4 Scattering Theory

Exercise 4.1: Delta-function potential

Consider a one-dimensional particle of mass m subject to the potential

V (x)= ~uδ(x). − (a) Show that the general solutions of the time-independent Schr¨odinger equation at energy ε = ~2k2/2m can be cast in the form

1 eikx + r(k)e−ikx x< 0, ψk(x) = √2π × t(k)eikx x> 0, 1 e−ikx + r′(k)eikx x> 0, ψ′ (x) = k √2π × t′(k)e−ikx x< 0, and ﬁnd explicit expressions for the amplitudes r(k), t(k), t′(k), and r′(k). What is the interpretation of the amplitudes r(k), t(k), t′(k), and r′(k)? (b) Show that one may, alternatively, cast the general solution to the stationary Schr¨odinger equation into the form 1 ψS,k(x) = cos(k x + δS), √π~v | | 1 ψA,k(x) = sign(x)sin(k x + δA), √π~v | |

with v = ~k/m. Calculate the “phase shifts” δS and δA. Can you explain your result for δA?

Exercise 4.2: Adiabatic switching on of interaction

The construction of the stationary retarded scattering state Ψ relies on a procedure in which the interaction Vˆ is adiabatically switched on, Vˆ (t) = |Veˆ ηt/~, where η is a positive inﬁnitesimal with the dimension of energy. The stationary retarded scattering state obeys the Lippmann-Schwinger equation

Ψ(t) = Φ(t) + Gˆ Veˆ ηt/~ Ψ(t) , (1) | | 0 | 1 where Gˆ =(E Hˆ +iη)−1 is the Green function or resolvent for the system with Hamiltonian 0 − 0 Hˆ0, i.e., in the absence of the interaction Vˆ , and Φ is the eigenstate of Hˆ0 that describes the initial state. This equation was derived under the| assumption that Ψ is a true stationary state, with a time dependence e−iEt/~. Clearly, this assumption can| not be strictly valid no matter how small η is chosen,∝ because the state Ψ(t) that solves Eq. (1) evolves from the initial state Φ(t) for t to the retarded| scattering state for times t 0, which is different from |Φ(t). The→ purpose −∞ of this exercise is to show, for a specific exampl∼ e, that | nevertheless the solution to Eq. (1) is the valid solution of the time-dependent Schrödinger equation with interaction Vˆ (t) = Veˆ ηt/~ in the limit η 0, and you will systematically include corrections to this solution for small but finite η. ↓ Hereto, we consider a particle of mass m in one dimension, subject to the potential V (x)eηt/~ = ~uδ(x)eηt/~. The wavefunction φ(x,t) of the initial state is taken to be − ~ 1 φ(x,t)= φ(x)e−iEt/ , φ(x)= eik0x, √2π

−1 where k0 = ~ √2mE. (a) Show that the Lippmann-Schwinger equation (1) gives a closed equation for the wavefunction ψ(0,t) at x = 0. Solve that equation and show that the full solution of Eq. (1) reads ηt/~ 1 −iEt/~ ik0x mue ikη|x| ψ(x,t)= e e ηt/~ e , (2) √2π − i~kη + mue 1 where kη = ~ 2m(E + iη).

Hint: The Green function Gˆ0 in position representation reads

′ m ′ ˆ ′ ikη|x−x | G0(x,x ; kη) = x G0 x = e 2 . | | i~ kη

(b) Show that in the limit η 0 (at ﬁxed x and t) your solution agrees with the retarded scattering state ψ(x,t) that↓ is obtained from a direct solution of the Schr¨odinger equation with a time-independent potential.

(c) The limit η 0 at ﬁxed x and t obscures the switching on of the potential. In order to see the evolution↓ of the wavefunction while the potential is switched on, one should take the limit η 0 while keeping the product ηt ﬁxed. Apply this limiting procedure to the solution to↓ the Lippmann-Schwinger equation (1) given in part (a) and interpret your answer.

2 The Lippmann-Schwinger equation (1) was derived from the time-dependent equation

t i ˆ ′ ~ ′ −iH0(t−t ) ˆ ηt/ ′ ψ(t) = φ(t) ~ dt e Ve ψ(t ) (3) | | − −∞ |

′ under the assumption that ψ(t) is a stationary state, ψ(t′) = ψ(t) eiE(t−t )/~. This assumption will not be made| in the remainder of this exercise| . |

(d) Specialize to the case of the delta-function potential, and show that Eq. (3) implies a closed (integral) equation for the full function ψ(0,t) of the wavefunction of the retarded scattering state at x = 0.

For a solution for the full wavefunction ψ(x,t) it is suﬃcient to ﬁnd a solution for the function ψ(0,t). This can be done using the ansatz

ψ(0,t)= e−iEt/~f(ηt; η), (4) where we anticipate that the function f, which describes to what extent the time dependence of ψ(0,t) deviates from that of a stationary function, varies with time t on the slow scale ~/η. One then solves for the function f as a power series in η,

f(ηt; η)= f (0)(ηt)+ ηf (1)(ηt)+ .... (5)

(e) Show that truncating the expansion at zero order reproduces the solution for ψ(0,t) you obtained from the Lippmann-Schwinger equation (1). Hint: Taylor expand f(ηt′; η) around t′ = t in order to transform the integral equation for f you obtained in (d) into an ordinary equation.

(f) Obtain an equation for the ﬁrst-order correction f (1), and solve that equation. Is the correction to f (0) signiﬁcant in the limit η 0? ↓ Exercise 4.3: Lippmann-Schwinger equation for advanced scattering states

R The stationary retarded scattering state Ψi (or simply Ψi ) is related to the stationary (free) initial state Φ through the Lippmann-Schwinger| equation| | i −1 R ˆ ˆ R Ψi = Φi + Ei H0 + iη V Ψi | | − | −1 = 1+ Ei Hˆ + iη Vˆ Φi , − | 3 R where η is a positive inﬁnitesimal. Show that the stationary advanced scattering state Ψf is related to the stationary (free) ﬁnal state Φ through the related equation | | f −1 A ˆ A Ψf = Φf + Ef H0 iη V Ψf | | − − | −1 = 1+ Ef Hˆ iη Vˆ Φf . − − |

Exercise 4.4: Optical theorem

The generalized optical theorem reads

T T ∗ = 2πi δ(E E )T T ∗ ba − ab − − i bi ai i = 2πi δ(E E )T T ∗ with E = E = E . − − f fa fb a b f Here the labels a and b specify initial and ﬁnal states.

(a) Show that the generalized optical theorem follows from the relation

T = Φ Vˆ ΨR ba b| | a R between the T matrix, the free state Φb , the scattering state Ψa and the interaction V , and the Lippmann-Schwinger equation.| |

(b) Show that the optical theorem,

Im T = π δ(E E ) T 2, ii − i − f | ﬁ| f follows from the generalized optical theorem.

S = δ 2πiT δ(E E ), ba ba − ba b − a is unitary.

4 Exercise 4.5: Analicity and Causality

In complex analysis you learn that the integral of a function f(z) over a contour γ in the complex plane can be deformed in those regions where f(z) is analytic. Speciﬁcally, you learn that (according to the residue theorem) the contour integral over a closed path γ of a function f(z), which is analytic up to some isolated points z ,...,z where f(z) has poles, { 0 N } can be calculated from the residues of f(z) at those points which are enclosed by γ,

γ dz f(z)=2πi( 1) Resz f, − i γ i=0,...,n where ( 1)γ = 1( 1) if γ encircles the enclosed poles z ,...,z counter-clockwise (clock- − − { 0 n} wise). The residue Reszi f is deﬁned through the Laurent expansion of f(z) near z = zi, ∞ f(z)= a (z )(z z )n, Res f = a (z ). n i − i zi −1 i n=−k(zi) If the coeﬃcient a = 0, the integer k(z ) is referred to as the “order” of the pole at −k(zi) i z = zi. (a) Calculate the integral ∞ 1 I(z0,z1)= dx , (x z )(x z ) −∞ − 0 − 1 where z0 and z1 are complex numbers with Im z0 > 0 and Im z1 < 0.

(b) Repeat the calculation of I(z0,z1) for complex numbers z0 and z1 with Im z0 > 0 and Im z1 > 0. (c) Calculate the integral

∞ −iE(t−t′)/~ ± ′ dE e F (Ek,t t )= , − 2π E E iη −∞ − k ± distinguishing the cases t>t′ and t

5 (e) Show that F +(E ,t t′)=[F −(E ,t′ t)]∗. k − k −

Exercise 4.6: One-dimensional free particle propagator

In the lecture you learned that the retarded, free particle propagator in a one dimensional system in the position-represenatation is

ik|x−x′| ′ −1 ′ e G0(x,x ; k)= x (Ek Hˆ0 + iη) x = , | − | ivk~

2 with k = 2mEk/~ and vk = ~k/m. (a) Show that the propagator satisﬁes the equation

~2 E + ∂2 G (x,x′; k)= δ(x x′). k 2m x 0 −

−1 (b) Show that the retarded propagator Gˆ0 =(Ek Hˆ0 + iη) is diagonal in the k representation, − δ(k k ) k Gˆ (k) k = G (k , k ; k)= 1 − 2 . 1| 0 | 2 0 1 2 E E + iη k − k1 Since Gˆ is diagonal in the k representation, we write G (k′; k) G (k′, k′; k). 0 0 ≡ 0 ′ ′ (c) Derive G0(x,x ; k) by Fourier transform of G0(k ; k),

∞ ′ ′ dk ′ ik′(x−x′) G0(x,x ; k)= G0(k ; k)e . −∞ 2π Hint: Use contour integration, see Ex. 4.5.

(d) Calculate the advanced propagator in position representation.

Exercise 4.7: Born series for the T matrix

6 Consider a one-dimensional system described by the Hamilton-operator ~2 Hˆ = ∂2 + V (ˆx) 2m x where V (x) = ~uδ(x) describes the scattering from a point-scatterer. Periodic boundary conditions with− period L are applied.

′ (a) Calculate the T -matrix elements Tkk′ = k Tˆ(Ek) k for the initial and ﬁnal states ′ | | k and k , corresponding to plane waves in position representation, φk(x)= x k = |L−1/2eikx| . For the calculation you may use the free particle propagator in the position- | representation, see Ex. 4.6, and the series expansion for the T -matrix,

∞ k Tˆ(Ek) = Vˆ [Gˆ0(Ek)Vˆ ] k=0

= Vˆ + Vˆ Gˆ0(Ek)Vˆ + Vˆ Gˆ0(Ek)Vˆ Gˆ0(Ek)Vˆ + ....

(b) Verify that the optical theorem,

Im T = π δ(E E ) T 2, ii − i − f | ﬁ| f is fulﬁlled.

(c) Express Tkk′ δ(Ek Ek′ )asa2 2-matrix in a basis of left- and right-moving states φ = k and φ′−= k , with× k > 0. | k | | k |− (d) If you express the delta function δ(E E ′ ) in terms of δ ′ and δ ′ , the matrix k − k kk k,−k elements you calculate in part (c) are dimensionless. From the T -matrix, derive the scattering matrix. What are the reﬂection and transmission coeﬃcients? Verify uni- tarity of the scattering matrix.

Exercise 4.8: One-dimensional scattering problem

Consider a one-dimensional particle of mass m subject to the potential if x< 0, V (x)= ∞ V0δ(x a) if x> 0, − 7 where a> 0, shown schematically in the ﬁgure below. Without the delta-function potential at x = a (i.e., for V0 = 0), the eigenstates of the Hamiltonian are of the form 1 φ (x)= e−ikx eikx , E √ ~ − 2π v with k = 2mE/~2 and v = ~k/m. These states are normalized with respect to energy,

∞ ∗ ′ ′ dxφE(x)φE (x)= δ(E E ). 0 −

V 0 a x

The wavefunction with the delta-function potential is denoted ψE(x). For x>a, ψE(x) can be written as 1 ψ (x)= e−ikx e2iδ(E)eikx , E √ ~ − 2π v where δ(E) is the “scattering phase shift”.

(a) The wavefunction ψE(x), as it is written above, represents a (retarded) scattering state. Can you explain why?

(b) Argue that the phase shift δ(E) is related to the T matrix T (E) of the delta-function potential as 1 T (E)= eiδ(E) sin δ(E). −π

(d) Construct the Lippmann-Schwinger equation for ψE(x). Solve your equation and ﬁnd an expression for δ(E).

Exercise 4.9: One-dimensional analogue of l =0 scattering

Show that any l = 0 scattering problem with a central potential V (r) in three dimensions is formally identical to that of symmetric states in a one-dimensional problem with the symmetric potential V ( x )= V (r). | | 8 Exercise 4.10: Particle in a spherically symmetric potential well

Consider a spinless particle of mass m in a central potential V (r) equal to

V if 0 r

~2 εψ(r)= ∂2ψ(r)+ V (r)ψ(r) −2m r

of the form ψ(r)= Ylm(θ,φ)Rlε(r), with Ylm(θ,φ) a spherical harmonic.

(a) Give the explicit form of the differential equation for the radial wavefunction Rl(r). (b) We restrict ourselves to eigenstates with m = l = 0 and positive energies ε> 0. Give an expression for the radial wavefunction R0ε(r) in this case. Compare the asymptotic form of R0ε(r) at large r to that of the spherical Bessel functions and give an expression for the scattering phase shift δ0(ε). (c) If the energy ε is sufficiently low, eigenstates with l> 0 are not affected by the potential well at the origin. What condition does ε have to satisfy for this to be the case? Does it matter whether V0 is positive or negative?

Exercise 4.11: Friedel sum rule

Consider a finite but large volume of linear dimension L which contains one spinless particle of mass m. Near the center of the volume (which we take as the origin of our coordinate system) there is a potential V (r), which falls off faster than 1/r upon going away from the center. Both with and without the potential V the particle’s energy spectrum is discrete. The cumulative number of states N(E) is defined as the number of discrete states with energy lower than E. The presence of the potential V changes the cumulative number of states by an amount δN(E) with respect to the situation without potential V .

9 (a) If the potential V (r) is central, i.e., V (r) depends on r only, show that, in the limit L , → ∞ ∞ (2l + 1)δ (E) δN(E)= l . (6) π l=0 This result is known as the Friedel Sum rule. Hint: You are free to choose the shape of the volume, as long as you show that your result for δN(E) converges in the limit L . → ∞ (b) Find an explicit expression for δN(E) for a Breit-Wigner resonance with width ~Γ and resonance energy E0. (c) Can you ﬁnd a formulation of the Friedel Sum rule for the case that the potential V (r) is not central? Hint: Use the fact that the scattering matrix for scattering from an arbitrary potential V (r) is unitary, so that there exists a basis of scattering states in which it is diagonal.

Exercise 4.12: Wigner time delay

Measuring time in quantum mechanics is problematic, since there is no such thing as a “time operator”. Wigner has shown that the scattering phase shifts can be used to define the time a particle is delayed because of interactions with a scatterer. In order to understand Wigner’s idea in the simplest possible setting, we consider a one- dimensional example. We compare two potential profiles labeled V1 and V2. Both potentials vanish for x> 0, V1(x)= V2(x)=0 if x> 0. The potentials differ for x< 0. For the first potential, we take

V (x) if x< 0. 1 → ∞ For the second potential, we take an arbitrary dependence on x, but with the condition that V2(x) if x . A schematic picture showing both potentials is shown in Fig. 1. Obviously,→ ∞ a→ particle −∞ incident from the right will reflect off either potential. However, the time after which it returns will be longer in the case of potential V2 than in the case of potential V1. The time the particle reflecting off potential V2 lags behind a particle reflecting off potential V1 is referred to as the “Wigner time delay” τW.

10 V V 1 2

0 x 0 x

Figure 1: Schematic drawing of the potentials V1 and V2. The time a particle incident from the right and reflecting off potential V2 lags behind a particle that reflects off potential V1 is known as the “Wigner time delay”.

(a) Model the incoming particle by a wavepacket, and ﬁnd an expression for τW in terms of the derivative of the scattering phase shift δ to the particle’s energy E.

(b) What is the maximal allowable energy uncertainty for the expression you derived under (a) to make sense? What does this imply for the relation between the minimal width of the wavepacket in the temporal domain and the delay time?

Note 1: Besides giving a quantitative relation between the energy-derivative of the scattering phase shift and the delay time, the result you ﬁnd under (a) proves a very important qualitative result: The scattering matrix is a fast function of energy if the projectile spends a long time in the scattering region, and it is a slow function of energy if the projectile spends only a short time in the scattering region. This duality between time and energy is closely related to the Heisenberg uncertainty principle. Note 2: Your analysis of part (b) shows that the Wigner delay time has little practical relevance as a delay time for particles that are well localized in the time domain. However, it plays an important role in solid state physics, where it is closely related to the “density of states”, a quantity that can be measured, e.g., using scanning probe techniques.

Exercise 4.13: Resonances

This exercise addresses properties of scattering states for resonant scattering inside the scattering region. Although the conclusions of this exercise will be general, we restrict our

11 calculations to the double barrier potential V (x) = ~u[δ(x + a)+ δ(x a)] with u v, where v is the projectile’s velocity. − − | | ≫

(a) Use the Lippmann-Schwinger equation to show that the phase shifts δA and δS for antisymmetric and symmetric states scattering oﬀ the double barrier potential are given by

2u cos(ak) iveiak e2iδS = e−2ika − , − 2u cos(ak)+ ive−iak 2u sin(ak) veiak e2iδA = e−2ika − . 2u sin(ak) ve−iak − (b) Show that the scattering from a double barrier has Breit-Wigner resonances if u v, ﬁnd an expression for the resonance width ~Γ, and show that ~Γ is much smaller| | ≫ than the spacing between resonance energies.

(c) Calculate the integral I = a dx ψR(x) 2 for symmetric or antisymmetric scattering −a | | states, as a function of the wavenumber k of the initial state. How do you interpret your result if k is oﬀ resonance? And how do you interpret your result if k is on resonance?

Exercise 4.14: Combining scattering matrices

Consider two spatially separated scattering potentials V1 and V2 in one dimension, as in Figure 2. Each potential can be characterized by a scattering matrix. Using a basis of left-moving and right-moving plane waves with normalization

′ ′ ′ ′ σ ,E σ, E = δ ′ δ(E E ), σ,σ = , | σσ − ± the scattering matrices of the two scatterers can be written as

′ ′ t1 r1 t2 r2 S1(Ef ; Ei)= ′ δ(Ef Ei), S2(Ef ; Ei)= ′ δ(Ef Ei), (7) r1 t1 − r2 t2 − where the matrix structure refers to the left-moving/right-moving indices σf and σi.

(a) Give an interpretation of the matrix elements in the above expressions for S1 and S2.

12 VV 1 2

Figure 2: Two spatially separated scattering potentials. The scattering matrix for the combined potential can be expressed in terms of the individual scattering matrices S1 and S2 of the separate potentials.

(b) Show that the two 2 2 matrices appearing in the expressions for S and S are unitary. × 1 2 (You may use the fact that S1(Ef ,Ei) and S2(Ef ,Ei), as they appear on the left-hand side of Eq. (7), are unitary operators.)

(c) Instead of treating both potentials separately, one may treat them as a single potential V = V1 + V2. Find an expression for the elements of the scattering matrix S of the two potentials combined in terms of the scattering matrices S1 and S2.

(d) Apply your expressions to the example of the double barrier, V1 = ~uδ(x + a), V = ~uδ(x a), and compare your answer with what you ﬁnd from a direct− solution 2 − − of the Lippmann-Schwinger equation for this problem.

Exercise 4.15: Advanced scattering states: ﬁction or reality?

Advanced scattering states play an important role in the formal scattering theory. An advanced scattering state has the appearance of a standard (retarded) scattering state, but with time running backward: the advanced scattering state satisﬁes a boundary condition for t ; its form for t is left to be determined by the dynamics of the scattering region.→ ∞ → −∞ Several experimental groups have tried to produce such advanced scattering states in the laboratory. Such attempts used classical waves, such as sound, not quantum mechanical waves, because classical waves are easier to control. In particular, for classical waves, it is possible to measure both the magnitude and the phase of the wavefunction. Still, the mathematics behind the experiment is the same as in the quantum mechanical case. You can ﬁnd a review of such an experiments in, e.g., M. Fink and C. Prada, Inverse Problems 17, R1-38 (2001).

13 input

output

Figure 3: Setup for creating an advanced scattering state with an acoustic cavity (left). Sound waves reﬂect multiple times oﬀ the cavity walls, so that an incoming pulse is stretched and deformed upon transmission (right).

In a typical experiment, a sound pulse is emitted from a speaker, scattered, and the scattered signal is then recorded by an array of microphones. The signal on the microphones is then played backward on an array of speakers and returned to the scatterer. In the end, a sound pulse emerges, propagating precisely in the direction of the original source. In this exercise you are asked to analyze a such an acoustic experiment. In order to keep the problem manageable, we consider an acoustic cavity that is connected to two acoustic waveguides that you may consider as being effectively one dimensional. In each waveguide there are speakers and microphones placed at equal distance from the cavity, see the figure below. After entering the cavity, a sound wave undergoes many reflections off the cavity walls before it exits the cavity through one of the waveguides. At each reflection, only a small fraction of the wave (corresponding to that part of the wavefront that overlaps with the opening of a waveguide) exits the cavity. This way, an incoming sound pulse is distorted and smeared out over a very long time interval. First we consider the situation in which the speakers generate sound waves at a single frequency ω. If the speaker in the left waveguide produces a sound wave with unit amplitude traveling towards the acoustic cavity, the amplitudes of the reflected and transmitted waves, as they are recorded by the microphones, are r(ω) and t(ω), respectively. Similarly, if the speaker in the right lead produces a sound wave with unit amplitude, the microphones in the right and left leads record waves with complex amplitude r′(ω) and t′(ω), respectively. (a) Argue that the matrix r(ω) t′(ω) S(ω, ω′)= δ(ω ω′) (8) t(ω) r′(ω) − serves as the “scattering matrix” of the acoustic cavity.

14 (b) What signal should one feed into the cavity in order to have an outgoing wave in the left channel only? The wave created this way is an example of an advanced scattering state.

The creation of an advanced scattering state is more spectacular if one works with sound pulses, rather than continuous waves. Even if the input signal is a short pulse, the output signal from the cavity is stretched out in space, because the sound waves undergo many reﬂections inside the acoustic cavity. Thus, for an advanced scattering state, the input signal will be stretched out over a long time period, whereas the output signal “magically” shrinks to a short pulse. The input signal for a pulse at center frequency ω and duration τ has a wavefunction

′ 2 2 ′ ψ(t) dω′e−(ω −ω) τ /4e−iω t. (9) ∝ (c) If a pulse with wavefunction (9) is emitted from the speaker in the left waveguide, what signal is recorded at the two microphones? Express your answer in terms of the (elements of the) scattering matrix S.

(d) What input signal do the two speakers in the waveguides need to emit in order to ensure that the output signal is precisely equal to the pulse (9)?

(e) In a real experiment, inevitably a part of the signal that leaves the cavity will be lost. This may happen, for example, because the array of detectors does not cover all directions in space, or because waves are absorbed inside the scattering region. In order to understand the eﬀect of losses, consider what happens in the situation you considered in parts (c) and (d) if the output signal is recorded (and played back) in the right waveguide only. Upon playing the recorded signal backward, do you still expect to see a narrow pulse in the left waveguide as output?

Exercise 4.16: Electrical transport through mesoscopic devices

The problem of electrical conduction can be formulated as a scattering problem. In this way, concepts from scattering theory can be applied in electronic circuits. Here we discuss a version of the scattering approach to electronic conduction that neglects the interactions between the conduction electrons. (The justiﬁcation why this can be done is provided by Landau’s “Fermi Liquid Theory” — see any course on solid state physics for details.)

15 We consider a two-dimensional sample that is connected to electrodes via “ideal leads”, see ﬁgure 4. (Two-dimensional electron gases exist in certain semiconductor heterostruc- tures.) We use coordinates x and y for the longitudinal and transverse directions in the leads, respectively, such that electrons moving towards the sample move in the positive x direction.

(+) y φ (−) φ x

Figure 4: Conducting sample (dotted) connected to two semi-inﬁnite ideal leads. The Landauer formula relates the sample’s electrical conductance to its scattering matrix s.

2 In the ideal leads, the conduction electrons are described by the Hamiltonian H0 = p /2m. The wavefunction satisﬁes the boundary condition ψ(r) = 0 at the lead boundaries at y =0 and y = W , where l is an index that labels the leads.1 The initial and ﬁnal states Φ(±) l | n,l,k are plane wave states in the leads. Their wavefunction is

±iknx 2 (±) e nπy 2 nπ 2 φn,l,k(r)= sin , r in lead l, kx + = k , (10) √2πvn Wl Wl 2 2 where Wl is the width of the lead, vn = ~kn/m, and E = ~ k /2m. The label “+” refers to electrons moving towards he sample and the label “ ” refers to electron moving away from (±) − the sample. The wavefunction φn,l,k vanishes zero outside lead l.

(a) Show that n can take the values n =1,...,Nl(k), where

Nl(k) = int(kW/π). (11)

Nl(k) is known as the “number of propagating modes” or “number of channels” in lead l at energy E = ~2k2/2m. (The integer part “int(x)” is the highest integer number smaller than or equal to x.)

1For a conduction experiment one needs at least two leads. However, it is possible to do meaningful experiments with samples connected to one lead only.

16 R (b) The scattering states Ψn,l,k are solutions of the Schrödinger equation for the full system (ideal leads and| sample) with the boundary condition that the incoming wave R R (−) component of Ψn,l,k is equal to that of Φn,l,k . Similarly, the scattering state Ψn,l,k is a solution of| the Schrödinger equation| such that its outgoing wave component| is equal to that of Φ(−) . The scattering matrix S(n′,l′, k′ n,l,k) is defined as | n,l,k | ′ ′ ′ (−) R S(n ,l , k n,l,k)= Ψ ′ ′ ′ Ψ . (12) | n ,l ,k | n,l,k Show that the scattering matrix takes the form

′ ′ ′ ′ S(n ,l , k n,l,k)= s ′ ′ δ(E E ), (13) | n ,l ;n,l k − k

where s is a unitary matrix of dimension N = l Nl. In the quantum transport literature, s (not S) is referred to as the scattering matrix. (c) For a sample with M leads, s can be decomposed as

r11 t12 ... t1M  t21 r22 ... t2M  s = . . . , (14)  . . .     tM1 tM2 ... rMM    where r and t are matrices of size N N and N N , respectively. Can you jj ij j × j i × j interpret the matrices rjj and tij appearing in this decomposition?

The Landauer formula states that the conductance G of the sample with two leads M = 2 reads 2e2 G = tr t t† . (15) h 12 12 Many other transport properties can also be expressed in terms of the scattering matrix s.

Exercise 4.17: Scattering of spin 1/2 particles

This exercise deals with elastic scattering of spin 1/2 particles from a spinless target. You are asked to consider scattering of spin 1/2 particles from a target with an interaction that is time-reversal symmetric and spin dependent. An example of a spin-dependent interaction that obeys time-reversal symmetry is the spin-orbit interaction. We consider a situation in

17 which the initial and ﬁnal states are labeled by a discrete quantum number n = 1,...,N, and energy, and write the scattering matrix as

′ ′ ′ S(n , k n, k)= s ′ δ(E E ), (16) | n ,n k − k 2 2 were Ek = ~ k /2m. The N N matrix s is unitary. An example of such a situation is given in the previous exercise,× where the index n labels the leads and/or transverse modes inside the leads. Another example would be a situation in which the target only couples to N different angular momentum states of the projectile. For spin 1/2 particles, the initial and final states are spinors. Since the elements of s connect initial and final spin 1/2 states, they can be seen as operators in spin space. Hence we write sn′n(s), where s is the spin operator. In the absence of the interaction with the target, spin is conserved, so that the “free” initial or final states can be chosen with definite spin (i.e., spin up or spin down). Without loss of generality, we may choose the wavefunctions of these free states to be real. Because the time-reversal operation amounts to complex conjugation of the orbital part of the wavefunction the time-reversal operation leaves the wavefunctions of these initial and final states unchanged. The time-reversal operation does, however, change the spin of the initial and final states.

(a) For this basis of initial/ﬁnal states, show that time-reversal invariance implies that

s ′ (s)= s ′ ( s). n ,n n,n −

(b) What does the condition sn′,n(s)= sn,n′ ( s) imply for the case N = 1 in which there is only one scattering channel? −

(c) What does the condition sn′,n(s)= sn,n′ ( s) imply for the case N = 2 in which there are two scattering channels? Which elements− of s are spin dependent, and which ones are not? This is the case relevant for electronic transport in a sample with two leads.

(d) If the initial state is in channel “1” and has no spin polarization, can the outgoing part of the beam in channel “2” have a spin polarization if N = 2? (In the context of electronic transport, this question reads: Can one use the spin-orbit interaction in a two-terminal device to create a spin-polarized current from an unpolarized source?)

(e) If the initial state is in channel “1” and has no spin polarization, can the outgoing part of the beam in channel “2” have a spin polarization if N = 3? (In the context of electronic transport, this question reads: Can one use the spin-orbit interaction in a three-terminal device to create a spin-polarized current from an unpolarized source?)

18 Exercise 4.18: Spin-orbit interaction

Consider a spin 1/2 particle subject to the Hamiltonian Hˆ = Hˆ0 + Vˆ , with p2 Hˆ = , Vˆ = V (r)+ ~−1V (r)σˆ ˆl. 0 2m pot so

Here σˆ = (ˆσx, σˆy, σˆz) and ˆl is the orbital angular momentum of the particle. The potentials Vpot(r) and Vso(r) have a ﬁnite range. In parts (a) and (b) below you are asked to construct a basis of eigenfunctions of Hˆ that 2 2 2 2 are also eigenfunctions of the operators ˆj , ˆl , and ˆjz, at eigenvalues ~ j(j + 1), ~ l(l + 1), and ~m, respectively, where j = l 1/2. The eigenvalue of Hˆ is written as ~2k2/2m. In the remainder of this exercise you then± use these eigenfunctions to construct the scattering state that corresponds to plane-wave initial conditions. In this problem, quantum states will be represented using spinors. (a) The spinor eigenfunction ψ (r)= ( r,σ j,l,m,k ) σ j,l,m,k | | σ

factorizes into a scalar wavefunction Rjlk(r) that depends on the radial coordinate r only and a spinor part that depends on the polar angles θ and φ. Show that the radial wavefunction Rjlk(r) satisﬁes the equation 1 l(l + 1) 2m d2r + V (r) R (r)= k2R (r), (17) −r r r2 − ~2 jl jlk jlk with Vpot(r)+ lVso(r) if j = l +1/2, Vjl(r)= Vpot(r) (l + 1)Vso(r) if j = l 1/2. − − The radial wavefunctions R (r) are normalized such that, asymptotically, for r , jlk → ∞ 1 R (r) e−i(kr−lπ/2) ei(kr−lπ/2)+2iδl±1/2,l(k) . l±1/2,l,k ∼−2ikr − Here δl±1/2,l(k) is a real “phase shift”, which will be determined by the explicit solution of Eq. (17) above. Hint: Make use of the identity ˆj2 = ˆl2 + ~ˆl σˆ +3~2/4 to write the potential Vˆ in terms of ˆj2 and ˆl2.

19 (b) Explain why the radial wavefunction Rjlk(r) does not depend on the magnetic quantum number m.

Using the expansion of eikz in spherical harmonics, this can be rewritten as

1 ∞ φ (r)= 4π(2l + 1)eiπl/2j (kr)Y (θ,φ) s . (18) i (2π)3/2 l l0 | i l=0

2 2 Each term in the expansion (18) is an eigenfunction of l and lz, but not of j and jz. Since 2 the radial wavefunctions Rjlk(r) you considered in (a) depend on the eigenvalue ~ j(j + 1) 2 2 of j , you must rewrite the expansion (18) in terms of eigenfunctions of j . As the Rjlk(r) do not depend on the eigenvalue ~m of jz, it is not necessary that the functions in your expansion are also eigenfunctions of jz. (c) Show that the operators

~(l +1)+ σˆ ˆl ~l σˆ ˆl Pˆ = , Pˆ = − + ~(2l + 1) − ~(2l + 1)

project the spinor wavefunctions Yl0(θ,φ) s onto their components at total angular momentum j = l 1/2. | ± R (d) Argue that the scattering state Ψi corresponding to the initial state Φi has spinor wavefunction | |

1 ∞ ψR(r)= 4π(2l + 1)eiπl/2R (r)P Y (θ,φ) s . i (2π)3/2 l±1/2,l,k ± l0 | i l=0 ± Hint: Compare the asymptotic r dependence for r of the radial wavefunctions → ∞ Rjlk(r) with that of the spherical Bessel functions jl(kr), 1 j (kr) e−i(kr−lπ/2) ei(kr−lπ/2) . l ∼−2ikr −

20 Asymptotically, for r , the spinor wavefunction ψR(r) can be written in the form → ∞ i 1 eikr ψR(r)= eikir + fˆ(k , k ) s , i (2π)3/2 f i r | i with ki = kez, where the “scattering amplitude” fˆ(kf , ki) is an operator in spinor space. (The scattering amplitude is related to the T matrix through the standard relation Tˆ(kf , ki) = ~v/(4π2k)fˆ(k , k ).) − f i (e) On the basis of your answer to (d), argue that the scattering amplitude for this problem has the form ki kf fˆk k = g(k,θ)+ h(k,θ)σˆ e , e = × , f , i ⊥ ⊥ k k | i × f | where

1 ∞ 4π g(k,θ) = k 2l +1 l=0 [(l + 1)eiδl+1/2,l(k) sin(δ (k)) + leiδl−1/2,l(k) sin(δ (k))]Y (θ,φ), × l+1/2,l l−1/2,l l0 i ∞ 4π h(k,θ) = k 2l +1 l=0 [eiδl+1/2,l(k) sin(δ (k)) eiδl−1/2,l(k) sin(δ (k))]∂ Y (θ,φ). × l+1/2,l − l−1/2,l θ l0

Hint: Recall that ki = kez and use the representation of the orbital angular momentum components ˆlx and ˆly in polar coordinates, ~ ~ ˆl = (cot θ cos φ∂ + sin φ∂ ), ˆl = (cot θ sin φ∂ cos φ∂ ). x − i φ θ y − i φ − θ

(f) Show that the total scattering cross section is

4π ∞ σ = [(l + 1)sin2(δ (k)) + l sin2(δ (k))]. k2 l+1/2,l l−1/2,l l=0

Exercise 4.19: Inelastic scattering of electrons

21 In this exercise you consider the inelastic scattering cross section for a beam of fast electrons impinging on a target. Here “fast” means that the kinetic energy of the scattered electrons is large in comparison to atomic energy scales. Since a fast electron interacts with the target only for a brief amount of time, one may take the interaction between the scattered electron and the electrons in the target as a perturbation and describe the scattering using the ﬁrst-order Born approximation.

(a) Find an expression for the inelastic differential scattering cross section d2σ/dΩdE for scattering off a single atom with atomic number Z. Here Ω is the solid angle corresponding to the change of the propagation direction of the scattered electron and E is the energy lost by the scattered electron. Express your answer in terms of the wavenum- bers ki and kf of the scattered electron before and after scattering, the wavefunctions r1,..., rZ i = ψi(r1,..., rZ ) and r1,..., rZ f = ψf (r1,..., rZ ) for the initial and final states| of the atom electrons, and the interaction|

Z e2 Ze2 U = r r − r j=1 | j − |

between the scattered electron at position r and the atom’s electrons at positions r1, ..., rZ . You may neglect exchange eﬀects resulting from the fact that the fast scattered electron is indistinguishable from the electrons bound by the atom.

(b) Show that your answer can be rewritten as

2 2 Z d σ 4kf = f eiqrj i δ(E E + E). 2 4 i f dEdΩ a0kiq | | − f j=1 2 2 where a0 = ~ /me is the Bohr radius and q = ki kf , Ei is the energy of the atomic state i , and E is the energy of the atomic state −f . | f | (c) Take the target to be a Hydrogen atom and consider the case that the incident electron is scattered in the forward direction (i.e., the electron does not change its propagation direction). Find an expression for d2σ/dEdΩ in the limit that the energy E lost by the scattered electron is much larger than the electronic binding energy Ei of the Hydrogen atom. You may take the initial state i to be the ground state− of the | Hydrogen atom,

1 −r1/a0 ψi(r1)= e . 3 πa0 22 (d) Sketch d2σ/dEdΩ as function of the energy loss E for the situation you considered in (c). How would your answer change if the target would consist of a diﬀerent element than H? Explain how a measurement of d2σ/dEdΩ can then be used to determine the chemical composition of a target. A qualitative discussion is suﬃcient.

Exercise 4.20: Inelastic scattering of electrons via an intermediate bound state

Consider a scattering process in which an electron combines with a (neutral) target atom in its ground state g to form an unstable (charged) bound state b . When the electron is emitted, the atom either| returns to its ground state, or to an excited| state e . The three | types of states are illustrated schematically in the ﬁgure below.

|b> k i |e> |g> k f

The Hilbert space for this process is spanned by the states k, g corresponding to an electron in a state with momentum ~k and the atom in the ground| state, the states k, e corresponding to an electron in a state with momentum ~k and the atom in the excitd| state, and the state b representing the unstable bound state. The Hamiltonian describing this process is | Hˆ = Hˆ0 + Hˆint, where Hˆ = (ε + ε ) k, g k, g + (ε + ε ) k, e k, e + ε b b , (19) 0 k g | | k e | | b| | k k 2 2 with εk = ~ k /2m, εg, εe, and εb denoting the energies of the free electron, the neutral atom in the ground state, the neutral atom in the excited state, and the charged bound state, respectively, and where γ γ Hˆ = g ( b k, g + k, g b )+ e ( b k, e + k, e b ) (20) int L3/2 | | | | L3/2 | | | | k k

describes the interaction between the electron and the atom. Here γg and γe are constants. The electron states are normalized using periodic boundary conditions in a volume of size L3. A beam of electrons with initial-state momentum ~ki is incident on a target consisting of atoms described as above. The atoms are in their ground state g . We are interested in inelastic scattering, in which an atom is in the excited state e after| the scattering process. | 23 (a) Inelastic scattering is possible only if the magnitude ~ki of the momentum of the incoming electrons is suﬃciently large. What is the threshold value of ~ki?

(b) The inelastic scattering cross section [dσ/dΩ]inel is defined as the total number of electrons scattered into a solid angle dΩ per unit time, divided by the flux of electrons incident on the target and the number of atoms in the target, under the condition that the atom undergoes the transition g e in the scattering process. Give an | →| expression for the inelastic scattering cross section [dσ/dΩ]inel in terms of the T matrix that describes the scattering of an electron off a single atom.

The “Born series” expresses the T matrix in terms of an expansion in the interaction Hint. Up to second order in Hint the Born series for the T matrix reads

Tˆ = Hˆ + Hˆ (E Hˆ + iη)−1Hˆ + ..., (21) int int − 0 int where E = εki + εg is the energy of the initial state and η a positive inﬁnitesimal.

(c) Argue that there is no contribution to Tˆ to first order in Hînt. (d) Calculate the matrix element of Tˆ relevant for inelastic scattering up to second order in Hînt.

(e) The T matrix you obtain from the Born series diverges for certain momenta ~ki of the incoming electrons. This divergence is an artifact of the perturbation expansion in powers of Hˆint used in the Born series. In an exact calculation the divergence is replaced by a “resonance peak” of ﬁnite magnitude and width. For what momenta ~ki does the resonance take place and what physics determines the width of the resonance?

Exercise 4.21: Screened Coulomb potential

2 2 A beam of particles with energy Ek = ~ k /2m is incident on a target with screened Coulomb potential e2 V (r)= er/a. 4πr (a) What is the angular dependence of the scattering amplitude f(θ, k) in the limit ka 1? (No calculation is necessary to answer this question.) ≪

24 (b) The scattering amplitude f(θ, k) can be used to calculate the diﬀerential scattering cross section dσ/dΩ. What is the general relation between f(θ, k) and dσ/dΩ? (c) Calculate the scattering amplitude f(θ, k) in the limit ka 1 in the Born approximation (i.e., to ﬁrst order in the potential V (r)). ≪ Hint: Use your answer to (a) to simplify your calculation. (d) Calculate f(k,θ) in the Born approximation, but now without the approximation ka 1. ≪

Exercise 4.22: Scattering and the WKB approximation

A particle with energy ε = ~2k2/2m scatters oﬀ a Gaussian potential V (x) in one dimension,

−x2/2a2 V (x)= V0e . (a) Calculate the reﬂection and transmission amplitudes r(k) and t(k) for this potential in the Born approximation. (b) In the limit ka 1 semiclassical methods should apply. In particular, the scattering state ψR(x) can≫ be calculated using the WKB approximation. Give the expressions for ψR(x) in this limit, and use them to obtain expressions for r(k) and t(k). Hint: Consider the cases V0 <ε and V0 >ε separately. (c) Compare your answer to (b) to your answer for (a).

Exercise 4.23: Diﬀraction grating

Consider a particle of mass m in three dimensions that is subject to the potential

u 2 2 V (r)= e−x /2a sin(qy). √2πa2 (Note that the potential extends over an inﬁnite range in the y and z directions.) Such a potential is referred to as a “grating”. Initially, the particle is moving towards the grating

25 in the positive x direction. The particle has energy ε = ~2k2/2m that is large enough that the conditions ka 1 and V (r) ε are obeyed for all r, so that backscattering from the potential can be neglected≫ (see Ex.≪ 4.22).

out

(a) The particle may pick up a momentum component in the y direction upon passing through the grating. Show that the y component ∆py of the momentum change is a multiple of ~q and calculate the probability P (n) that ∆py = n~q in the Born approximation. (b) What can you say about P (n) if higher orders in perturbation theory are used? In particular: How does P (n) scale with the strength u of the grating potential? (c) In the limit q k one can ﬁnd the scattering state in the so-called “eiconal approximation”: The≪ scattering state ψR(r) is approximated as

1 x ′ ′ ψR(x,y,z)= eikxei R−∞ dx [k(x ,y,z)−k], √2π with k(r) the position-dependent wavenumber used in the WKB approximation, k(r)= k2 2mV (r)/~2. − Use the eiconal approximation to calculate P (n) and compare your answer to items (a) and (b). You may assume that u /a ε, so that the energy of the incident particle is much larger than the potential| V| at≪ all times.

Exercise 4.24: Life time of electrons in an impure metal

26 In a simple model for electrons in a weakly disordered metal, we neglect the regularly ar- ranged lattice ions, such that one is left with an electron interacting with the impurity ions only. In the absence of impurities, the electrons are described by plane waves (we use periodic boundary conditions) 1 ikr r k = ψk(r)= e , | L3/2 characterized by a wave-vector k. In the presence of impurities, the electron will not remain in a given momentum state k , but will decay into other momentum states k′ due to impurity scattering. The mean| time until a scattering event is referred to as the| “life time” of an electron with momentum k. The goal of this excercise is to calculate the life time of 2 an electron with momentum k and energy ǫk = k /2m, scattering from a collection of Nimp randomly distributed, identical, point-like scatterers with potential

Nimp

V (ˆr)= V δ(ri ˆr). 0 − i=1 Within the Born approximation the inverse life time is given by

1 2π ′ 2 = Γk→k′ , Γk→k′ = k Vˆ k δ(ǫk ǫk′ ). τk ~ | | | | − k′=k

(a) Calculate the transition rate Γk→k′ . You will ﬁnd an expression which still depends on the locations r , r ,..., r of the impurities. { 1 2 Nimp } (b) Calculate the inverse life-time by performing the summation over all ﬁnal states k′ . Argue that the result is independent of the positions r , r ,..., r if the distance| { 1 2 Nimp } between impurities is much larger than the electron’s wavelength, ri rj 1/ k , | − | ≫3 | | and express you answer in terms of the impurity concentration nimp = Nimp/L .