Introduction Historically, the basic experiments and interpretations for Quantum Theory are:. Diraction (Young 803, von Laue 92) 2. Black-body radiati

Transcription

1 Quantum Mechanics II Th.A. Rijken Institute for Theoretical Physics, University of Nijmegen January 8, 998

2 Introduction Historically, the basic experiments and interpretations for Quantum Theory are:. Diraction (Young 803, von Laue 92) 2. Black-body radiation (Planck 900) 3. Photoelectric eect (Einstein 904) 4. Compton eect (923) 5. Combination principle (Ritz-Rydberg 908) 6. Specic heat (Einstein 907, Debye 92) 7. Franck-Hertz experiment (93) 8. Stern-Gerlach experiment (922) 9. Electron diraction (Davidson-Germer 927) See textbooks on Quantum-Mechanics for the description of these experiments. The Davidson-Germer experiment for example used a 75V electron beam on Ni and observed diraction of particles for the rst time. This conrmed the de Broglie relation = h=p. This is the second course in Quantum Mechanics for students in their 3hrd year. The material of this syllabus contains, in a very concise form, the topics covered in these lectures. This course is based on two rather recent books on Quantum Mechanics: a. Introduction to Quantum Mechanics, B.H Bransden & C.J. Joachain b. Modern Quantum Mechanics, J.J. Sakurai The primary purpose of this course is to acquaint the student with modern techniques, such that he/she is prepared to read the current research literature. Therefore, the very general formalism of Dirac using the 'bra' and 'ket' notation is employed. This general framework is very suited to deal with any particular system in setting up a quantum description. Furthermore, recoupling schemes are treated, Clebsch-Gordan coecients dened, which occur very frequently in practice. There are treatments of e.g. : Perturbation Theory, both time-independent and time-dependent, and Scattering Theory. To close this introduction, we recommend the interested student, to read (parts) of the classic: The Principles of Quantum Mechanics, by P.A.M. Dirac, and the modern discussion on the foundations of Quantum Mechanics: Speakable and unspeakable in quantum mechanics, by J.S. Bell. This, in particular during leisure time, like weekends and vacations! 2

3 2 Quantum Mechanics and Complex Numbers For the general development of Quantum Theory, i.e. to be able to deal with general physical conditions, such as boundary values for general geometries and congurations, it is important to have a wave equation. This way one can go beyond the simple cases where a single harmonic wave is sucient to describe the system. In 926 Schrodinger put forward his famous equation, which we now review with the emphasis on its basic assumptions.. Requirements Schrodinger wave equation: (i) linear: superposition principle, wave packets, quantum interference; (ii) coecients wave equation contains only physical constants, not energy, momentum, etc. 2. Non-relativistic: free particle relations: p = hk, E = h!, E = p 2 =2m $ h! = h 2 k 2 =2m. 3. Plane waves should be solution of the wave equation: cos(kx!t), sin(kx!t), e i(kx!t), e i(kx!t). This in order to be capable to describe the diraction experiments by Davidson & Germer(927), Thomson(928). 4. Consider @x 2 The only type of plane wave for this equation is exp i(kx!t). Then = ih, which is satisfactory in view of (ii), see above. For the other type 2m of harmonic waves it is clearly impossible to satisfy this wave equation. So, one arrives at the one-dimensional Schrodinger wave = 2 2 Note that the most common one-dimensional wave equation, describing transversal waves on a string, sound waves in a gas, and light ) =!2 2 k 2 and so does not satisfy the above criterium (ii). = E2 p 2 = p2 4m 2 The foregoing arguments lead, as we have seen, to a complex dierential equation and complex waves:. 5. Three-dimensional: p = hk. The extension to 3-dimensios is straightforward. The wave equation = h2 2m r2 3 ; E = p2 2m :

4 This suggests the correspondence; ; p! h i r i.e. operators in the space of the wave functions, the so-called Schrodinger representation. 6. The generalization to systems with forces is which leads to the wave equation E = p2 2m + V = h2 2m r2 + V : 7. The physical interpretation of the wavefunction is : probability amplitude: probability density P = j j 2 = (Born 926,927). 8. Relativistic: Then the energy-momentum relationship is altered: E 2 = p 2 c 2 + m 2 0 c4 Proceeding in the same way as in the non-relativistic case gives; ; p = h i r (adoption) 2 = h2 c 2 r 2 + m 2 0 c4 i.e. the Klein-Gordon equation. 9. In the case of photons m 0 = 0: c 2 r2! = 0 ; and again E ; complex! 4

5 3 Quantum Physics In this section we photons and their polarization properties in the framework of quantum physics. We will discuss a prototype of a Hilbert-space, where the base vectors are directly related to the photon states. The quantum theory of photons started with Planck and Einstein. Introduced were the light quanta and the energy wavelength relation for photons in a single mode E! = nh! = nhc=. As written beautifully by Dirac in his famous book 'The Principles of Quantum Mechanics', the polarization properties of light can be understood as quantum phenomena. Consider a beam of polarized photons that passes through a lter, for example a tourmaline crystal, E x = E y = E # x-lter E x = E E y = 0 The beam energy is halved by lter: probability point of view is forced! Namely, each photon has a probability to pass. 2 In general: the fraction that passes is je x j 2 je x j 2 + je y j 2 = je xj 2 jej 2 = P = probability per photon: So, if E x = jej cos! P = cos 2. One can describe this law by using state vectors: photon state, polarization. Representing this by the state vector! x j i = ; j x j 2 + j y j 2 = x = s y V 8h! E x ; y = s V 8h! E y Here we used that for one photon: jej 2 V = 8h!. Next we use the following notation for the state vectors belonging to the dierent polarizations: jxi = 0 jyi = 0!! jri = p 2 i jli = p 2 i : x-polarization : y-polarization!! : right circular-polarization : left circular-polarization 5

6 Next to the column vector j i we introduce the corresponding row vector h j, j i = {z} 00 ket 00 and the scalar hermitean inner-product: x y! ; h j {z} x ; y 00 bra 00 (; ) = x x + y y = ( ; ) hj i = h ji This turns the space of the polarization states into a 2-dimensional Hilbert-space. Note that the states we introduced are normalized: h j i =, and orthonormal: hxjxi = hyjyi =, hxjyi = 0. The space of polarizations is two-dimensional. Possible bases are jxi; jyi or jri; jli. According to the superposition principle any state can be expressed as a linear combination of either basis. In terms of the plane wave polarization states: j i = x y! = x jxi + y jyi Expressing this in terms of the circular polarization basis gives j i = x jxi + y jyi = R jri + L jli = x i p y jri + x + i p y jli 2 2 ; hxj i = x hxjxi + y hxjyi = x etc j i = jxihxj i + jyihyj i = jrihrj i + jlihlj i Considering again the x-lter polarization 'measurement', the probability to nd the photon on the back side of the tourmaline crystal is given by probability = j x j 2 j x j 2 + j y j 2 = j xj 2 = jhxj ij 2 : Here, hxj i is the so-called probability amplitude. Note that the measurement leads to the famous wave function reduction! The assertion is that: all experimental facts with polarization photons can be accounted for in terms of probability amplitudes, using a description of the polarized photons in a two-dimensional Hilbert-space. 6

7 4 Dirac -functie In this course, we will make use of the Dirac -functions quite extensively. Therefore, we review the most important properties of these so-called 'generalized functions'. The -functie (x a) is a linear functional which e.g. any continuous function f(x) dened in the interval ( ; +) maps onto his value in x = a. The usual notation for this mapping is f(a) = (x a)f(x)dx (4.) Since?? has to hold for any arbitrary f, it is easy to see that (x a) = 0 for x 6= a. According to the classical integration theory the -function can not be an ordinary function. One can on the other hand consider the -function as a limit of a set of contineous classical functions, for each of which the normal mathematical analysis applies: (x a) = lim "!0 + "(x a) (4.2) where the "-functions become more and more sharply peaked when " approaches zero. So, equation?? strictly taken has to be read as f(a) = lim "(x a)f(x)dx (4.3) "!0 + Since according?? for the special function f(x) = the integral gives, we choose the normalization "(x a)dx = (4.4) We note that the set of functions f"g is not unique. Examples are: "(x) = " x 2 + " 2 ; p " e x2 =" ; sin(x=") x (4.5) In these examples the support of these functions is not nite. In contrast, for the next choice this the case: (x) = [(x + ") (x ")] (4.6) 2" 7

8 where the -function is dened as: (x) = for x > 0, and (x) = 0 for x < 0. Properties of the -functions are (i) (ii) (x a)dx = 0 (x a)f(x)dx = f 0 (a) (iii) (x 2 a 2 ) = [(x a) + (x + a)] X 2jaj (iv) (g(x)) = i jg 0 (x i )j (x x i); g(x i ) = 0 (v) ( x) = (x) (vi) (ax) = jaj (x) (vii) x(x) = 0 (viii) 0 (x) = (x) (4.7) The extension to more dimensions is straightforward. For Rl 3 the 3-dimensional -functie is dened as such that 3 (x a) = (x a )(x 2 a 2 )(x 3 a 3 ) (4.8) V 3 (x a)d 3 x = (4.9) if x = a in V and else zero. from this follows the generalization of?? f(a) = d 3 x 3 (x a)f(x) : (4.0) Applications in quantum mechanics are all over the place. For example, the wave function for a particle that is precisely located at the position a is a (x) = (x a). The reader will see much more in these lecture notes. Also it is instructive to see that dierentiations performed on normal functions can lead to 'generalized functions', especially when these functions are not contineous. For example, the following identity holds; r 2 jxj! = 4 3 (x) (4.) It is easy to verify that the l.h.s. is zero for jxj 6= 0 is voor jxj 6= 0. Then, application of the theorem of Gauss to a sphere with radius a around the point x = 0, gives lim a!0! d 3 x r 2 f(x) = lim jxj a!0 da r O! f(x) = 4f(0) (4.2) jxj 8

9 where we used the Taylor-expansion of f(x). This has important applications in Classical Electrodynamics, but also in Scattering Theory. 4. Fourier Analysis (i) Orthogonal functions and expansions: Let fu n (); n = ; 2; : : :g be a set real or complex valued functions. Orthogonality over the interval (a; b) means that b a U n() U m ()d = n;m (4.3) We here consider in particular sets which are 'complete' for the on the interval (a; b) quadratically integrable functions f(), i.e. f() = X n= a n U n () (4.4) where the r.h.s. converges, possible except for an ensemble of point having zero measure. Because?? the coecients a n are given by a n = b a U n()f()d (4.5) Substitution of?? in?? gives the relation b ( ) X f() = Un( 0 )U n () f( 0 )d 0 (4.6) a n= Since this must hold for all and f(), it follows that X n= the so-called 'completeness-relation'. U n( 0 )U n () = ( 0 ) ; (4.7) In Quantum Mechanics, many applications of this theory can be found. We recall the Legendre-, the Hermite-, the Laquerre-, and Bessel-expansions. A very important example of this theory is the theory of the Fourier expansion. This we will review in some detail in the rest of this section. (ii) Fourier series: Take (a; b) = ( a ; a ) and the functions 2 2 s 2 U n;o (x) = U n;e (x) = 9 2nx a sin a s 2 2nx a cos a ; : (4.8)

10 Note that the index n (n; o) of n (n; e), respectively for the odd and even part of the function which is expanded. The Fourier expansion is f(x) = 2 A 0 + X n= A n cos 2nx a 2nx + B n sin a (4.9) where A n = 2 a B n = 2 a a 2 a 2 a 2 a 2 2nx f(x) cos a 2nx f(x) sin a dx (4.20) dx (4.2) (iii) Innite interval: When (a; b)! ( ; ) then the set fu n ()g becomes a continuum of functions and the Kronecker m;n in de orthogonality relation becomes a -function. Consider for the Fourier analysis the set of functions U n (x) = p a e i ( 2nx a ) ; n = 0; ; 2; : : : (4.22) which is equivalent to??. Then, with f(x) = p a A n = p a 2 a a 2 X n= A n e i( 2nx a ) (4.23) e i ( 2nx a ) f(x)dx (4.24) Before taking the limit a!, we introduce the quantities s 2n 2 a! k ; A n! A(k) (4.25) a and rewrite?? as follows, with n =, f(x) = p a X n= = (2) =2 X A n e i ( 2nx a ) n= 0 r a 2 A n e i ( 2nx a ) 2n : a

11 Since in this last expression we clearly have a Riemann-sum, we nd in the limit for a! that the previous expression becomes (iv) Fourier Integrals: f(x) = (2) =2 A(k)eikx dk (4.26) f(x) = (2) =2 A(k)eikx dk (4.27) A(k) = (2) =2 e ikx f(x)dx (4.28) The orthogonality relation (??) becomes 2 and the completenessrelation (??) appears as 2 e i(k k0 )x dx = (k k 0 ) (4.29) eik(x x0) dk = (x x 0 ) (4.30) (??) and (??) can be understood also as follows: from (??) we see that sin x (x) = lim!0 x = lim sin Lx L! x L = lim e ixt dt = L! 2 L 2 which immediately leads to (??) and (??). e ixt dt (4.3) The generalization to 3-dimensions is simple. We get in that case f(x) = (2) 3=2 A(k)e ikx d 3 k A(k) = (2) 3=2 e ikx f(x)d 3 x (4.32) and e i(k k0 )x dx = (k k 0 ) ; (2) 3 0 ) dk = (x x 0 ) (4.33) (2) 3 eik(x x

12 5 Scalar product, dual space, completeness, bra's and ket's In this section we give the connection between bra's h j and the space of linear functionals dened on the space of the kets j i. Consider the vector space V, with nite dimension N, having an orthonormal basis e ; : : : ; e N, i.e. scalar product: (e i ; e j ) = ij. For any x 2 V : x = x j e j, x j = (e j ; x). Next we introduce the dual space V : the space of linear functionals on V 2V x 2 V x! x (y) 2 Cl ; Rl The dimension of this space is also N. It is easy to proof that a basis for the dual space V : e ; : : : ; e N is provided by the linear functionals e i ; i = ; ::; N with e i (e j ) (e i ; e j ) = ij An extremely important operator is identity operator O on V, where NX O = e n e n = (formal sum) Proof: Ox = X n = X n n= e n e n X j e n x n = x Therefore X x j e j A = e n x j e n(e j ) {z } n;j nj 8x 2 V NX e n e n = : n= Tranlating the foregoing in the notation of Dirac, we have e i! jii : ket e i! hij : bra X e i (e j )! hijji = ij = e n e X n ) jnihnj = n n x = x j e j! jxi = x j jji x j = (e j ; x)! x j = hjjxi x = x 0 je j! hxj = x 0 jhjj x (x) ) X x 0 j x j = ; jjxjj = ) x 0 j = x j ; i.e. complex conjugation. In the next section we will extend the notions of this section to the innite dimensinal case. 2

13 5. Ket's, Bra's, and the continuum We consider an one-dimensional system: e.g. particle on x-axis. space is H = f (x); wavefunctions; sol:s:e:g = linear complex vector space A basis of H is provided by the generalized functions: ' x (x 0 ) = (x x 0 ) ; The hermitean inproduct is dened by (' x ; ' x 0) = x = label 2 Rl dx 00 ' x(x 00 )' x 0(x 00 ) dx 00 (x x 00 )(x 0 x 00 ) = (x x 0 ) The Hilbert The physical interpretation is as follows ' x (x 0 ): is the (idealized) state where the particle is precisely localized at point x. All (x) 2 H can be expanded in basis f x ; x 2 Rl g: (x) = dx 0 (x x 0 ) (x 0 ) = dx 0 (x 0 )' x 0(x) The correspondence between the wave functions and the state space, using the Dirac notation: ' x (x 0 )! jxi (x 0 )! j i " " H H The linear functional hxj on H is dened by: hxj i (' x ; ) = (x) ; which for the special case j i = jx 0 i reads: Proof: hxjx 0 i = (' x ; ' x 0) = (x x 0 ) O = (x x 0 ) = hxjx 0 i = = = hxj dx 00 jx 00 ihx 00 j = 8j i 2 H dx 00 (x x 00 )(x 00 x 0 ) dx 00 hxjx 00 ihx 00 jx 0 i = hxjojx 0 i 3 dx 00 jx 00 ihx 00 j jx 0 i

14 For all ; 2 H it is easy to demonstrate that: O =. Proof: j i = h j i = ( ; ) = dx 0 (x 0 )jx 0 i dx 0 (x 0 ) (x 0 ) = h j = R dx 0 (x 0 )hx 0 j Check: h j i = = dx 0 dx 00 dx 0 j (x 0 )j 2 = (x 0 ) (x 00 ) hx 0 jx 00 i {z } (x 0 x 00 ) h joji = = dx 0 dx 00 dx 0 (x 0 )(x 0 ) = ( ; ) = h joji = h ji (x 0 )(x 00 ) hx 0 j[ R dxjxihxj]jx 00 i {z } hx 0 jx 00 i=(x 0 x 00 ) q.e.d. The generalization to the three-dimensional case is straightforward: H = f (x) ; sol.s.e.g ' x (x 0 ) = (x x 0 )! jxi hxj : hxj i (' x ; ) = (x) hxjx 0 i = (' x ; ' x 0) = (x x 0 ) d 3 xjxihxj = j i = h j = d 3 x (x)jxi d 3 x (x)hxj 4

15 5.2 Fourier analysis in Dirac notation The wave functions corresponding to the states of a frre particle with momentum p i.e. p op jpi = pjpi ; p op! h i r : satisfy the dierential equation with the solution: h i r xhxjpi = phxjpi hxjpi = (2) 3=2 e i h px = (2) 3=2 e ikx The orthogonality of the plane wave solutions reads: d 3 x e +i(k k0 )x = (k k 0 ) (2) 3 which can be seen as follows (2) 3 d 3 xe i(k k0 x = d 3 xhk 0 jxihxjki = hk 0 jki = (k k 0 ) : The completeness of the plane wave solutions follows from d 3 ke i(x x0 )k = (x x 0 ) (2) 3 or d 3 khxjkihkjx 0 i = (x x 0 ) : Since this identity holds for arbitrary x and x 0 it can be inferred that d 3 kjkihkj = In the remainder of this section we want to elaborate a litle on the relation of the correspondence p op! ihr x and the postulate [x op ; p op ] = ih. We start with the commutation relation (postulate) and take the matrix element [x op ; p op ] = ih hxj[x op ; p op ]jx 0 i = ih(x x 0 ) which gives for the matrix elements of the p op the equation (x x 0 )hxjp op jx 0 i = ih(x x 0 ) : 5

16 The solution is The check runs as follows: hxjp op jx 0 i = (x x0 ) : z = x x 0 ; z d dz (z)? = (z) l.h.s. = = dz 0 (z)zf(z) = dz(z)f(z) = r.h.s. dz(z)[zf] 0 q.e.d. Using this result for the matrix elements op the p op we get p op jpi = pjpi ; hxjp op jpi = phxjpi Employing, as so often, = R dx 0 jx 0 ihx 0 j we get l.h.s. = dx 0 hxjp op jx 0 ihx 0 jpi = h dx 0 0 (x x 0 )hx 0 jpi i = ( ) 2 h dx 0 (x x 0 ) d i dx 0 hx0 jpi = h d i dx hxjpi : This leads to the dierential equation The solution of this equation is h d hxjpi = phxjpi : i dx hxjpi = (2) =2 e i h px : We note that there is here no arbitrary constant allowed. This can be seen by using the box normalization and requiring that the solution is normalized. In the continuum case we require that hp 0 jpi = (p 0 p). The generalization into 3-dimensions is: hxjp op jx 0 i = h i r x(x x 0 ) p op jpi = pjpi h i r xhxjpi = phxjpi hxjpi = (2) 3=2 e i h px 6

17 6 Coordinate representation The basic state jxi describes the situation where the particle is exactly located at the point x. This means the eigenvalue for the =bfx op is =bfx. So, and For a general state hxjx op j i = hxjv (x op ) i = x op jxi = xjxi hxjx op jx 0 i = x 0 (x x 0 ): = dx 0 hxjx op jx 0 ihx 0 j i dx 0 x 0 (x x 0 ) (x 0 ) = x (x) V (x op )jx 0 i = V (x 0 )jx 0 i hxjv (x op )jx 0 i = V (x 0 )(x x 0 ) = dx 0 hxjv (x op )jx 0 i (x 0 ) dx 0 (x x 0 )V (x 0 ) (x 0 ) = V (x) (x) The time dependent states satisfy the Schrodinger j (t)i = H opj (t)i ; (x; t) = hxj (t)i : The consequences for the Schrodinger wave function (=bf x; t) can be worked out as hxj (t)i = hxjh opj (t)i = dx 0 hxjh op jx 0 ihx 0 j (t)i " # = d 3 x 0 h2 2m r2 x (x0 x) + V (x)(x 0 x) # = " h2 2m r2 x + V (x) (x; t) (x 0 ; t) (x; t) 7

18 6. Momentum (impuls) representation Complementary to the basic states of the previous section are those where the particle is in a state with an exactly dened momentum jpi: the particle has omentum p. For an arbitrary state j (t)i the momentum space wave function is given by hpj (t)i = e (p; t) : Similarly as above, from the time-dependent Schrodinger equation, we derive for the momentum space wave function: e (p; t) = d 3 p 0 hpjh op jp 0 i e (p 0 ; " p = d 3 p 0 2 2m (p p0 ) + dx dx 0 hpjxihxjv op jx 0 ihx 0 jp i# 0 e(p 0 ; t) = p2 e (p; t) + d 3 p 0 d 3 x d 3 x 0 e ipx V (x)(x x 0 )e ip0 x e 0 (p 0 ; t) 2m = p2 e (p; t) + d 3 p 0 d 3 x e i(p p0 )x V (x) e(p 0 ; t) 2m e (p; t) + d 3 p 0 V e (p p 0 ) e (p 0 ; t) = p2 2m = e (p; The relation between the coordinate and the momentum representation is given by the Fourier transform. Using again = R d 3 xj=bfxih=bfxj, we have e (p) hpj i = d 3 xhpjxihxj i : Using hpjxi = ( e' p ; ' x ) = (' x ; e' p ) = hxjpi = exp( i h p x)(2h) 3=2 we nd that e (p) = (2h) 3=2 (x) = (2h) 3=2 d 3 x e i h px (x) ; d 3 p e i h px e (p) ; which shows the Fourier relation between the coordinate and the momentum space wave functions. 7 Wave packets With Dirac -functions one describes particles either with a precise position or with a precise momentum. These are, of course, idealized states. In reality there 8

19 will be spreading both in position and momentum. Wave packets are states where the particles are located rather sharply both in x-space and in p-space. This of course in accordance with the Heisenberg uncertainty relation x p h=2. Wave packets are typical the states of the particles in a beam of modern particle accelarators. If the spreading of the wave packets is small during the 'live' of a particle in a scattering experiment, one can deal with them, as far as the accelerator aspects is concerned, very well as classical particles. The state that corresponds to a wave packet, as any other state, can be written as a superposition of plane wave states: j i = d 3 p a(p)jp; ti ; i.e. plane waves weighted by a prole (= `test') function a(=bf p). products are hj i = d 3 p f d 3 p i a (p f )a (p i ) hp f jp i i {z } (pf pi) The scalar So: in practice the -functions always occur in combination with `test'-functions! The normailization of the wave packets is h j i = d 3 p a (p)a (p) = The plane waves are very convenient both for calculations and theoretical considerations. Also, having the results for plane waves is completely adequate to describe the experimental results in conjunction with the wave packet prole functions. The details of the prole functions will not be important, apart from the x and p values of these functions. The latter are determined by the collimaters used in the actual experiments. 7. Gaussian packets The gaussian wave packet at the origin at t = 0 is dened by 0(x; 0) = e h i p ix F (x)! 3! F (x) = =2p exp x2 d 2d 2 F (x) = hxi = 0 ; hpi = p i d 3 k e ikr e F (k) ; p = hk 9

20 0 (x; 0) = d 3 p e i h px e F (p pi ) From 0 (x; 0) = hxj 0 (0)i we infer that the state corresponding to the wave packet is j 0 (0)i = (2h) 3=2 d 3 p F e (p =bfp i ) jpi : From the time-dependent Schrodinger equation we have j 0 (t)i = exp i h H 0t j 0 (0)i Combining this with the representation in terms of jpi above, and using that E p = p2 we nd 2m 0(x; t) = Writing =bfp = =bfp i + =bfp 0 one gets d 3 p exp i h (p x E pt) F e (p p i ) : {z } p 0 p x p2 2m t = (p0 + p i ) x (p0 + p i ) 2 2m = p i x p2 i t 2m + p0 (x vt) p02 2m t : Assuming p 02 t=2m (machine design!), then i 0(x; t) exp h (p i x E i t) d 3 p 0 e h i (x v p0 i t) F e (p 0 ) = e i h (p ix E i t) F (x v i t) i.e. under this condition we have a non-spreading wave packet! In a typical scattering set up one extracts the particles from an accelerator. The particles pass several collimators, who dene the parameters of the wave packets. The distance from the collimators to the targets is such that one may neglect the spreading of the wave packets. The conditions for non- spreading are that p 02! 0 t = h E(p0 )t = h 2m t h 2 h 2md t = h 2 2md t ; 2 where used is that the relavant momenta in the wave packet come from the region p 0 h=d. If the distance from the particle source to the target is L, the time of ight is t = L v i, which should be much less than = md 2 =h in order not to have spreading. 20 t

21 8 Formalism Requirements physical theory:. description physical states at time t 2. description observables and measurements 3. equations of motion: time evolution of states and observables Axioms QM:. states $ state vectors 2 H 2. observables $ hermitean linear operator 3. superposition principle 4. expectation value in state jai: hai a = hajajai=hajai 5. time j (t)i = H opj (t)i 6. measurement postulate (Copenhagen interpretation) State: quantum systems can be represented in a (separable) Hilbert space by a unit ray /noindent Pure state: eigen vector of a complete set of independent and commuting observables (`observation maximum') Observables: linear operators A Eigenstate: state jai with Ajai = ajai Hilbert space H: complex vector space with an inner (scalar) product separable: with a denumerable orthonormal base ray: j i = f all vectors jai 2 H diering merely a phase exp i g = fall jai 2 H ; jai = e i jai 0 ; jai 0 ; 0 < 2g Superposition principle: H is a linear space: ja i; ja 2 i {z } phys:state Dual space H : space of linear functionals on H H: hermitean inner product jai hbj! hbjai = (b; a) 2 Cl hbjc a + c 2 a 2 j = c hbja i + c 2 hbja 2 i jai; jbi 2 H! (a; b) = (b; a) 2 Cl 2 H ) c ja i + c 2 ja 2 i 2 H {z } phys:state Theorem: For any jai 2 H, 9haj 2 H such that (a; b) = hajbi for all jbi 2 H ad. Operators: A; B; : : : 2

22 . A = B if Ajai = Bjai for all jai 2 H 2. (A + B)jai = Ajai + Bjai, Ajai = Ajai 3. AB 6= BA in general 4. adjoint of A: A y dened by: haja y is dual of Ajai for all jai 2 H so: hbjajai = haja y jbi, jbi 2 H 5. A hermitean if A = A y 6. if A y = A: eigenvalues real, eigenvectors for dierent eigenvalues are orthogonal Ajai = ajai ha 0 ja y = ha 0 ja = ha 0 ja 0 ha 0 jajai = aha 0 jai ha 0 jajai = a 0 ha 0 jai 0 = (a a 0 )ha 0 jai % a = a (a = a 0 ) & ha 0 jai = 0 (a 6= a 0 ) 7. normalized eigenvectors of hermitean A: Ajni = a n jni ; hnjni = for any jbi: jbi = X n c n jni hmjbi = X n c n hmjni = c m jbi = X n hnjbijni = X n jnihnjbi 0 = X n jnihnj = 8. if [A; B] = 0 ) 9 a complete base of simultaneous eigenvectors jn; mi: Ajn; mi = a n jn; mi, Bjn; mi = b m jn; mi. Ajai = ajai ) ABjai = BAjai = abjai ) Bjai eigenvector of A with same eigenvector One has to distinguish two cases: (i) a nondegenerate: Bjai = jai jai simultaneous eigenstate (ii) a is generate: set ja i; : : : ; ja N i: Aja i i = aja i i 22

23 Bjai = B We next try to nd c i. NX i = jai c i ja i i = c Bja i + : : : c N Bja N i Conditions (Bjai = jai; ha i jbjai = ha i jai ) because space fja i ig invariant under B; ha i jbja j iha j jai = ha i jai or B ij c j = c i ) (B ij ij )c j = 0) c (B ) + c 2 B 2 : : : c N B N = 0 c 2 B 2 + c 2 (B 22 ) : : : c N B 2N = 0 B = ha jbja i ; B ij = ha i jbja j i Observables are, hermitean operators: restriction to B y = B: 2-dimensional case: N = 2, condition for solutions: B 2 B B 2 B 22 = 0 B = B B 2 = B 2 B 22 = B 22 : 2 sol's = 2 orthonormal eigenvalues of B in subspace spanned by ja i and ja 2 i (q.e.d.) 9 >= >; : sol: = 2 (B + B 22 ); B B 22 ; B 2 = B 2 = 0. 2 sol's:. c 6= 0; c 2 = 0 2. c = 0; c 2 6= 0 ja i and ja 2 i eigenvalues of B q.e.d. c (B ) + c 2 B 2 = 0 c B 2 + c 2 (B 22 ) = 0 9. Observation maximum: maximal set of commuting observables: A () ; A (2) ; : : : ; A (p) ; eigenvalues jn ; : : : ; n p i : 23

24 0. Matrix representation: Ajni = X m A mn jmi ; A mn = hmjajni :. Base transformation: Aja n i = a n ja n i Bjb m i = b m jb m i [A; B] 6= 0 ) ja n i = X m S mn jb m i ha n ja k i = nk ; hb m jb`i = m` : U = X k jb k iha k j Uja n i = X k kn jb k i = jb n i X U y U = ja k ihb k jb`iha`j k;` = X k ja k iha k j = ) U is unitary! Continuous spectrum Example: p z in L 2 (z) no eigenvector. Plane wave e ikz is kind of eigenvector with p z = hk, < k <. In general A has continuous and discrete spectrum. 2 spectrum if R A = (A ) not dened. Discrete ja n i (n = : : :), continuous ji: X n ha n ja m i = mn ; hnji = 0 ; hji = ( ) (orthogonality) ja n iha n j + djihj (completeness) X j i = ha n j ija n i + dhj iji n Schrodinger and Heisenberg pictures: There are dierent ways to describe the time dependence of systems. 24

25 . Schrodinger picture: Suppose: = 0 j ; ti = Hj ; ) j ; ti = exp i h Ht Schrodinger picture j ; 0i Schrodinger picture: states time dependent, operators not (unless explicitly, like for example an external laser eld probing atoms.) 2. Heisenberg picture: Denition: A H = e iht=h Ae iht=h It follows that i.e. A H = i h [H; A H] j i H e iht=h j ; ti = j ; 0i For the expectation values one has which should be the case, of course. Hh ja H j i H = h ; tjaj ; ti ; 25

26 9 Propagator Theory Consider the Schrodinger equation = H with a time independent Hamiltonian H. For the complete set eigenfunctions of H: H n (x; t) = E n n (x; t), can assume the property of orthonormality: ( n ; m ) = n;m, and completeness: P n n(x; t) n (x 0 ; t) = (x x 0 ). Seperating the time-dependence, we write the solutions in terms of the timeindependent eigenfunctions, n (x; t) = u n (x)e i=h Ent. For a general state j (t)i one has j ; ti = exp = X n This gives for the wave functions hx 0 j ; ti = X n {z } (x 0 ;t) = X n = i h H(t t 0) jnihnj ; t 0 i exp hx 0 jnihnj ; t 0 i exp(: : :) hx 0 jni d 3 x 00 X n j ; t 0 i i h E n(t t 0 ) d 3 x 00 hnjx 00 ihx 00 j ; t 0 i exp(: : :) hx 0 jnihnjx 00 i exp(: : :) hx 00 j ; t 0 i {z } (x 00 ;t 0 ) From this result we recognize Huygens principle, expressed by the inegral equation (x 0 ; t) = d 3 x 00 K(x 0 ; t; x 00 ; t 0 ) (x 00 ; t 0 ) t > t 0 The /bf kernel of this integral equation, the so-called `propagator', K(x X 0 ; t 0 ; x; t) = hx 0 jnihnjxi exp i n h E n(t 0 t) (t 0 t) X = u n(x 0 )u n (x) exp i h E n(t 0 t) (t 0 t) Now, it is easy to derive that because of the introduced (t 0 t)- factor in the propagator, the following dierential equation holds " 0 H(x0 ; t 0 ) K(x 0 ; t 0 ; x; t) = ih(t 0 t)(x 0 x) ; which will be useful at the end of this section. 26

27 Specializing to the case of a free particle: H = p2 2m 'propagator' explicitly: " i d 3 p exp K(x ; t ; x 2 ; t 2 ) = (2h) 3 = m 2ih(t t 2 ) h p(x x 2 ) i h! 3=2 " # im(x x 2 ) 2 exp 2h(t t 2 ) one can evaluate the # (t t 2 ) 2m p2 Physical interpretation: the propagator as a transition amplitude (x; t) = hxj ; ti S = hxje iht=h j i H hx; tj i H Here jx; ti = exp i h Htjxi: eigenfunction x-op in Heissenberg picture. We can rewrite the propagator as a transition amplitude for the j=bf x; ti-states: K(x ; t ; x 2 ; t 2 ) = X n = X n hx jnihnjx 2 ie i=h E n(t t 2 ) hx je iht =h jnihnje iht 2=h jx 2 i hx ; t jx 2 ; t 2 i i.e. amplitude for the position transition (x 2 ; t 2 )! (x ; t ). 9. Huygens-principle and the Path-integral Starting from the identities (x f ; t f ) = K(x f ; t f ; x i ; t i ) (x i ; t i )d 3 x i = d 3 xjxihxj = d 3 xjx; tihx; tj ; for all t (x f ; t f ) = hx f ; t f j i H = d 3 x i hx f ; t f jx i ; t i i hx {z } i ; t {z i j i H} K(xf ;t f ;xi;t i ) (xi;t i ) one divides the time interval (t f ; t i ) in small (and equal) pieces: K(x f ; t f ; x i ; t i ) = hx f ; t f jx i ; t i i = = d 3 x : : : d 3 x n hx f ; t f jx n ; t n ihx n ; t n jx n ; t n i : : : : : : hx ; t jx i ; t i i ; t j+ t j = ; n large : 27

28 This formula clearly can be interpreted as being build up of the QM amplitudes by a summation over all path's! (Feynman formulation QM) To generalize the propagator theory to systems with time-dependent Hamiltonians, one procedes as follows. General: denition K(x 00 ; x 0 ): (i) t 00 > t 0 : " 00 H(x00 ; t 00 ) K(x 00 ; t 00 ; x 0 ; t 0 ) = ih(t 00 t 0 )(x 00 x 0 ) (ii) t 00 < t 0 : K(x 00 ; t 00 ; x 0 ; t 0 ) = 0 From (i) one can derive by applying to this equation the operation + t2 =t + t 2 =t : K(x 2 ; t + ; x ; t ) K(x 2 ; t ; x ; t ) = (x 2 x )! K(x 2 ; t ; x ; t ) = (x 2 x ) (9.34) because the second term on the l.h.s. is zero, due to the (t 0 t-function. 28

29 Also, one observes that when 0 (2) d 3 x K(2; ) (); t 2 t that 0 satises the S.E., and for t 2 = t, because (eq.??), 0 (x; t ) = (x; t ). Therefore, 0 = So: (i) and (ii) are adequate to dene K(2; ) for general H(t). 0 Symmetry in Quantum Theory A symmetry operation is a transformation of a physical system, state j i! j 0 i, such that all transition probabilities are conserved: jh 0 j 0 ij 2 = jhj ij 2 where j i; j 0 i 2 H, and the U mapping j i! j 0 i is. Synopsis (/bf active viewpoint):. consider a xed coordinate system 2. apparatus A prepares j Ai 3. transformation such A! A 0, A 0 prepares j A 0i. e.g. A = source, emitting electrons through a (small) hole A 0 = same source, rotated 90 o, moving with velocity v. 4. measuring device M and transformed M 0 : M! M invariance: M( A ) = M 0 ( A 0). 6. Quantum Mechanical Measurement: determination probability that physical system! some state ji i.e. jhj Aij 2. e.g. probability electron has momentum p = jh p j Aij 2 = jh p 0j A 0ij2 = probability to nd for fa 0 ; M 0 g momentum p 0 = p. 7. Wigner theorem: j 0 i = U()j i ; 8j i 2 H jh 0 j 0 ij 2 = jhj ij 2 U() is unitary or anti-unitary (V. Bargmann, J. Math. Phys. 5, 862 (964)). 0. Space Translations Suppose that we have a particle at the position x. We move the particle instantly to the position x 0 = x + a, leaving all other properties of the particle the same. So, x 0 = x + a ; p 0 = p 29

30 j 0 i = U(a)j i jxi! U(a)jxi I = jx + ai I : active! hxj 0 i = d 3 x 0 hxju(a)jx 0 i hx {z } 0 j i hxjx 0 +ai=(x x 0 a) 0 (x) = (x a) = (T [a]x) Applied to the x op this gives A 0 = U(a)AU (a) x 0 op = U(a)x op U (a) = x op a ; which we will verify explicitly below. From this relation we see that h 0 jx 0 opj 0 i = d 3 x 00 d 3 x 0 h 0 jx 0 ihx 0 jx 0 opjx 00 ihx 00 j 0 i = d 3 x 0 d 3 x 00 0 (x 0 ) (x 00 a) 0 (x 00 )(x 0 x 00 ) = = d 3 x 0 (x 0 a)(x 0 a) (x 0 a) d 3 x (x)(x) (x) = h jx op j i showing the consistency with the translation invariance of the x measurement. The translation-operation in the conguration space induces the corresponding unitary operation U(a) in the Hilbert space. Here, we mean the Hilbert space of the states: kets j i. The unitary operator for translations is given by: U(a) = exp + i h p a ; p! h r i : The explicit demonstration that this operator has the desired properties is rather straightforward: hxj 0 i = hxju(a)j i =? hxje i=h pa j i i = d 3 x 0 hxj exp h p a jx 0 ihx 0 j i 0 (x) = d 3 x 0 hxj exp (a r x 0) jx 0 i (x 0 ) and therefore 0 (x) = hxja rjx 0 i ) +a r x 0(x 0 x) ; = d 3 x 0 (x 0 x) exp( a r 0 ) (x 0 ) X n! ( a r)n (x) = (x a) n=0 30

31 q.e.d. We note that from this formula it follows that for the Hilbert space of the wave functions (x), we have for the translation operator: U T (a) (x) = exp i h a p op From the fact that [p i ; p j ] = 0 ) (x) ; or U T (a) = exp U(a )U(a 2 ) = U(a + a 2 ) ; U(0) = ; which implies that the translations form a commutative group. 0.2 Space Rotations i h a p op We next consider the rotations in three-dimensions. Let the set fe i ; i = ; 2; 3g denote a orthonormal base in R 3. Then, for each 3-dimensional vector we can write x = x i e i. The rotation R applied to x is x 0 = Rx = x i Re i = x i e 0 i where the e 0 i are the rotated base vectors : e 0 i = Re i = X k e k R ki : Rotations are dened by the requirement that all distances are invariant under rotations. To satisfy this, it is sucient that the rotated set fe 0 i ; i = ; 2; 3g is also orthonormal: ij = e 0 i e 0 j = X kl e k R ki e l R lj = X kl R ki R lj kl = X k R ki R kj = X k er ik R kj = ( e RR) ij = ij err =, R: orthogonal. The orthogonal matrices in 3-dimensions with detr = form a group: SO(3). Also here, we work in the active viewpoint: x rotates x 0 = Rx = x i e 0 i = x 0 ie i ; with x 0 i = R ik x k. Similar to translations, rotation invariance is expressed by: h jx i;op j i = h 0 jx 0 i;opj 0 i 3

32 Now x 0 i;op = R ik x k;op ) hx 0 j i = U(R)x i;op U (R) = R ik x k;op U(R)jxi = jrxi (active) hx 0 ju(r)jxi hxj i {z } hx 0 jrxi=(x 0 Rx) = (R x 0 x) ; det R = : 0 (x) = (R x) The procedure to nd the operators in the Hilbert space of the wave functions which correspond to the rotations in R 3 is a little more complicated as in the case of the translations. We consider rst rotations around z-axis: n = e 3, and in particular innitesimal ones: R(e 3 ) = ' 0 0 cos sin sin cos C A C A ; Then, for active rotations we have 0 (x) = (R x), so 0 (x) = (x x) ' (x) ( x {z} 2 This leads to the equation r + x r {z} 2 ) (x) x x 2 ' (x) (^n x r) (x) = (x) i (^n x p) (x) h = (x) i (^n L) (x) : h d 0 (x; ) d = i h (^n L) 0 (x; ) ; 0 (x; 0) = (x) In order to convert this into a dierential equation w.r.t., we note rst that for rotations around the z-axis, which commute, R ( + ) = Ri () R (). Furthermore, by denoting ((x; ) = (R ()x), we see that by the same procedure as above we reach the dierential equation d 0 (x; ) d = i h (^n L) 0 (x; ) ; 0 (x; 0) = (x) ; 32

33 with the solution 0 (x; ) = exp i h ^n L (x) : This formula is, obviously, valid for any rotation axis ^n, since it is easy to see that the above analysis can be carried out for any axis. Notice the role of the angular momentum operators L i (i = ; 2; 3) as generators of the rotations. So, in order to nd the operators in Hilbert- space, we have to nd the representations of the angular momentum operators in Hilbert space. Before we study the representation of the L i operators in Hilbert space, we rst determine the commutators. We recall that L = r p ; L i = ijk x j p k ; where in the second expression we introduced the Levi-Civita symbol, and of course the Einstein convention about repeated indices is understood. One easily can check the following relations: [L i ; L j ] = j`k [L i ; r l p k ] [L i ; r`] = ijk [x j p k ; x`] = ih ijk x j k` = ih i`j x j [L i ; L j ] = ik` jmn [x k p`; x m p n ] ) hi ijk L k [L i ; p`] = ijk [x j p k ; p`] ) ih i`k p k = jlk f[l i ; r l ]p k + r l [L i ; p k ]g = jlk ih f ilm x m p k + ikm x l p m g = ih f( ji km jm ki )x m p k ( ji lm jm il )x l p m g = ih( x j p i + x i p j ) = ih ijk L k Repeating that ijk L k = ijk kmn x m p n = ( im jn in jm )x m p n = x i p j x j p i j 0 i = U(R)j i ; U(R) = exp( i ^n LR) ; ) h we demonstrate the ease with which one can calculate, using the Levi-Civita symbol: hxj i h ^n Ljx0 i = i h ^n i imn hxjx opm p opn jx 0 i = i h ^n i imn x m h i r n(x x 0 ) = +^n i imn x 0 mr 0 n (x x0 ) 33

34 hxj i h ^n Lj i = + d 3 x 0 ^n i imn x 0 mr 0 (x n x0 ) (x 0 ) = ^n i imn x m r n (x) = (^n x r) (x) d 0 () d = i h (^n L) 0 (0) etc: The commutation relations for the orbital angular momentum operators are [L i ; L j ] = ih ijk L k ; [L 2 ; L z ] = 0 : A complete set of commuting operators for the Lie-algebra above is provided by L 2 and L 3 = L z. A complete set of eigenstates is given by: j`; mi : L 2 j`mi = `(` + )h 2 j`mi L z j`mi = mhj`mi ` = 0; ; 2; : : : ; m = `; : : : ; +` In the next subsection we will proof this for the angular momentum in general. A complete set of eigenfunctions on the unit sphere, using polar coordinates, are the well known harmonic functions: h; j`mi = Y `m(; ) = ( ) m s 2` + 4 vu u t (` m)! (` + m)! P m` (cos )e im 34

35 Angular Momentum Theory The general angular momentum operators J i ; (i = ; 2; 3), are dened by the commutation relations [J i ; J j ] = ih ijk J k and the additional requirement that J y i = J i. Introducing the raising and lowering operators: J + = J x + ij y ; J = J x ij y The J 2 -operator can be written as follows J 2 = J 2 x + J 2 y + J 2 z = 2 (J +J + J J + ) + J 2 z The following commutation relations are important for the demomstrations below [J i ; J 2 ] = 0 ; [J z ; J ] = hj ; [J + ; J ] = 2hJ z A maximal set of independent commuting operators is J 2 and J z. The eigenstates of J 2 and J z we denote by jj; mi in Hilbert space, where J 2 jj; mi = j(j + )h 2 jj; mi ; J z jj; mi = mhjj; mi : The eigenvalue j(j + ) of J 2 is semi-positive. This because for any j i we have h jj 2 j i = 3X i= h jj 2 i j i = 3X i= jjj i j ijj 2 0 : which is due to the hermitean character of the angular momentum operators. Using [J + ; J ] = 2J z we get the useful identities J J + = J 2 J z (J z + ) ; J + J = J 2 J z (J z ) : From these identities we nd J J + jj; mi = [j(j + ) m(m + )]jj; mi = (j m)(j + m + )jj; mi ; J + J jj; mi = [j(j + ) m(m )]jj; mi = (j = m)(j m + )jj; mi : Thus the norms of the states J jj; mi are hj; mjj J + jj; mi = (j m)(j + m + )hj; mjj; mi ; hj; mjj + J jj; mi = (j = m)(j m + )hj; mjj; mi : From a fundamental axiom of Hilbert space these norms can not be negative, so (j m)(j + m + ) 0 ; (j + m)(j m + ) 0 ; 35

36 which implies that j m j : Furthermore, in a Hilbert space the vanishing of the norm implies that the vector (state) is the null-vector. So, and likewise J + jj; mi = 0 if; andonlyif (j m)(j + m + ) = 0 ; J jj; mi = 0 if; andonlyif (j + m)(j m + ) = 0 : Because we know that m is in the interval ( j; +j), the vanishing conditions reduce to m = j and m = j respectively. If m 6= j, J + jj; mi is a eigenstate of J 2 and J z eigenvalues j(j + )h 2 respectively mh: J 2 J + jj; mi = J + J 2 jj; mi = j(j + )jj; mi and using J z J + = J + (J z + ), we get J z J + jj; mi = J + (J z + )jj; mi = (m + )J + jj; mi : Similarly, one nds for J jj; mi the eigenvalues j(j + )h 2 and (m )h for respectively J 2 and J z. We know already that = j(j + ) 0. Taking for j the positive solution of the equation j(j + ) =, we start from the maximum m-value: i.e. m = j. Then we generate a set of states applying the lowering operator successively. This sequence of non-null vectors must end, since otherwise we would violate the condition that m j. Let us say that after p-steps we have J jj; p ji = 0. We have seen before that this can only happen for m = j, i.e. j p = j, or p = 2j, which is an integer. So, j = 0; ; ; 3 ; 2; :::. The number of states in the 2 2 multiplet, characterized by j is 2p + = 2j +. We now summarize these results in the following fundamental theorem: Theorem: (i) The only possible eigenvalues of J 2 are j(j+)h 2, where j is a non-negative, integral or half-integral numbers: j = 0; 2 ; ; 3 2 ; 2; : : : ; (ii) Given a j, the only possible eigenvalues of J z integral or half-integral numbers: are mh, where m are the m = j; j + ; : : : ; j ; j : Eigenstates: jjmi; j = 0; ; ; : : :; m = j; : : : ; +j. 2 J 2 jj; mi = h 2 j(j + )jjmi ; J z jj; mi = hmjjmi : 36

37 . Condon & Shortley Phase Convention We assume that the states jj; mi are normalized. Then, we have seen above that jjj jj; mijj 2 = [j(j + ) m(m )]h 2 : To x the relative phases in a j-multiplet we will use the Condon & Shortley phase convention: q J jjmi = h j(j + ) m(m )jjm i This phase convention is generally adopted in the literature. Therefore, all tables of e.g. Clebsch-Gordan coecients (see below) depend on this convention!.2 SO(3), Euler angles etc. We have seen that the rotation in Rl 3 are orthogonal transformations: e 0 i = e 0 i = X k e k R ki ; R e R = ; det R = + : where we use an orthonormal base e i e j = ij. These rotations form an SO(3) Lie-group. Including also the reections P : det P =, the group becomes the Orthogonal group O(3) = SO(3) [ P SO(3). The irreducible representations U(R) in the Hilbert space spanned by the states fjj; mi ; m = j; : : : ; +jg satisfy the homomorphic law: U(R )U(R 2 ) = U(R R 2 ) The rotations can be described by dierent sets of parameters:. The rotation axis ^n and the rotation angle. We note that R^n = ^n ; + 2 cos = Tr R (0 < 2) 2. The Euler angles: In Fig. the Euler angles ; ; are indicated. The rotations are dened in terms of three successive rotations over the Euler angles: a. a rotation around the z-axis R(e 3 ), b. a second rotation around the new y-axis b 2 = R(e 3 ) over an angle : R(b 2 ), and c. a third rotation around the rotated original z-axis b 0 3 = R(b 3 ) over an angle : R(b 0 3). 37

38 So, we have the prametrization of the rotations as follows: R(; ; ) = R(b 0 3)R(b 2 )R(e 3 ) : Notice that we used here three sets of orthonormal vectors: e k ; b k = R(e 3 )e k ; b 0 k = R(b 2 )b k (see gure) Now, it can be shown that the above rotation is identical to R(; ; ) = R(e 3 )R(e 2 )R(e 3 ) the so-called standard form of the rotations in terms of Euler angles. 0 < 2 ; 0 ; 0 < 2 : the so-called standard form of the rotations in terms of Euler angles. To demonstrate the equivalence of these two representations of R(; ; ) we note the following:. if n 0 = Rn, then R(n 0 ) = RR(n)R : R R(n)R n 0 = RR(n)n = Rn = n 0 2. Tr RR(n)R = Tr R(n) = Tr R(n 0 )! ' 0 = ' Then, one can rather easy derive, using some algebra, that R(b 0 3) = R(b 2 )R(b 3 )R (b 2 ) ; R(b 2 ) = R(e 3 )R(e 2 )R (e 3 ) ; b 3 = e 3 Using these relations one can work out in a straightforward manner that indeed.3 Wigner matrices R(b 0 3)R(b 2 )R(e 3 ) = R(; ; ) In terms of the Euler angles, the representation in Hilbert space of the rotations are readily obtained: U(R) = U(; ; ) = e ijz=h e ijy=h e ijz=h The representation of the angular momentum operators, we have established in the previous section. The irreducible representatations are spanned by the states jj; mi ; m = j; : : : ; +j. From [J 2 ; U] = 0 J 2 Ujjmi = h 2 j(j + )Ujj; mi 38

39 it is clear that this (2j + )-dimensional space in invariant under U(R). Furthermore, we know that we can reach every state in this (2j + )-dimensional subspace by using the raising and lowering operators J, so the representation is rreducible. Therefore, U(R)jj; mi can be expanded as follows U(; ; )jjmi = `X m 0 = ` From this we nd for the expansion coecients jj; m 0 id (j) m 0 m () : D (j) m 0 m () = hjm0 ju()jjmi = hjm 0 je ijz=h e ijy=h e ijz=h jjmi = e im0 hjm 0 je ijy=h jjmie im e im0 e im d (j) m 0 m() : The d()-matrices are called Wigner matrices..4 Rotations in Pauli-spinor space Rotation in spinor space: j = s =! 2-dimensionaal. U(R): unitary matrices 2 in 2D, with det U = + : group SU(2) Writing the relation U = U y gives U = d c b a U = a b c d!! ; ad bc = = U y = a c b d!! U = with jaj 2 + jbj 2 =. A basis for the 2-dimensional matrices is given by!!!, x =, y = i, i z = The i -matrices satisfy the Lie-algebra [ i ; j ] = 2i ijk k ;! a b b a ; From this we see that a 2-dimensional representation of the angular momentum Lie algebra is given as follows L i! U(L i ) = 2 h i :!, 39

40 Using this so-called spinor-representation, we get the spinor- representation for the rotations: U(; ; ) = e 2 i z e i 2 y e i 2 z : From e i 2 z = i 2 z n! ( i 2 z) n + : : : = i 2 z 2! ! = cos 2 i z sin 2 ; 8 3 z : : : we obtain the explicit rotation matrices U(; ; ) = = e i 2 k = cos 2 i k sin 2 ;! e i 2 0 cos sin 0 e i sin cos 2 2! cos 2 e i 2 (+) sin 2 e i 2 ( ) sin e i 2 ( ) cos e i 2 (+) 2 2 The parameter space for the unitary group SU(2): range ; ; : 0 < 4 0 < 2 0 < 2!! e i e i 2 9 >= D(2; 0; 0) = D(0; 0; 0) >; It is rather easy to see that for the correspondence between SO(3) and SU(2) one has R 2 SO(3)! 2 elementsof SU(2): We close this paragraph by remarking that the relation between the irreducible representations of SO(3) and SU(2) is matrices: irreducible projective representations SO(3) $ irreducible vector representations SU(2).5 Addition Angular Momenta, Clebsch-Gordan Coef- cients In this paragraph we review the theory for the addition of angular momenta J and J 2. For deniteness we choose J = L, and J 2 = S. The commutator algebra 40

41 is for this case [L i ; L j ] = ih ijk L k [S i ; S j ] = ih ijk S k [L i ; S j ] = 0 From these one readily obtains the commutation relations for the total angular momentum J = L + S : [J i ; J j ] = ih ijk J k We note that: L 2, S 2 are `scalar' for L, S, and J J 2 not `scalar' for L or S L S not `scalar' for L or S, but [J k ; L S] = 0. Here, L 2 is scalar for L means that [L i ; L 2 ] = 0, etc. The Hilbert space for the combined system is: H = H L H S. There are two complete sets of independent, commuting operators: I II L 2 ; S 2 ; L z ; S z L 2 ; S 2 ; J 2 ; J z The corresponding set of eigenstates in these two cases are:. eigenstates set I: 2. eigenstates set II: `; s; j; mi `m` s m s = j`; s; m`; m s i h`0; s 0 ; m 0`m0 sj`; s; m`m s i = `0` s0 s m 0`m` m 0 sm s h`0; s 0 ; j 0 ; m 0 j`; s; jmi = `0` s0 s j0 j m0 m So, we have two orthonormal bases: there exists a unitary transformation: X s j j`; s; jmi = m s m j`; s; m`; m s i C`m` m`;m s C`m` s m s j m = h`; s; m`m s j`; s; jmi where the transformation coecients are the so-called Clebsch-Gordan coef- cients. The Condon-Shortley convention: unitary matrix! real orthogonal matrix: X m`;m s C`m` s m s j m C`m` 4 s m s j 0 m 0 = jj 0 mm 0

42 ofrom X C`m` j;m C`m` s m s j m C`m 0` s m 0 s j m = m`m 0`m sm 0 s s m s j m = 0 for m 6= m` + m s J z j`s; jmi = mhjls; X jmi s j jls; jmi = m s m j`s; m`m s i m`m s C`m` we nd by applying J z to the l.h.s. and the r.h.s. X m`m s C`m` s m s j m (m` + m s m)j`s; m`m s i = 0 which implies, in view of the independence of the jl; s; m l m s i states, that (m` + m s m)c`m` s m s j m = 0 etc: Recurrence relations: Here we consider two angular momenta J and J 2. Consider the state and using the following items jj j 2 ; jmi = C j m j 2m2 j m jj j 2 ; m m 2 i J + = J x + ij y ; J = J x ij y we have [J z ; J + ] = J + ; [J z ; J ] = J ; [J + ; J ] = 2J z q J jjmi = j(j + ) m(m )jjm i J + jj j 2 ; jmi = [j(j + ) m(m + )] =2 jj j 2 ; jm + i = C j j 2m2 j m m fj + + J 2+ gjj m ijj 2 m 2 i = C j m 0 j 2 m 0 2 j mf[j (j + ) m 0 (m 0 + )] =2 jj j 2 ; m 0 + m 0 2i + [j 2 (j 2 + ) m 0 2(m )] =2 jj j 2 ; m 0 m0 2 + ig : Taking next the inproduct with jj j 2 ; m m 2 i we obtain: [j(j + ) m(m + )] =2 hj j 2 m m 2 jj j 2 jm + i = j 2 m 0 2 n = C j j m 0 m [j (j + ) m 0 (m 0 + )] =2 m m 0 + m2 m 0 2 o + [j 2 (j 2 + ) m 0 2(m )] =2 m m 0 m 2 m

43 which gives the recurrence relation q j(j + ) m(m )C j + q q j (j + ) m (m )C j j 2 (j 2 + ) m 2 (m 2 )C j m Analogously one can proceed, using J. m j 2m2 jm = j 2 j m m 2 m j 2m2 Vector oprerators: We close this section by making some remarks vector operators. We have seen the commutation relations: j m [J i ; J j ] = ih ijk L k [L i ; L j ] = ih ijk L k [L i ; x j ] = ih ijk x k [L i ; p j ] = ih ijk p k A vector operator, is usually dened as a quantity with 3-components, and these three components have similar transformation properties under rotations as the components of the x-vector. An alternative, equivalent to the former denition, is to say that a vector operator, by denition has the following commutation relations with the components of the angular momentum operator J: [J i ; V j ] = ih ijk V k This implies for a nite rotation the formula U(R)V i U (R) = V k R ki = R ik V k Explicitly this rule reads e i h ^nj V i e i h ^nj = R ik V k For innitesimal transformations this is easy to check to be in accordance with the 'usual' denition: for the l.h.s.: ' V i i h ^n p[l p ; V i ] and for the r.h.s.: R ki ' ki ^n j jki, etc. 43