XIV Contents 27 Bloch Oscillations 277 27.1 Unitary Transformation on a Quantum System . . . . . . . . . . . . . . 277 27.2 Band StructureinaPeriodicPotential 277 27.3 The Phenomenon of Bloch Oscillations . . . . . . . . . . . . . . . . . . . . . 278 27.4 Solutions 281 27.5 Comments 285 Author Index 287 Subject Index 289 Summary of Quantum Mechanics In the following pages we remind the basic definitions, notations and results of quantum mechanics. 1 Principles Hilbert Space The first step in treating a quantum physical problem consists in identifying the appropriate Hilbert space to describe the system. A Hilbert space is a complex vector space, with a Hermitian scalar product. The vectors of the space are called kets and are noted |ψ. The scalar product of the ket |ψ 1 and the ket |ψ 2 is noted ψ 2 |ψ 1 . It is linear in |ψ 1 and antilinear in |ψ 2 and one has: ψ 1 |ψ 2 =(ψ 2 |ψ 1 ) ∗ . Definition of the State of a System; Pure Case The state of a physical system is completely defined at any time t by a vector of the Hilbert space, normalized to 1, noted |ψ(t). Owing to the superposition principle, if |ψ 1 and |ψ 2 are two possible states of a given physical system, any linear combination |ψ∝c 1 |ψ 1 + c 2 |ψ 2 , where c 1 and c 2 are complex numbers, is a possible state of the system. These coefficients must be chosen such that ψ|ψ =1. 2 Summary of Quantum Mechanics Measurement To a given physical quantity A one associates a self-adjoint (or Hermitian) operator ˆ A acting in the Hilbert space. In a measurement of the quantity A, the only possible results are the eigenvalues a α of ˆ A. Consider a system in a state |ψ. The probability P(a α ) to find the result a α in a measurement of A is P(a α )= ˆ P α |ψ 2 , where ˆ P α is the projector on the eigensubspace E α associated to the eigenvalue a α . After a measurement of ˆ A which has given the result a α , the state of the system is proportional to ˆ P α |ψ (wave packet projection or reduction). A single measurement gives information on the state of the system after the measurement has been performed. The information acquired on the state before the measurement is very “poor”, i.e. if the measurement gave the result a α , one can only infer that the state |ψ was not in the subspace orthogonal to E α . In order to acquire accurate information on the state before measurement, one must use N independent systems, all of which are prepared in the same state |ψ (with N 1) . If we perform N 1 measurements of ˆ A 1 (eigenval- ues {a 1,α }), N 2 measurements of ˆ A 2 (eigenvalues {a 2,α }),andsoon(with p i=1 N i = N ), we can determine the probability distribution of the a i,α , and therefore the ˆ P i,α |ψ 2 .Ifthep operators ˆ A i are well chosen, this determines unambiguously the initial state |ψ. Evolution When the system is not being measured, the evolution of its state vector is given by the Schr¨odinger equation i¯h d dt |ψ = ˆ H(t) |ψ(t) , where the hermitian operator ˆ H(t) is the Hamiltonian, or energy observable, of the system at time t. If we consider an isolated system, whose Hamiltonian is time-independent, the energy eigenstates of the Hamiltonian |φ n are the solution of the time independent Schr¨odinger equation: ˆ H|φ n = E n |φ n . They form an orthogonal basis of the Hilbert space. This basis is particu- larly useful. If we decompose the initial state |ψ(0) on this basis, we can immediately write its expression at any time as: 1 Principles 3 |ψ(0) = n α n |φ n →|ψ(t) = n α n e −iE n t/¯h |φ n . The coefficients are α n = φ n |ψ(0), i.e. |ψ(t) = n e −iE n t/¯h |φ n φ n |ψ(0) . Complete Set of Commuting Observables (CSCO) A set of operators { ˆ A, ˆ B, , ˆ X} is a CSCO if all of these operators commute and if their common eigenbasis {|α,β, ,ξ} is unique (up to a phase factor). In that case, after the measurement of the physical quantities {A,B, ,X}, the state of the system is known unambiguously. If the measurements have given the values α for A, β for B, , ξ for ˆ X, the state of the system is |α,β, ,ξ. Entangled States Consider a quantum system S formed by two subsystems S 1 and S 2 .The Hilbert space in which we describe S is the tensor product of the Hilbert spaces E 1 and E 2 respectively associated with S 1 and S 2 . If we note {|α m } a basis of S 1 and {|β n } a basis of S 2 , a possible basis of the global system is {|α m ⊗|β n }. Any state vector of the global system can be written as: |Ψ = m,n C m,n |α m ⊗|β n . If this vector can be written as |Ψ = |α⊗|β,where|α and |β are vectors of E 1 and E 2 respectively, one calls it a factorized state. In general an arbitrary state |Ψ is not factorized: there are quantum correlations between the two subsystems, and |Ψ is called an Entangled state. Statistical Mixture and the Density Operator If we have an incomplete information on the state of the system, for instance because the measurements are incomplete, one does not know exactly its state vector. The state can be described by a density operator ˆρ whose properties are the following: • The density operator is hermitian and its trace is equal to 1. • All the eigenvalues Π n of the density operator are non-negative. The den- sity operator can therefore be written as ˆρ = n Π n |φ n φ n | , 4 Summary of Quantum Mechanics where the |φ n are the eigenstates of ˆρ and the Π n can be interpreted as a probability distribution. In the case of a pure state, all eigenvalues vanish except one which is equal to 1. • The probability to find the result a α in a measurement of the physical quantity A is given by P(a α )=Tr ˆ P α ˆρ = n Π n φ n | ˆ A|φ n . The state of the system after the measurement is ˆρ ∝ ˆ P α ˆρ ˆ P α . • As long as the system is not measured, the evolution of the density operator is given by i¯h d dt ˆρ(t)=[ ˆ H(t) , ˆρ(t)] . 2 General Results Uncertainty Relations Consider 2N physical systems which are identical and independent, and are all prepared in the same state |ψ (we assume N 1). For N of them, we measure a physical quantity A, and for the N others , we measure a physical quantity B. The rms deviations ∆a and ∆b of the two series of measurements satisfy the inequality ∆a ∆b ≥ 1 2 ψ|[ ˆ A, ˆ B]|ψ . Ehrenfest Theorem Consider a system which evolves under the action of a Hamiltonian ˆ H(t), and an observable ˆ A(t). The expectation value of this observable evolves according to the equation: d dt a = 1 i¯h ψ|[ ˆ A, ˆ H]|ψ + ψ| ∂ ˆ A ∂t |ψ . In particular, if ˆ A is time-independent and if it commutes with ˆ H, the expec- tation value a is a constant of the motion. 3 The Particular Case of a Point-Like Particle; Wave Mechanics TheWaveFunction For a point-like particle for which we can neglect possible internal degrees of freedom, the Hilbert space is the space of square integrable functions (written in mathematics as L 2 (R 3 )). 3 The Particular Case of a Point-Like Particle; Wave Mechanics 5 The state vector |ψ is represented by a wave function ψ(r). The quantity |ψ(r)| 2 is the probability density to find the particle at point r in dimensional space. Its Fourier transform ϕ(p): ϕ(p)= 1 (2π¯h) 3/2 e −ip·r/¯h ψ(r) d 3 r is the probability amplitude to find that the particle has a momentum p. Operators Among the operators associated to usual physical quantities, one finds: • The position operator ˆ r ≡ (ˆx, ˆy, ˆz), which consists in multiplying the wave function ψ(r)byr. • The momentum operator ˆ p whose action on the wave function ψ(r)isthe operation −i¯h∇. • The Hamiltonian, or energy operator, for a particle placed in a potential V (r): ˆ H = ˆp 2 2M + V (ˆr) → ˆ Hψ(r)=− ¯h 2 2M ∇ 2 ψ(r)+V (r)ψ(r) , where M is the mass of the particle. Continuity of the Wave Function If the potential V is continuous, the eigenfunctions of the Hamiltonian ψ α (r) are continuous and so are their derivatives. This remains true if V (r)isastep function: ψ and ψ are continuous where V (r) has discontinuities. In the case of infinitely high potential steps, (for instance V (x)=+∞ for x<0andV (x)=0forx ≥ 0), ψ(x) is continuous and vanishes at the discontinuity of V (ψ(0) = 0), while its first derivative ψ (x) is discontinuous. In one dimension, it is interesting to consider potentials which are Dirac distributions, V (x)=gδ(x). The wave function is continuous and the discon- tinuity of its derivative is obtained by integrating the Schr¨odinger equation around the center of the delta function [ψ (0 + ) − ψ (0 − )=(2Mg/¯h 2 ) ψ(0) in our example]. Position-Momentum Uncertainty Relations Using the above general result, one finds: [ˆx, ˆp x ]=i¯h → ∆x ∆p x ≥ ¯h/2 , and similar relations for the y and z components. 6 Summary of Quantum Mechanics 4 Angular Momentum and Spin Angular Momentum Observable An angular momentum observable ˆ J is a set of three operators { ˆ J x , ˆ J y , ˆ J z } which satisfy the commutation relations [ ˆ J x , ˆ J y ]=i¯h ˆ J z , [ ˆ J y , ˆ J z ]=i¯h ˆ J x , [ ˆ J z , ˆ J x ]=i¯h ˆ J y . The orbital angular momentum with respect to the origin ˆ L = ˆ r × ˆ p is an angular momentum observable. The observable ˆ J 2 = ˆ J 2 x + ˆ J 2 y + ˆ J 2 z commutes with all the components ˆ J i . One can therefore find a common eigenbasis of ˆ J 2 and one of the three components ˆ J i . Traditionally, one chooses i = z. Eigenvalues of the Angular Momentum The eigenvalues of ˆ J 2 are of the form ¯h 2 j(j + 1) with j integer or half integer. In an eigensubspace of ˆ J 2 corresponding to a given value of j, the eigenvalues of ˆ J z are of the form ¯hm , with m ∈{−j, −j +1, ,j − 1,j} (2j + 1 values) . The corresponding eigenstates are noted |α, j, m,whereα represents the other quantum numbers which are necessary in order to define the states completely. The states |α, j, m are related to |α, j, m±1 by the operators ˆ J ± = ˆ J x ±i ˆ J y : ˆ J ± |α, j, m = j(j +1)− m(m ±1) |α, j, m ±1 . Orbital Angular Momentum of a Particle In the case of an orbital angular momentum, only integer values of j and m are allowed. Traditionally, one notes j = in this case. The common eigenstates ψ(r)of ˆ L 2 and ˆ L z can be written in spherical coordinates as R(r) Y ,m (θ, ϕ), where the radial wave function R(r) is arbitrary and where the functions Y ,m are the spherical harmonics, i.e. the harmonic functions on the sphere of radius one. The first are: Y 0,0 (θ, ϕ)= 1 √ 4π ,Y 1,0 (θ, ϕ)= 3 4π cos θ, Y 1,1 (θ, ϕ)=− 3 8π sin θ e iϕ ,Y 1,−1 (θ, ϕ)= 3 8π sin θ e −iϕ . 5 Exactly Soluble Problems 7 Spin In addition to its angular momentum, a particle can have an intrinsic angular momentum called its Spin. The spin, which is noted traditionally j = s,can take half-integer as well as integer values. The electron, the proton, the neutron are spin s =1/2 particles, for which the projection of the intrinsic angular momentum can take either of the two values m¯h: m = ±1/2. In the basis |s =1/2 ,m= ±1/2, the operators ˆ S x , ˆ S y , ˆ S z have the matrix representations: ˆ S x = ¯h 2 01 10 , ˆ S y = ¯h 2 0 −i i0 , ˆ S z = ¯h 2 10 0 −1 . Addition of Angular Momenta Consider a system S made of two subsystems S 1 and S 2 , of angular momenta ˆ J 1 and ˆ J 2 . The observable ˆ J = ˆ J 1 + ˆ J 2 is an angular momentum observable. In the subspace corresponding to given values j 1 and j 2 (of dimension (2j 1 + 1) × (2j 2 + 1)), the possible values for the quantum number j corresponding to the total angular momentum of the system ˆ J are: j = |j 1 − j 2 | , |j 1 − j 2 | +1, ··· ,j 1 + j 2 , with, for each value of j,the2j + 1 values of m: m = −j, −j +1, ··· ,j. For instance, adding two spins 1/2, one can obtain an angular momentum 0 (singlet state j = m = 0) and three states of angular momentum 1 (triplet states j =1,m =0, ±1). The relation between the factorized basis |j 1 ,m 1 ⊗|j 2 ,m 2 and the to- tal angular momentum basis |j 1 ,j 2 ; j, m is given by the Clebsch-Gordan coefficients: |j 1 ,j 2 ; j, m = m 1 m 2 C j,m j 1 ,m 1 ;j 2 ,m 2 |j 1 ,m 1 ⊗|j 2 ,m 2 . 5 Exactly Soluble Problems The Harmonic Oscillator For simplicity, we consider the one-dimensional problem. The harmonic po- tential is written V (x)=mω 2 x 2 /2. The natural length and momentum scales are x 0 = ¯h mω ,p 0 = √ ¯hmω . By introducing the reduced operators ˆ X =ˆx/x 0 and ˆ P =ˆp/p 0 , the Hamil- tonian is: 8 Summary of Quantum Mechanics ˆ H = ¯hω 2 ˆ P 2 + ˆ X 2 , with [ ˆ X, ˆ P ]=i. We define the creation and annihilation operators ˆa † and ˆa by: ˆa = 1 √ 2 ˆ X +i ˆ P , ˆa † = 1 √ 2 ˆ X − i ˆ P , [ˆa, ˆa † ]=1. One has ˆ H =¯hω ˆa † ˆa +1/2 . The eigenvalues of ˆ H are (n +1/2)¯hω, with n non-negative integer. These eigenvalues are non-degenerate. The corresponding eigenvectors are noted |n. We have: ˆa † |n = √ n +1|n +1 and ˆa|n = √ n |n −1 if n>0 , =0 ifn =0. The corresponding wave functions are the Hermite functions. The ground state |n =0 is given by: ψ 0 (x)= 1 π 1/4 √ x 0 exp(−x 2 / 2x 2 0 ) . Higher dimension harmonic oscillator problems are deduced directly from these results. The Coulomb Potential (bound states) We consider the motion of an electron in the electrostatic field of the proton. We note µ the reduced mass (µ = m e m p /(m e + m p ) m e ) and we set e 2 = q 2 /(4π 0 ). Since the Coulomb potential is rotation invariant, we can find a basis of states common to the Hamiltonian ˆ H,to ˆ L 2 and to ˆ L z . The bound states are characterized by the 3 quantum numbers n, , m with: ψ n,,m (r)=R n, (r) Y ,m (θ, ϕ) , where the Y ,m are the spherical harmonics. The energy levels are of the form E n = − E I n 2 with E I = µe 4 2¯h 2 13.6eV. The principal quantum number n is a positive integer and can take all integer values from 0 to n − 1. The total degeneracy (in m and ) of a given energy level is n 2 (we do not take spin into account). The radial wave functions R n, are of the form: 6 Approximation Methods 9 R n, (r)=r P n, (r)exp(−r/(na 1 )) , with a 1 = ¯h 2 µe 2 0.53 ˚ A . P n, (r) is a polynomial of degree n − −1 called a Laguerre polynomial. The length a 1 is the Bohr radius. The ground state wave function is ψ 1,0,0 (r)= e −r/a 1 / πa 3 1 . 6 Approximation Methods Time-Independent Perturbations We consider a time-independent Hamiltonian ˆ H which can be written as ˆ H = ˆ H 0 + λ ˆ H 1 . We suppose that the eigenstates of ˆ H 0 are known: ˆ H 0 |n, r = E n |n, r ,r=1, 2, ,p n where p n is the degeneracy of E n . We also suppose that the term λ ˆ H 1 is sufficiently small so that it only results in small perturbations of the spectrum of ˆ H 0 . Non-degenerate Case. In this case, p n = 1 and the eigenvalue of ˆ H which coincides with E n as λ → 0 is given by: ˜ E n = E n + λ n| ˆ H 1 |n + λ 2 k=n |k| ˆ H 1 |n| 2 E n − E k + O(λ 3 ) . The corresponding eigenstate is: |ψ n = |n + λ k=n k| ˆ H 1 |n E n − E k |k + O(λ 2 ) Degenerate Case. In order to obtain the eigenvalues of ˆ H at first order in λ, and the corresponding eigenstates, one must diagonalize the restriction of λ ˆ H 1 to the subspace of ˆ H 0 associated with the eigenvalue E n , i.e. find the p n solutions of the “secular” equation: n, 1|λ ˆ H 1 |n, 1−∆E n, 1|λ ˆ H 1 |n, p n . . . n, r|λ ˆ H 1 |n, r−∆E . . . n, p n |λ ˆ H 1 |n, 1 n, p n |λ ˆ H 1 |n, p n −∆E =0. The energies to first order in λ are ˜ E n,r = E n +∆E r , r =1, ,p n . In general, the perturbation is lifted (at least partially) by the perturbation. . . . 27 8 27 .4 Solutions 28 1 27 .5 Comments 28 5 Author Index 28 7 Subject Index 28 9 Summary of Quantum Mechanics In the following pages we remind the basic definitions, notations and results of quantum. XIV Contents 27 Bloch Oscillations 27 7 27 .1 Unitary Transformation on a Quantum System . . . . . . . . . . . . . . 27 7 27 .2 Band StructureinaPeriodicPotential 27 7 27 .3 The Phenomenon of. can take either of the two values m¯h: m = ±1 /2. In the basis |s =1 /2 ,m= ±1 /2 , the operators ˆ S x , ˆ S y , ˆ S z have the matrix representations: ˆ S x = ¯h 2 01 10 , ˆ S y = ¯h 2 0 −i i0 , ˆ S z = ¯h 2 10 0