, on the imaginay axis. They ae the tue bound states. The second kind lie on the lowe halfplane, which coespond to esonances in the scatteing amplitude. Now in ode to deive some moe impotant popeties of S-matix element, we wite the function S l in tems of the enegy E instead of k. Recall that the enegy of the fee paticle is just E = 2 k 2 2m The time dependence of the plane wave state is just like in odinay quantum mechanics ψ(t) e ik S(E)e ik] e iet/ (4) Now instead of consideing the eigenstates of the Hamiltonian, which by definition ae states infinitely spead ove all the position space, we conside wave packets, which ae localized in space. We can conside linea combinations of enegy eigenstates ψ in (t) de F (E )e ik ie t/ In esponse to the incoming paticles scatteing on taget, we have the outcoming wave functions ψ out (t) de G(E )e ik ie t/ Because ou Schödinge equation is linea, theefoe ou esponse must be linea, and we can define a Geen function, o popagato connecting the outcoming wave function with the incoming wave function ψ out (t) = (2) (3) (5) (6) H(t t )ψ in (t ) dt (7)
whee H is a function which is independent of eithe the incoming state o the outgoing state. Thee is obviously constaints on the kenel H. One is causality, which equies that H(τ) = fo τ <. This is natual because thee can be no outgoing wave when thee is no incoming wave. We can Fouie tansfom the integation kenel and wite the above equation into ψ out (t) = dt dω exp iω(t t ) ] h(ω)ψ in (t ) (8) Compaing this with ou scatteing wave function we can identify that which means that e 2ik S(E) = h(ω) = h(e/ ) = 2π e2ik S(E) (9) e ieτ/ H(τ) dτ = e ieτ/ H(τ) dτ () The second equality is due to causality. Because we ae discussing the analytic popeties, hee the agument E will be complex E = Re E + Im E. If Im E we have exponential decay in the above integand, and the integal will convege in geneal, and the left hand side will not have any singulaities. Theefoe we have the impotant esult that causality foces S(E) to be analytic at Im E >. Notice that this egion coesponds to the fist quadant of the complex k plane. It also coesponds to the thid quadant of the k plane but it is on the second pat of the Riemann sheet and it is not physical fom a QFT point of view. So S(k) will be analytic in the fist quadant in k plane, and we immediately know that it has no poles in both the fist and second quadant, and it has no zeoes in the thid and fouth quadant. This is in accodance with ou discussion above, and it is connected to causality in a vey basic level.. Coulomb Scatteing Now we conside Coulomb scatteing. The scatteing potential is In Bon appoximation we have V = Q Q 2 () ( 2mQ Q 2 f(θ) = 2 ) q 2 = Q Q 2 4E sin 2 θ/2 (2) theefoe we have dσ dω = Q2 Q2 2 6E 2 sin 4 θ/2 Notice that if we integate ove all solid angle we will get a divegent total coss section. This is also the classical esult of Ruthefod scatteing. We define the quantum mechanical quantity γ = Q Q 2 v = mq Q 2 2 k This paamete is the only meaningful paamete in Coulomb scatteing and it povides a convenient scale in the poblem. The Bon appoximation coesponds to γ while the classical limit coesponds to (3) (4) 2
γ, so the fact that the Bon esult coesponds to the classical limit means that the scatteing amplitude should be constant fo any γ. This peculiaity will be consideed in a moment. Let s conside fist an inteacting femion gas. The gound state of the system will be a Femi sea whee all the enegy levels below the Femi level will be occupied wheeas none of the levels above is occupied. We denote the Femi enegy by E F and the coesponding momentum p F and we have the expession fo Femi momentum p F = (3π 2 n) /3 (5) whee n is the numbe density of the femions. If we have shot ange inteaction V = λδ(x x 2 ), then we will have potential enegy pe paticle equal to λn because each paticle sees a potential popotional to the numbe density at that point. The kinetic enegy pe paticle will be of the ode E F which goes like n 2/3. Theefoe when we incease the density the potential enegy contibution gets moe impotant, which is in accodance with ou intuition because if we incease the density it will not emain a gas, but will become liquid and solid. Now if we conside Coulomb inteaction V = Q Q 2 /, then γ will be γ = e2 v = e2 m 2 k F n /3 (6) So contay to the above intuition, when the numbe density becomes lage enough we have an ideal gas, and only when the numbe density is low will the inteaction be impotant. Let s come back to Coulomb scatteing. Remembe we have the tansition opeato T and we have the opeato equation T = V + V G + T, G + = (7) E H + iɛ The scatteing amplitude is just the matix element of the tansition opeato f(k, k) = 4π2 m 2 k T k (8) The Bon appoximation is just T = V. This gives us the nice esult which is equal to the classical esult, howeve we know fom the above equation that T = V is not an exact solution. But this seems to contadict with ou pevious obsevation that the fist ode is aleady the classical esult. The esolution hee is that dσ/dω is the absolute value squaed of the scatteing amplitude. In fact it is tue that the amplitude does not get coected in petubation theoy, howeve the scatteing phase is not petubative, and it has vey nontivial dependence on γ. Remembe we had the following asymptotics ψ + (x) (2π) 3/2 e ik x + ] eik f (9) which we justify by equiing that V when. We found an equation fo u kl which is just R kl () d 2 u d 2 + k 2 2m l(l + ) V () 2 2 ] u = (2) We agued that we can discad the potential tem at infinity because of the above assumption, and we also can discad the centifugal baie because. Only then do we have the simple equation that 3
u + k 2 u =. Howeve now we need to conside this in moe detail. Let us wite u kl = expiw(k)] and we can get (w ) 2 + iw + k 2 2m l(l + ) V 2 2 = (2) To kill the k 2 tem we can just shift w by k and define w = k + w, so that we have 2kw + (w ) 2 + iw = 2m l(l + ) V + 2 2 (22) Now if V / +ɛ then w will also be popotional to /+ɛ and we will get w ɛ Howeve fo Coulomb case we have V /. Then we have (23) w = w ln(k) (24) Theefoe we have a poblem. Due to this logaithmic divegence we can t apply ou pevious techniques and we ae foced to wite u kl e i(k+γ ln k+... ) (25) and the logaithm in the exponential is called the Coulomb logaithm. Since the cuent is j = i m ψ ψ ψ ψ ] (26) so the logaithm will cancel in the exponent and because its deivative is / so its contibution is not impotant in limit. This is why we don t have this effect in the classical limit. We can wite the Schödinge equation as ( 2 + k 2 + 2γk ) ψ = (27) It is convenient to sepaate vaiables in paabolic coodinates (ζ, η, ϕ) whee ζ = + z, η = z, tan ϕ = y x If we plug this into the above equation and solve it then we get the following esult ] { ψ k (, θ) = + iγ 2 e ikz iγ lnk( cos θ)] + f(θ) } ln 2k eik+iγ k( cos θ) (28) (29) whee the scatteing amplitude is γ Γ( iγ) f(θ) = 2k sin 2 θ/2 Γ( + iγ) e2iγ ln(sin θ/2) (3) Recall some impotant identities of the Γ function Γ(s) = e t t s dt, Γ() =, Γ(s + ) = sγ(s), Γ(s ) = Γ(s) (3) 4
So the scatteing amplitude is only coected by a complicated phase of Γ functions. If we wite out the scatteing phase shift then we get e 2iδ l Γ(l + iγ) = (32) Γ(l + + iγ) Note that Γ function has poles at s =,, 2,... This means that the above combination has poles at iγ = n l +. If we wite this condition out using the definiton of γ then we will have γ = e2 m 2 k = ka B = k n = ina B = E n = 2 2ma 2 B n2 = E n 2 (33) So we ecove all of the hydogen spectum. By only analyzing the scatteing poblem and the analytic stuctue of the scatteing amplitude, we ecoveed the whole spectum of the bound states of the Coulomb potential. 5