Quantum Mechanics II SS Roser Valentí Institut für Theoretische Physik Universität Frankfurt

Transcription

1 Quantum Mechanics II SS 2014 Roser Valentí Institut für Theoretische Physik Universität Frankfurt

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3 Contents 1 Scattering theory Importance of collision experiments Elastic scattering Scattering cross section Calculation of the scattering cross section Asymptotic form of stationary scattering states. Scattering amplitude Relation between σ(θ,φ) and f(θ,φ) Partial wave analysis Example: Quantum hard-sphere scattering Phase shift Phase shift for the 3-dimensional case The Born approximation Integral form of the Schrödinger equation Solving the Green s function The first Born approximation Example: Yukawa potential The Born series Collisions with absorption Formal scattering theory Propagator theory and the Lippmann-Schwinger equation Meaning of the Lippmann-Schwinger equation Position and momentum representation Retarded and advanced Green s functions Interpretation of the Lippmann-Schwinger equation and its expansion in terms of Feynmann s diagrams Theory of angular momentum Importance of angular momentum Orbital angular momentum in Quantum Mechanics Spin Generalization: angular momentum J Special properties of J = S = 1/ Non-relativistic description of a spin-1/2 particle Angular momentum and rotations

4 -2 CONTENTS Definition of rotation Orthogonal group Infinitesimal rotations Rotation operators in state space Rotation operators in terms of angular momentum observables System of several spinless particles Rotation of observables Rotation invariance Invariance of physical laws Rotation operators for a spin 1/2 particle Unitary unimodular group SU(2) Addition of angular momenta Addition of two spins 1/ Formal theory of angular-momentum addition Clebsch-Gordan coefficients Recursion relations for the Clebsch-Gordan coefficients Clebsch-Gordan coefficients and rotation matrices Bell s inequality Symmetry in quantum mechanics Symmetries, conservation laws, degeneracies Symmetries in classical physics Symmetries in quantum mechanics Discrete symmetries: parity Wavefunctions under parity Example: double-well potential Discrete symmetry: lattice translation Tight-binding approximation Bloch s theorem Discrete symmetry: time-reversal Antiunitary transformation Time-reversal operator Transformation of operators under time reversal Wavefunctions under time reversal Kramers degeneracy Approximation methods: perturbation theory Non-degenerate time-independent perturbation theory Two-state Hamiltonian General perturbation expansion Renormalization of the wave-function Example Degenerate time-independent perturbation theory First-order approximation Second-order approximation (λ 2 )

5 CONTENTS Example I: linear Stark effect Example II: fine structure Time-dependent Hamiltonian: the Dirac picture Dirac picture / Interaction picture Time-dependent perturbation theory Operator formalism: Dyson series Concept of transition probability Perturbation expansion Example: A constant perturbation Fermi s golden rule Example: harmonic perturbation Example: interaction of a bound electron with a classical radiation field Many-particle systems. Second quantization Identical particles and permutation theory Totally symmetric and totally antisymmetric states Bosons Creation and annihilation operators Particle-number operator Single- and many-particle operators Fermions Creation-annihilation operators Single- and many-particle operators Momentum representation Hamiltonian in second-quantized form Spin-1/2 fermions The Fermi sphere, excitations Electron gas with Coulomb repulsion Relativistic quantum mechanics The Klein-Gordon equation Derivation The continuity equation Free solutions of the Klein-Gordon equation Dirac equation Derivation of the Dirac equation Continuity equation Properties of the Dirac matrices α k and β Covariant form of the Dirac equation Non-relativistic limit of the Dirac equation Physical interpretation of the solution to the Dirac equation

6 0 CONTENTS Literature 1) J. J. Sakurai: Modern Quantum Mechanics, Addison-Wesley Publishing Company ) J. J. Sakurai and J. J. Napolitano: Modern Quantum Mechanics, Pearson Ed. Limited ) W. Nolting: Grundkurs Theoretische Physik 5/2. Quantenmechanik-Methoden und Anwendungen, Springer ) F. Schwabl: Advanced Quantum Mechanics, Springer ) C. Cohen-Tannoudji, B. Diu, F. Laloke: Quantum Mechanics II, Wiley-VCH ) M. Tinkham: Group Theory and Quantum Mechanics, McGraw-Hill, Inc ) L. H. Ryder: Quantum Field Theory, Cambridge University Press, Inc Acknowledgements I would like to thank Kateryna Foyevtsova for her help in typing the present manuscript.

7 Chapter 1 Scattering theory 1.1 Importance of collision experiments Many experiments in physics ( atomic and nuclear physics, solid state physics and high energy physics) consist of directing a beam of particles (1) onto a target of particles (2) (see Fig. 5.4), and studying the resulting collisions. This is possible since the various particles constituting the final state of the system (state after the collision) are detected and their characteristics (direction of emissiom, energy, etc.) are measured. Aim of such experiments: Determine the interactions that occur between the various particles entering the collision. The collisions give rise to reactions: (1)+(2) (3)+(4)+(5)+... In quantum mechanics, we can only speak of probabilities of the possible states coming out of the collision. Among the reactions possible under given conditions, scattering reactions are defined as those in which the final state and the initial state are composed of the same particles (1) and (2). In an elastic scattering reaction, none of the particles internal states change during the collision. Figure 1.1: The collision experiment 1.2 Elastic scattering Let us focus on the elastic scattering of the incident particles (1) by the target particles (2). In classical mechanics, we would determine the deviations in the incident particles trajectories (1) due to the forces exerted by particles (2). 1

8 2 CHAPTER 1. SCATTERING THEORY In quantum mechanins, we study the evolution of the wave function associated with the incident particles under the influence of their interactions with the target particles, i.e., scattering of particles (1) by particles (2). In order to analyze the scattering process, we will consider the following simplifying hypotheses: 1. we assume that particles (1) and (2) have no spin; 2. we do not consider the internal structure of particles (1) and (2); this assumption excludes therefore inelastic scattering phenomena, where part of the kinetic energy of (1) is absorbed in the final state by the internal degrees of freedom of (1) and (2); 3. we assume that the target is thin enough to neglect multiple scattering processes, i.e., processes during which a particular incident particle is scattered several times before leaving the target; 4. we neglect coherence between waves scattered by the different particles of the target; thisisjustified ifthespreadofthewave packet associated with(1)issmall compared to the average distance of particles in (2). An example that we are neglecting is the coherent scattering by a crystal (Bragg diffraction). We concentrate on the scattering of particle (1) of the beam by particle (2) of the target; 5. we assume that the interactions between particles (1) and (2) can be described by a potential energy: V( r 1 r 2 ) = V( r). In the center of mass reference, 1 m = 1 m m 2 m : relative mass, the problem reduces then to the study of a single particle with mass m scattered by the potential V( r). 1.3 Scattering cross section To introduce the concept of scattering cross section, we count the particles scattered in a certain direction in space. The process is illustrated in Fig. 1.2, and the following notation is used: Oz: direction of the incident particle of mass m; dn: number of particles scattered per unit time into the solid angle dω about the direction (θ, φ); Figure 1.2

9 1.4. CALCULATION OF THE SCATTERING CROSS SECTION 3 where θ and φ is the angular dependence in spherical coordinates. F i : flux of incident particles; it is the number of particles which traverse per unit time a unit surface to Oz in the region where z takes on very large negative values; V( r): potential localized around the origin O; D: detector situated at a distance from O, which is large compared to the linear dimensions of the potential s zone of influence, measures the number dn of particles scattered per unit time into solid angle dω. The fraction of scattered particles should be proportional to the flux of incident particles, therefore: dn dω is proportional to F i: Dimensions: dn = F i σ(θ,φ)dω σ(θ,φ): coefficient of proportionality. (1.1) [dn] = T 1 [F i ] = (L 2 T) 1 [σ(θ,φ)] = (L 2 ) dimensions of a surface σ(θ, φ) is the differential scattering cross section in the direction (θ, φ). Units: 1 barn = cm 2 The number of particles per unit time which reach the detector is equal to the number of particles which would cross a surface dσ = σ(θ,φ)dω placed to Oz in the incident beam. The total scattering cross section σ: σ = dωσ(θ,φ). Differential cross section σ(θ,φ) = dσ/dω Important: The concept of cross section is not limited to the case of elastic scattering. 1.4 Calculation of the scattering cross section In quantum mechanics, in order to describe the scattering of a given particle by the potential V( r), it is necessary to study the time evolution of the wave packet representing the state of the particle. We have that

10 4 CHAPTER 1. SCATTERING THEORY 1. the wavefunctions are known at t for Oz < 0 far from V( r) and 2. the evolution of the wave packet can be obtained if it is expressed as a superposition of stationary states (because V( r) V( r,t)), therefore, we study the eigenvalue equation of the Hamiltonian: P H = H 0 +V( r) H 0 = 2 : describes the particle s kinetic energy 2m Stationary scattering states Ψ( r) are defined as follows. Ψ( r,t) = Ψ( r)e iet/ evolution of a particle in the potential V( r), where Ψ( r) is the solution of the eigenvalue equation: ] [ 2 2m +V( r) Ψ( r) = EΨ( r). (1.2) We assume that V( r) decreases faster than 1/r as r approaches (this hypotheses excludes the Coulomb potential). Then, if we write the kinetic energy of the incident particle, E, as E = 2 k 2 2m and the potential, V( r), as eq. (1.2) becomes V( r) = 2 2m U( r), [ +k 2 U( r)]ψ( r) = 0. (1.3) Boundary conditions: conditions that must be imposed on the solutions of eq. (1.3) if they are to be used in the description of a scattering process Asymptotic form of stationary scattering states. Scattering amplitude The wavefunction Ψ k ( r) representing the stationary scattering state associated with a given energy, E = 2 k 2 /2µ, is a superposition of the plane wave e ikz and the scattered wave. For large r, the scattered wave approches an asymptotic function whose radial dependence in a given direction (θ,φ) is of the form eikr (spherical wave). It is worth noticing r that:

11 1.4. CALCULATION OF THE SCATTERING CROSS SECTION 5 it is an outgoing wave which has the same energy as the incident wave; the factor 1 results from the fact that we have three spacial directions, and Ψ 2 r must go like 1 to conserve probability: r 2 ( +k 2 )e ikr is not zero, but ( +k 2 ) eikr r where r 0 is the range of the potential. For r, Ψ k ( r) therefore behaves as = 0 for r r 0. Ψ k ( r) e ikz +f k (θ,φ) eikr r, (1.4) f(θ, φ): scattering amplitude, depends on the potential V( r). Analogous is the scattering of waves in classical mechanics, f.i., water waves encountering a rock Relation between σ(θ,φ) and f(θ,φ) We define dv as the volume of incident beam that passes through an area dσ in the time dt with velocity v, dv = dσvdt The whole problem for a scattering process reduces to determine the scattering amplitude, f k (θ,φ), whose squared value gives the probability of scattering in a given direction (θ,φ) and, as we will see, is related to the differential cross section. Remembering that Ψ incident = Ae ikz, and } Ψ scattered = A {f(θ,φ) eikr, r the probability that the incident particle, travelling at speed v, passes through the infinitesimal area dσ in time dt, dp = Ψ incident 2 dv = A 2 (vdt)dσ

12 6 CHAPTER 1. SCATTERING THEORY is equal to the probability that the particle scatters into the corresponding solid angle dω (see Fig. 1.2), It follows that: σ(θ,φ) = f(θ,φ) 2 dp = Ψ scattered 2 dv = A 2 r 2 f 2 (vdt)r 2 dω. }{{} dσ = σ(θ,φ)dω = f 2 dω, differential cross section. The differential cross section (which is what the experimentalist measures) is equal to the absolute square of the scattering amplitude (which is obtained by solving the Schrödinger equation). This equation relates directly observation with theory. There are two important techniques for calculating the scattering amplitude: partial wave analysis; Born approximation. dv 1.5 Partial wave analysis In the special case of a central potential, V(r), the orbital angular momentum, L, of the particle is a constant of motion. Therefore, there exist stationary states with well-defined angular momentum, i.e., eigenstates common to H, L 2 and L z. And, as we found in QM I, the Schrödinger equation of a spherically symmetrical potential V(r) has separable solutions of the form: Ψ(r,θ,φ) = R(r)Y m l (θ,φ), where Yl m (θ, φ) are spherical harmonics and u(r) = rr(r) satisfies the radial equation: ] 2 d 2 u [V(r)+ 2mdr + 2 l(l +1) u = Eu. 2 2m r 2 where the term proportional to 1/r 2 is the centrifugal potential energy. We consider first the asymptotics of this equation since in section 1.4 we reasoned how the wavefunction of the particle should behave at large distances away from the influence of the potential. For r (scattered region), V(r) 0 and the centrifugal term goes to 0. Then d 2 u dr 2 k2 u, has the general solution: u(r) = Ce ikr }{{} outgoing +De }{{ ikr }. incoming

13 1.5. PARTIAL WAVE ANALYSIS 7 An incoming wave to describe the scattered wave makes no sense and therefore D = 0. Then, at very large r, R(r) = u(r) r eikr r what in optics is called as the radiation zone. (valid for kr 1) Since we assume that the potential is localized, in the intermediate region the equation is: d 2 u l(l +1) u = k 2 u (V can be ignored but not the centrifugal term) dr2 r 2 Please, note that this cannot be applied to the Coulomb potential since 1/r goes to zero slowlier than 1/r 2 as r and the centrifugal term does not dominate this region. In this sense, the Coulomb potential is not localized, and partial wave analysis is inappropriate. Other potentials, like quantum hard-sphere scattering or a finite depth potential, fulfill this condition. In the intermediate region the solution of the equation: is a linear combination d 2 u dr l(l+1) u = k 2 u 2 r 2 u(r) = Arj l (kr)+brn l (kr) of the spherical Bessel functions (see Arfken,mathematical Methods): ( ) l 1 j l (x) = ( x) l d sinx xdx x, ( ) l 1 n l (x) = ( x) l d cosx xdx x, The centrifugal term

14 8 CHAPTER 1. SCATTERING THEORY and j 0 (x) = sinx x, n 0(x) = cosx x. Since none of them represents an outgoing (or incoming) wave and we should have the correct asymptotics for the scattering process, we consider spherical Hankel functions which are the linear combinations of j 0 (x) and n 0 (x): h (1) l h (2) l h (1) l (x) j l (x)+in l (x), h (2) l (x) j l (x) in l (x), h (1) 1 = h (1) 0 = i eix x, h (2) 0 = i e ix x, ( i ( h (2) i l = x 2 1 x ) e ix, ) e ix, x 1 2 x } 1 x ( i)l+1 e ix for x >> 1. 1 x (i)l+1 e ix For outgoing waves, we should choose the solution (boundary conditions): R(r) h (1) l (kr), which gives the correct asymptotics and the wavefunction outside the scattering region (V(r) = 0) is: Ψ(r,θ,φ) = A e ikz }{{} incident plane wave+ l,m c l,m h (1) l (kr)yl m (θ,φ). }{{} scattered wave We assume that the potential is spherically symmetric, and Ψ is independent of φ, which means that m = 0 (Y m l e imφ ). Then, P l : the lth Legendre polynomial. We redefine the expansion coefficients as: Y 0 l (θ,φ) = 2l+1 4π P l(cosθ), c l,0 i l+1 k 4π(2l+1)a l.

15 1.5. PARTIAL WAVE ANALYSIS 9 Then { Ψ(r,θ) = A e ikz +k a l : lth partial wave amplitude. For r, l=0 i l+1 (2l+1)a l h (1) l (kr)p l (cosθ) }, then and h (1) l (kr) ( i) l+1eikr kr ; } Ψ(r,θ) A {e ikz +f(θ) eikr, r f(θ) = l=0 (2l+1)a lp l (cosθ) scattering amplitude, f(θ), in terms of the partial wave amplitude, a l These results confirm more rigorously the general structure of the wavefunctions postulated previously. The differential cross section σ(θ) is then: σ(θ) = f(θ) 2 = l l (2l+1)(2l +1)a l a l P l(cosθ)p l (cosθ), and the total cross section: σ = 4π l=0 (2l+1) a l 2, where we used the orthogonality of the Legendre polynomials: 1 1 P l (x)p l (x)dx = ( ) 2 δ ll. 2l+1 What to do next?: All that remains is to determine the partial wave amplitudes, a l, for the potential in question. This is accomplished by solving the Schrödinger equation in the interior region (where V(r) is distinctly non-zero) and matching this to the exterior solution using the appropriate boundary conditions. Now, a problem with notation arises since we have: spherical coordinates for the scattered wave; cartesian coordinates for the incident wave.

16 10 CHAPTER 1. SCATTERING THEORY Let s rewrite the wavefunction in a more consistent notation: e ikz satisfies the Schrödinger equation with V = 0; on the other hand, the general solution to the Schrödinger equation with V = 0 can be written as: [A l,m j l (kr)+b l,m n l (kr)]yl m (θ,φ). l,m Therefore, let us express e ikz in this way: z = rcosθ has no φ dependence, only m = 0 occurs for Y m l ; e ikz is finite at z = 0 therefore n l (kr) is not allowed since it diverges at r = 0. Therefore, e ikz = i l (2l+1)j l (kr)p l (cosθ) Rayleigh s formula l=0 Then, the wave function in the exterior region of the potential can be expressed entirely in terms of r and θ: Ψ(r,θ) = A [ ] l=0 il (2l+1) j l (kr)+ika l h (1) l (kr) P l (cosθ) Example: Quantum hard-sphere scattering We consider the following potential: { for r b; V(r) = 0 for r > b. And we want to calculate the scattering cross section for elastic scattering under the influence of this potential. Boundary conditions: Ψ(b,θ) = 0 The potential is central, therefore we consider the partial wave analysis. Then, Ψ(b,θ) = 0 = l=0 and the total cross section is: [ ] i l (2l+1) j l (kb)+ika l h (1) l (kb) P l (cosθ) = 0, σ = 4π k 2 j l (kb) a l = i kh (1) l (kb), (2l+1) j l (kb) 2 h (1) (kb). l=0

17 1.5. PARTIAL WAVE ANALYSIS 11 What do we learn out of that? First, let s consider the low-energy scattering limit: kb 1. In this limit, the wavelength is much greater than the radius of the sphere: j l (x) h (1) l (x) = and hence, j l (x) j l (x)+in l (x) ij l(x) n l (x) i 2l l!x l /(2l+1)! (2l)!x l 1 /2 l l! = i 2l+1 σ 4π k 2 l=0 [ ] 1 2 l 4 l! (kb) 4l+2. 2l+1 (2l)! [ 2 l l! (2l)! ] 2 x 2l+1, Since kb 1, the higher powers are negligible, and only l = 0 will be significant. This means that the differential cross section for low-energy hard-sphere scattering is independent of θ, as in the classical case: σ = 4πb 2. Remarks: the scattering cross section is four times the geometrical cross section; σ is the total surface area of the sphere. This larger effective size of σ is characteristic of long-wavelength scattering (this is also in optics); the waves feel their way around the whole sphere, whereas classical particles only see the head-on cross section Phase shift Phase shift is a very useful variable in scattering theory since it often reduces the complexity of the calculations of scattering processes. In order to introduce this concept in scattering theory, let s consider a one-dimensional scattering of a particle from a localized potential V(x) on the half line x < 0: V = { V(x) for a < x < 0; for x = 0. (bounded on the right by an infinite well) Incident wave coming from : Ψ i (x) = Ae ikx (x < a), reflected wave: Ψ r (x) = Be ikx (x < a). Independently of what happens in the interaction region ( a < x < 0), the amplitude of the reflected wave has got to be the same as that of the incident wave (conservation of probability), but it need not have the same phase.

18 12 CHAPTER 1. SCATTERING THEORY If there were no potential at all, V(x) = 0 (only the well at x = 0), then B = A, we would have total reflection, since the total wavefunction must vanish at the origin (Ψ(0) = 0) and: Ψ 0 (x) = A ( e ikx e ikx). For V(x) 0, Ψ(x) = A ( e ikx e i(2δ kx)) for x < a. The whole theory of scattering reduces to the problem of calculating the phase shift δ for a specific potential. The advantage of working with phase shift (as opposed to calculating the complex amplitude B) is that it makes use of the physics: conservation of probability: all the potential can do is to shift the phase of the reflected wave; simplifies mathematics extremely Phase shift for the 3-dimensional case The incident plane wave, Ae ikz, carries no angular momentum (m = 0) but includes all values of the total angular momentum (l = 0,1,2,...). In a spherically symmetrical potential, angular momentum is conserved. Each partial wave (labelled with l) scatters independently with no change of amplitude, only in phase. When V(r) = 0, Ψ 0 = Ae ikz and the lth partial wave is: where we used the Rayleigh formula For x = kr 1, j l (x) = 1 2 and the l th partial wave: Ψ (l) 0 = Ai l (2l+1)j l (kr)p l (cosθ); (e ikz = i l (2l+1)j l (kr)p l (cosθ)). l=0 [ ] h (1) l (x)+h (2) l (x) 1 [ ( i) l+1 e ix +i l+1 e ix], 2x Ψ (l) (2l +1) [ 0 A e ikr ( 1) l e ikr] P l (cosθ). 2ikr For V(r) = 0 the first term in the square brackets represents the outgoing wave and the second term the incoming wave. When V(r) 0 the first term picks up a phase shift δ l : The h (2) l Ψ (l) A (2l+1) 2ikr [ e i(kr+2δ l ) ( 1) l e ikr] P l (cosθ). (1.5) component in e ikz is phase shifted 2δ l and emerges as an outgoing spherical wave.

19 1.6. THE BORN APPROXIMATION 13 Connection between phase shift δ l, partial wave amplitude a l and scattering cross section σ: We had as general wavefunction in the exterior region: Ψ(r,θ) = A l=0 we consider the asymptotic behavior of eq. (1.6): { (2l+1) Ψ (l) A 2ikr Comparing eq. (1.5) and eq. (1.7): [ ] i l (2l+1) j l (kr)+ika l h (1) l (kr) P l (cosθ). (1.6) [ e ikr ( 1) l e ikr] + 2l+1 a l e }P ikr l (cosθ). (1.7) r a l = 1 ( e 2iδ l 1 ) = 1 2ik k eiδ l sinδ l. It follows that: and f(θ) = 1 k (2l+1)e iδ l sinδ l P l (cosθ), l=0 σ = 4π k 2 (2l+1)sin 2 δ l. l=0 The advantage of working with phase shifts (as opposed to partial wave amplitudes) is that they are easier to interpret physically. δ l exploits conservation of angular momentum and reduces solving a complex quantity a l to a simple real one δ l. 1.6 The Born approximation We consider now the Born approximation which is suitable for scattering processes where the scattering potential is weak. First we introduce the integral form of the Schrödinger equation Integral form of the Schrödinger equation The stationary Schrödinger equation, 2 2m 2 Ψ+VΨ = EΨ, can be written (analogous to the Helmholtz equation) as: ( 2 +k 2) Ψ = Q,

20 14 CHAPTER 1. SCATTERING THEORY where k 2mE, and Q 2mV 2 Ψ. Note that it has only the form of the Helmholtz equation since Q depends on Ψ. We remind about the concept of a Green s function, G( r): G( r) is the solution of the Helmholtz equation, ( 2 +k 2) G( r) = δ (3) ( r), (1.8) when the source is a delta function. It can be shown that Ψ can be expressed in terms of G as an integral: Ψ( r) = G( r r 0 )Q( r 0 )d 3 r 0, which satisfies the Schrödinger equation: ( 2 +k 2) [( Ψ( r) = 2 +k 2) G( r r 0 ) ] Q( r 0 )d 3 r 0 = δ (3) ( r r 0 )Q( r 0 )d 3 r 0 = Q( r), The Green s function for the Helmholtz equation, G( r r 0 ), is the response to a delta function source Solving the Green s function The Fourier transform of G( r), G( r) = 1 (2π) 3/2 e i s r g( s)d 3 s, (1.9) transforms the differential equation Eq. 1.8 into an algebraic equation. g( s) can be now obtained by multiplying both sides of Eq. (1.9) by ( 2 +k 2 ): ( 2 +k 2) G( r) = 1 (2π) 3/2 [( 2 +k 2) e i s r] g( s)d 3 s. Then, using that and 2 e i s r = s 2 e i s r, δ (3) ( r) = 1 e i s r d 3 s, (2π) 3 ( 2 +k 2) G( r) = δ (3) ( r),

21 1.6. THE BORN APPROXIMATION 15 we arrive at: and therefore: 1 (2π) 3/2 ( s 2 +k 2 )e i s r g( s)d 3 s = 1 (2π) 3 g( s) = 1 (2π) 3/2 (k 2 s 2 ). e s r d 3 s, We now introduce g( s) into eq. (1.9): G( r) = 1 (2π) 3 e i s r 1 (k 2 s 2 ) d3 s. and solve the integral for G( r). We choose spherical coordinates (s, θ, φ), with the polar axis along r as shown in the figure. With this choice of the coordinate frame, Coordinate frame s r = sr cosθ, d 3 s = dφs 2 sinθdθds, dφ = 2π, Thus, π 0 e isrcosθ sinθdθ = eisrcosθ isr π 0 = 2sin(sr). sr G( r) = 1 2 s sin(sr) (2π) 2 r 0 k 2 s ds = π 2 r s sin(sr) ds. (1.10) k 2 s2 We consider the relation: and we rewrite eq. (1.10) as: G( r) = = { + sinx = eix e ix, 2i i se isr + } 8π 2 r (s k)(s+k) ds se isr (s k)(s+k) ds i 8π 2 r (I 1 I 2 ). (1.11) We can now solve the integrals I 1 and I 2 by using the Cauchy s integral formula: { f(z) 2πif(z0 ) for z dz = 0 within the contour; z z 0 0 otherwise. (z complex variable)

22 16 CHAPTER 1. SCATTERING THEORY For Eq the integration is along the real axis, and it passes right over the pole singularity at ±k. One can decide different ways to skirt the poles. We choose to go over the pole at k and under the one at +k (some other conventions will give different Green s Skirting the poles in the contour integral functions, but they are all equally valid). For each integral in eq. (1.11), we have to close the contour in such a way that the semicircle at doesn t contribute. Thus, in the integral I 1, I 1 = + se isr (s k)(s+k) ds, e isr goes to zero when s has a large positive imaginary part: e i(re(s)+iim(s))r, and therefore we close the integral from above. The contour encloses only the singularity at s = +k and: [ ] [ se isr 1 se isr I 1 = ds = 2πi = iπe s+k s k s+k] s=k ikr. For the integral I 2, I 2 = + se isr (s k)(s+k) ds, the factor e isr goes to zero when s has a large negative imaginary part, and therefore we close from below: and the contour encloses the singularity at s = k. Since it goes around in the clockwise direction, we pick up a ( ) sign: [ ] se irs I 2 = s k 1 ds = 2πi s+k [ ] se isr s k s= k = iπe ikr.

23 1.6. THE BORN APPROXIMATION 17 Then, G( r) = i [( ) iπe ikr ( iπe ikr)] = eikr 8π 2 r 4πr. (1.12) This is the expression for the Green s function for the Helmholtz equation. We can add to G( r) any function, G 0 ( r), that satisfies the homogeneous Helmholtz equation, ( 2 +k 2) G 0 ( r) = 0, and the result, (G+G 0 ), will still satisfy the initial equation, which reflects the ambiguity of how to skirt the poles: a different choice of how to skirt the poles amounts to picking a different G 0 ( r). With the explicit expression for G( r), (eq. (1.12)), the general solution of the Schrödinger equation, ( 2 +k 2) Ψ = Q, with Q 2m 2mE VΨ andk, 2 in then: Ψ( r) = Ψ 0 ( r) m e ik r r 0 2π 2 r r 0 V( r 0)Ψ( r 0 )d 3 r 0 (1.13) where Ψ 0 satisfies thefree-particle Schrödinger equation. This equation is the Schrödinger equation in integral form The first Born approximation We will now make use of the previous equation to derive the Born approximation. We assume that V( r 0 ) is localized around r 0 = 0 and calculate Ψ( r) at points far away from the scattering center. Since r >> r 0, ( r r 0 2 = r 2 +r0 2 2 r r 0 = r r r ) 0, r 2

24 18 CHAPTER 1. SCATTERING THEORY and hence r r 0 = r ˆr r 0. with ˆr = r r Then We define: and where we used: x = 2 r r 0 r r r 0 r 2 = 1 x 1 x 2 (forx << 1). k kˆr ; then, e ik r r 0 = e ikr e i k r 0 and therefore, e ik r r 0 r r 0 = eikr r e i k r 0. In a scattering process the incident wave is: Ψ 0 ( r) = Ae ikz, which is the solution of the free-particle Schrödinger equation. For large r and using the integral form of the Schrödinger equation, the total wavefunction outside the region of the potential is: Ψ( r) = Ae ikz m e ikr 2π 2 r e i k r 0 V( r 0 )Ψ( r 0 )d 3 r 0, which can be compared to the equation: } Ψ(r,θ,φ) A {e ikz +f(θ,φ) eikr r (large r), so that, This result is exact. f(θ,φ) = m 2π 2 A e i k r 0 V( r 0 )Ψ( r 0 )d 3 r 0. Now, we invoke the Born approximation which corresponds to a weak potential approximation. In the Born approximation, one assumes that the incoming plane wave is not substantially altered by the potential. Then: Ψ( r 0 ) Ψ 0 ( r 0 ) = Ae ikz 0 = Ae i k r 0, with k kẑ.

25 1.6. THE BORN APPROXIMATION 19 i.e we assume that the wavevector keeps the direction of z, while the norm k = k is the norm of the incident wavevector since we have energy conservation. This would be the exact wave function, if V were zero. Otherwise, it is a weak potential approximation. In the Born approximation, f(θ, φ) then becomes: f(θ,φ) = m e i( k k) r 0 V( r 2π 2 0 )d 3 r 0. While the partial wave analysis is useful when the incident particle has low energy (since only the first few terms of the series contribute significantly), the Born approximation applies when the potential is weak compared to the incident energy or, in other words, when the deflection is small. for the low energy scattering (long wavelengths), the exponential factor is essentially constant over the scattering region, and the Born approximation simplifies to: f(θ,φ) m 2π 2 V( r 0 )d 3 r 0. - k points in the incident direction - - k points in the scattered direction - - ( k k) = q: momentum transfer - - 2ksin θ 2 = q for a spherically symmetric potential, V( r) = V(r), the Born approximation can also be reduced. In this case we have: and q k k, ( k k) r 0 = qr 0 cosθ 0. We define the polar axis for r 0 along q. Then, f(θ) m e iqr 0cosθ 0 V(r 2π 2 0 )r0sinθ 2 0 dr 0 dθ 0 dφ 0. Since we obtain: dφ 0 = 2π and π f(θ) 2m 2 q 0 e iqr 0cosθ 0 sinθ 0 dθ 0 = 2sin(qr 0) qr 0, 0 r 0 V(r 0 )sin(qr 0 )dr 0 (1.14) Example: Yukawa potential

26 20 CHAPTER 1. SCATTERING THEORY very rapidly for r >> 1/µ. The Yukawa potential, V(r) = α 2e µr r, (1.15) describes, in a very crude way, the binding force in an atomic nucleus. It was introduced by Yukawa in 1938 to describe the nuclear interaction between protons and neutrons due to the pion exchange. In 1940 Yukawa got the Nobel Prize for predicting the existence of pions. Often this potential is written with α 2 = V 0 µ and 1 µ corresponds to the range of the potential. V(r) goes to zero We first want to understand where the expression Eq comes from. Let us consider the electrostatic potential about a charged point particle: which fulfills the Laplace equation Φ = e 4πǫ 0 r 2 Φ = 0 Φ describes the potential for a force mediated by massless particles, the photons and it has long-ranged behavior. We consider now a relativistic particle with mass m, momentum p and energy E, then: E 2 = p 2 c 2 +m 2 c 4 This equation can be converted into a wave equation by using the correspondence principle: ( 2 2 Φ t = 2 (m2 c 4 2 c 2 2 )Φ E i t, p x i ). x Φ is the potential field (pion field in our case). In the stationary case, ( ) 2 m2 c 2 Φ = 0, which reduces to: for the massless case. For a point source with spherical symmetry: and the differential equation is then: 2 2 Φ = 0, 2 Φ 1 r d 2 dr 2(rΦ), d 2 dr 2(rΦ) = m2 c 2 2 rφ,

27 1.6. THE BORN APPROXIMATION 21 with the solution: Φ = α 2e µr r, where α is the coupling strength and 1 µ = is the range of the potential. This potential mc has the form of a screened Coulomb interaction. We assume now that α is sufficiently small for the Born approximation Eq to be valid. The scattering amplitude f(θ) is then: f(θ) 2mα2 2 q to perform this integral we used: we have that Im[ 0 0 e rµ sin(qr)dr = 2mα2 2 (µ 2 +q 2 ) e µr e iqr dr] = Im 1 µ+iq = q 2 = 4k 2 sin 2θ 2 = 2k2 (1 cosθ) (1.16) q µ 2 +q 2 (1.17) Then, in the first Born approximation, the differential cross section for scattering by a Yukawa potential is given by: σ(θ) = ( 2mα2 ) 2 1 (1.18) 2 [µ 2 +2k 2 (1 cosθ)] 2 Note that as µ 0, the Yukawa potential is reduced to the Coulomb potential, provided that α 2 is fixed, f.i. α 2 = ZZ e 2 in the limiting process. Then the differential cross section in the first Born approximation becomes: σ(θ) = (2m)2 (ZZ e 2 ) k 4 sin 4 (θ/2) which is the Rutherford scattering cross section that can be obtained classically. The term 2mα 2 2 ( m2 c 2 +q 2 ) = 2mα2 m 2 c q 2. 2 can be interpreted as a function of the propagator term, which forr c = 1 and = 1 is: (1.19) 1 q 2 +m 2 It arises from the exchange of a virtual boson (pion) whose rest mass is m The Born series

28 22 CHAPTER 1. SCATTERING THEORY θ = tan 1 (I/p) for small θ p: incident momentum In order to understand the Born approximation, one may compare it with the impulse approximation in classical scattering theory. In the impulse approximation, it is assumed that the particle after scattering continues undeflected. The delivered transverse momentum is calculated as: I = F dt. In the zeroth-order Born approximation, the incident plane wave passes by with no modifications; in the first-order Born approximation, the model is improved by introducing the first-order correction. This idea can be iterated to generate a series of higher-order corrections, which should converge to the exact value. Let us recall the integral form of the Schrödinger equation derived earlier: Ψ( r) = Ψ( r 0 )+ g( r r 0 )V( r 0 )Ψ( r 0 )d 3 r 0, with g( r) = m e ikr, (1.20) 2π 2 r Ψ 0 : incident wave; g( r): the Green s function (multiplied by the factor 2m/ 2 ), g( r) = 2G( r)m 2 ; V: scattering potential. Schematically, Eq. (1.20) can be written as: Ψ = Ψ 0 + gvψ. Now, we take this expression for Ψ and plug it into the integral sign of its own: Ψ = Ψ 0 + gvψ 0 + gvgvψ. In this way, we generate a formal series: Ψ = Ψ 0 + gvψ 0 + gvgvψ 0 + gvgvgvψ (1.21)

29 1.7. COLLISIONS WITH ABSORPTION 23 In each integrand of Eq. (1.21), only the incident wavefunction, Ψ 0, appears together with more and more powers of gv. Diagrammatically, it can be represented as: zeroth order: Ψ 0 is untouched by the potential; first order: Ψ 0 is kicked once and then propagates out in some new direction; second order: Ψ 0 is kicked, propagates to a new location, is kicked again, and then propagates out, and so on. In the context of the Born series, the Green s function is called the propagator (the solution we found for the Yukawa potential was g in momentum space). It describes how the disturbance propagates between one interaction and the next one. The Born series was the original source for Feynman s formulation of relativistic quantum mechanics which is expressed in terms of vertex factors, V, and propagators, g, connected together in Feynman s diagrams. 1.7 Collisions with absorption Up to now we considered elastic scattering of particles. Collisions between particles can also be inelastic and lead to other reactions (creation or destruction of particles, etc.), particularly, if the energy of the incident particles is high. During such processes, some particles disappear : they are said to be absorbed. In the partial wave method, the effect of the potential is to multiply the wavefunction by e 2iδ l. For elastic scattering, δl is real since: the amplitude of the outgoing wave is equal to that of the incoming wave; the probability is conserved during the scattering; the total number of particles is constant. This suggests to consider the absorption phenomena by giving the δ l an imaginary part, we define η l : η l = e 2iδ l, with η l < 1. e 2i(Reδ l+iimδ l ) = e 2iReδ l e 2Imδ l and η l 2 = e 2Imδ l 2

30 24 CHAPTER 1. SCATTERING THEORY Then, we can define the absorption cross section as: σ abs = π (2l+1)[1 η k 2 l 2 ]. If η l 2 = 1 there is no absorption. We had that the elastic cross section was (now expressed in terms of η l ): therefore, the total cross section will be: l=0 σ el = π (2l+1) 1 η k 2 l 2, l=0 σ tot = σ el +σ abs = 2π k 2 (2l+1)[1 R e η l ]. (1.22) l=0 The total cross section is the number of particles which, per unit time, participate in one or another of the possible reactions, divided by the incident flux. Optical Theorem: For θ = 0, we are considering the scattering in the forward direction. f(0) = (2l+1)a l The imaginary part of the elastic scattering amplitude in the forward direction is: l=0 with Then, Imf(0) = (2l+1)Ima l l=0 a l = 1 2ik (η l 1) Imf k (0) = 1 k Comparing Eq and Eq we have: l=0 (2l+1) 1 R eη l 2 (1.23) σ tot = 4π k Imf k(0). which is the optical theorem. It is the interference in the forward direction between the incident plane wave and the scattered wave which accounts for the attenuation of the transmitted beam due to the scattering of particles in all directions of space.

31 1.8. FORMAL SCATTERING THEORY Formal scattering theory So far we considered the scattering problem in the position representation. We represented the incoming particle by a plane wave, and, since this is an eigenstate of the free-particle hamiltonian, p 2 2m Ψ = EΨ, and the collision potential is time independent, we were able to reduce the problem to the stationary case. We will consider now a more abstract representation, which is more general that what we did up to now Propagator theory and the Lippmann-Schwinger equation As we have seen, the scattering problem consists of finding the solution of the Schrödinger equation with particular boundary conditions for r, namely, an incoming wave and anangledependent scatteredwave, Ψ = Ψ in +Ψ scatt, forr. Theproblemisseparated into: 1. a general solution of the Schrödinger equation with open boundary conditions; 2. a particular solution which specifies the appropriate boundary conditions. This can be achieved systematically with the propagator theory. General solution As we are interested in the stationary problem, we consider the Schrödinger equation with fixed E. The resolvent operator ( Green s operator inverse operator) of the Schrödinger equation is defined as the solution of the following Schrödinger equation: H = H 0 +V; P H 0 = 2 : the free Hamiltonian; 2m V: scattering potential; 1: unit operator in Hilbert space. (E H)Ĝ = 1 representation-free form (1.24) Note: we will come to the known equation for the Ĝ with the δ function once we choose a representation. For the free system, the Green s operator is Ĝ0: (E H 0 )Ĝ0 = 1.

32 26 CHAPTER 1. SCATTERING THEORY The full Green s operator, Ĝ, can be expressed in terms of Ĝ0 by multiplying eq. (1.24) from the left with Ĝ0: Ĝ 0 (E H)Ĝ = Ĝ01, Particular solution Ĝ 0 EĜ Ĝ0HĜ = Ĝ0, Ĝ 0 EĜ Ĝ0(H 0 +V)Ĝ = Ĝ0, Ĝ 0 (E H 0 ) Ĝ Ĝ0VĜ }{{} = Ĝ0, 1 Ĝ = Ĝ0 +Ĝ0VĜ. Let s now find the particular solution, Ψ, for the scattering boundary problem: (a) (E H 0 ) Ψ = V Ψ (b) (E H 0 ) Φ 0 = 0 solution of the free problem By substracting (b) from (a), we get: (E H 0 ) Ψ (E H 0 ) Φ 0 = V Ψ, which is then to be multiplied by Ĝ0 from the left side: Formally, Ĝ 0 (E H 0 )( Ψ Φ }{{} 0 ) = Ĝ0V Ψ, 1 Ĝ 0 = Ψ = Φ 0 +Ĝ0V Ψ. (1.25) 1 (E H 0 ) Finally, combining eq. (1.25) and (1.26), we arrive at: Ψ = Φ 0 + which is called the Lippmann-Schwinger equation. Remarks: ( ) (E H 0 )Ĝ0 = 1. (1.26) 1 E H 0 V Ψ, This equation is very similar to the one describing the time evolution of a state in the context of time-dependent perturbation theory. The only difference is that here we consider the stationary case only. Both are called Lippmann-Schwinger equation; the free solution, Φ 0, accounts for the incoming plane wave Ψ in ;

33 1.8. FORMAL SCATTERING THEORY 27 the equations Ĝ = Ĝ0 +Ĝ0VĜ and Ψ = Φ E H 0 V Ψ differ from each other in that in the latter the boundary conditions are implemented via Φ 0. Both equations generate a perturbation series for Ĝ and Ψ, respectively, via iterations: Ĝ = Ĝ0 +Ĝ0VĜ0 +Ĝ0VĜ0VĜ = Ĝ0 +Ĝ0TĜ0 = Ĝ0SĜ0, T : = V +VĜ0V +VĜ0VĜ0V +... T-matrix S : = Ĝ 1 0 +T = E H 0 +T S-matrix Since the matrix elements of S are transition probabilities between incoming and outgoing states, S has a direct relation to quantities that are measured in scattering theory, e.g., scattering amplitude and scattering cross section. With boundary conditions, the wavefunctions can be expressed as: Ψ = Φ +Ĝ0V Φ 0 +Ĝ0VĜ0V Φ = Φ 0 +Ĝ0T Φ Meaning of the Lippmann-Schwinger equation In the Lippmann-Schwinger equation, Ψ = Φ E H 0 V Ψ, Ψ reduces to Φ 0 as V vanishes. However, without prescriptions for dealing with a singular operator 1 E H 0, it has no meaning. Then, in order to make the Lippmann-Schwinger equation more workable, one uses the trick of making E slightly complex: Ψ (±) = Φ E H 0 ±iη V Ψ (±). This is the conventional Lippman-Schwinger equation. In a moment, we will discuss the meaning of ±.

34 28 CHAPTER 1. SCATTERING THEORY Position and momentum representation Position representation The Lippmann-Schwinger equation is a ket equation independent of a particular representation. We now consider the position basis by multiplying it with r from the left: r Ψ (±) = r Φ 0 + d 3 r 1 r E H 0 ±iη r r V Ψ (±). This is an integral equation for scattering because the unknown ket Ψ (±) appears under an integral sign. If Φ 0 stands for a plane wave state with momentum p, we can write: r Φ 0 = ei p r/ (2π ) 3/2 (not normalizable and is not a vector in Hilbert space). The normalization is such that: d 3 r p r r p = δ (3) ( p p ). Momentum representation The Lippmann-Schwinger equation in the momentum representation, p Ψ (±) 1 = p Φ 0 + p V Ψ (±), E ( p 2 /2m)±iη }{{} is derived using the following arguments: 1 = p p p Ĝ 0 p d 3 p (2π) 3e i p r e i p r = δ (3) ( r r ), 1 = (E H 0 )Ĝ0 p 1 p = p (E H 0 ) p p p Ĝ0 p δ p p = }{{} diagonal p (E p ) 2 δ p p 2m }{{} diagonal Ĝ 0 p p. Solving for Ĝ0 p p : Ĝ 0 p p = 1 E ( p 2 /2m)±iη. Since Ĝ0 p p is diagonal in p-representation, it is often identified with its diagonal element: Ĝ 0 p (E) = 1 E ( p 2 /2m)±iη.

35 1.8. FORMAL SCATTERING THEORY Retarded and advanced Green s functions Physical significance of the poles in Ĝ: E = ω, Ĝ (±) 0 p (E) = 1 E p2 2m ±iη, p 2 2m = ǫ p E = ǫ p are the energy eigenvalues of H! Ĝ(±) 0 p ( ω) = 1 ω ǫ p ±iη, To treat the poles, e.g., in the integration over E, they are shifted in the complex plane E away from the real axis by an infinitesimal positive or negative imaginary part ±iη in the form: The relevance of ±iη becomes clear by transforming the Lippmann-Schwinger equation into the time domain: p Ψ (±) (t) = p Φ 0 (t) + dt Ĝ (±) 0 p (t t ) p V Ψ (±) (t ). (1.27) We Fourier transform Ĝ(±) 0 p (t t ): Ĝ (±) 0 p (t t ) = = dω ) Ĝ 2π e iω(t t 0 p ( ω) dω e iω(t t ) 2π ω ǫ p ±iη. The integral is calculated with the method of residues, f(z) (z z 0 ) dz = 2πif(z 0), as:

36 30 CHAPTER 1. SCATTERING THEORY Ĝ (±) 0 p (t t ) = { iθ(t t )e i ǫp(t t ) iθ(t t )e i ǫp(t t ) +iη; iη. Using this result, we rewrite eq. (1.27): p Ψ (±) (t) = p Φ 0 (t) + + dt Ĝ r/a 0 p (t t ) p V Ψ (±) (t ). Ĝ r/a is the retarded/advanced propagator because it propagates the wavefunction from t to t with t > t (retarded, causal) or t < t (advanced, acausal). Position representation Let us consider the position basis: or, explicitly, Ĝ (±) ( r, r ) 2 2m r 1 E H 0 ±iη r, Ĝ (±) ( r, r ) = 1 e ±ik r r 4π r r, with E 2 k 2 2m. (1.28) Remember that we already had such an expression! Let us derive eq. (1.28). 2 2m r 1 E H 0 ±iη r = 2 2m where H 0 acts on p. Now, d 3 p d 3 p r p p 1 ( ) p p r, E ±iη p 2 2m (1.29) p 1 ( ) p = δ(3) ( p p ) ( ), E ±iη E ±iη p 2 2m r p = ei p r/ (2π ) 3/2, p r = e i p r / (2π ) 3/2. p 2 2m

37 1.8. FORMAL SCATTERING THEORY 31 The integral in eq. (1.29) thus becomes: 2 2m d 3 p (2π ) 3 E i p ( r r )/ e ( ) p 2 2m. ±iη We write and set We can now perform the integral: E = 2 k 2 2m p q. We used: 1 2π q 2 dq (2π) 3 0 = 1 8π 2 1 i r r = 1 e ±ik r r 4π r r dφ d(cosθ)e i q r r cosθ 1 k 2 q 2 ±iη dqq ( e iq r r e ) iq r r = Ĝr/a 0 ( r, r ). q 2 k 2 iη ( ) iη q = ±k 1± ±k ±iη. This is the Green s function of the Helmholtz equation, ( 2 +k 2) Ĝ (±) ( r, r ) = δ (3) ( r r ), k 2 r Ψ (±) = r Φ 0 2m 2 d 3 r e ±ik r r 4π r r r V Ψ (±), which is the equation we have already derived! Eq Interpretation of the Lippmann-Schwinger equation and its expansion in terms of Feynmann s diagrams Ĝ = Ĝ0 + dr Ĝ 0 ( r, r )V( r )Ĝ( r, r )

38 32 CHAPTER 1. SCATTERING THEORY