QuantumPhysics.dvi

Note that the scale of the cross section is set by the momentumtransferq, and the mass. This formula exhibits singular behavior ...

The fact that on the right hand side, the scattering amplitude is being evaluated for equal initial and final momenta is respons ...

symmetry of the problem then allows one to decouple the system into partial waves, producing conceptual and practical simplifica ...

For real values ofρ, the functionjℓ(ρ) is real, buthℓ(ρ) is complex. The asymptotic values of these functions are as follows. As ...

Now, we know the solutionjℓ(kr) for U = 0, and consider the full solution in a perturbative expansion inU. We set Rℓ(r) =jℓ(kr) ...

Differentiating a second time, and substituting the result into the defining equation (12.77) for Gℓ(r,r′), we obtain, ( d^2 dr^ ...

The first term represents a radially incoming wave, while thesecond represents a radially outgoing wave. Because the potentialV( ...

The result is given by σtot= ∫ 4 π dΩ|f(k′,k)|^2 = 4 π k^2 ∑∞ ℓ=0 (2ℓ+ 1) sin^2 δℓ (12.98) It is readily checked that the optica ...

the above relation becomes, e^2 iδℓ(k)=− h∗ℓ(kb) hℓ(kb) (12.106) The phase shifts for the lowest values ofℓmay be computed expli ...

At high energy, the particles behave classically and see only the aspect disk with areaπb^2 , but at low energy, thes-wave scatt ...

Comparison of the asymptotics of this formula with the definition of the phase shifts in (12.93), we find, after some rearrangem ...

Sinceδℓ(k) is dimensionless, we need indeed two dimensionful parameters ER, the position of the resonance, and Γ, the width of t ...

13 Time-dependent Processes So far, we have studied problems in which the Hamiltonian is time-independent. This is suitable for ...

The time-dependent Schr ̈odinger equation for|ψ(t)〉then reduces to the following set of equations fora±, ia ̇+(t) = ω 0 2 a+(t) ...

13.2 The interaction picture We now turn to time dependent problems in the general case whereH=H 0 +V(t). We shall assume that t ...

then we define the corresponding state|ψa;t〉Iin the interaction picture by factoring out the time dependence induced byH 0 alone ...

13.3 Time-dependent perturbation theory A systematic way of organizing time-dependent perturbation theory is through the use the ...

13.4 Switching on an interaction We consider perhaps the simples of time-dependent potentials, V(t) =Vθ(t) (13.36) whereθ(t) is ...

It is best to consider this problem mathematically first. We setx= (E−Ei)/(2 ̄h), and thus seek to obtain the following largetbe ...

13.5 Sinusoidal perturbation Another very important special case, analogous to the problem of spin magnetic resonance, is when t ...