Symmetries. Chapter Introduction. 8.2 Transformations. Refs.: G, S, MS, MZ

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Chapter 8

Symmetries

Refs.: G, S, MS, MZ

8.1 Introduction

Uma hipótese fundamental subjacente a todas as aplica¸cões da Mecânica Quântica é que o espa¸co usual é sujeito às leis da geometria Euclidiana e que é fisicamente homogêneo e isotrópico. Deste modo, pode-se mover o aparato (i.e.,o sistema f´ısico objeto de estudo eo aparelho de medida) de um lugar a outro sem afetar o resultado da experiência. Ou seja, não há posi¸cões ou orienta¸cões espaciais privilegiadas.

Uma transforma¸cão que preserva as rela¸cões mútuas de aspectos fisicamente rele- vantes do sistema é chamada de opera¸cão de simetria. Apresentaremos neste Cap´ıtulo algumas opera¸cões de simetria em Mecânica Quântica. A discussão de rota¸cões nos levará a recuperar a álgebra do momento angular, já obtida e explorada no Cap. 7. Veremos também que, como em Mecânica Clássica, simetrias de um sistema f´ısico dão origem a leis de conserva¸cão.

Estudaremos diversas transforma¸cões e faremos também uma pequena introdu¸cão à Teoria de Grupos.

8.2 Transformations

Devemos ter em mente que o estudo de transforma¸cões envolve considera¸cões em três n´ıveis distintos, porém interligados. Na Fig. 8.1 mostramos um sistema f´ısico S que se encontra em um estado quântico | i, e um aparelho de medida, M, que mede um observável A; as localiza¸cões de S e de M são dadas, respectivamente, pelos vetoresr_S e rM. Uma transforma¸cão sobre o sistema e o medidor é descrita matematicamente por: (1) transforma¸cões nos vetores,r_S!r⁰_S e r_M !r⁰_M; (2) transforma¸cões no estado quântico de S, | i ! | ⁰i; (3) transforma¸cões no medidor M, A ! A⁰. É importante frisar que, enquanto vetores sofrem transforma¸cões no espa¸co Euclidiano tri-dimensional, as transforma¸cões em estados e observáveis são efetuadas no espa¸co de Hilbert.

137

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Figure 8.1: XXX

No que se segue, discutiremos aspectos gerais de transforma¸cões nos três n´ıveis men- cionados, estabelecendo diversos v´ınculos a que elas estão sujeitas. Por exemplo, a im- posi¸cão de um isomorfismo entre as transforma¸cões atuando nos espa¸cos tri-dimensional e de Hilbert nos permitirá inferir propriedades dos geradores, tais como rela¸cões de comuta¸cão e espectros.

8.2.1 Transformation of Coordinates

Ao discutir transforma¸cões de coordenadas podemos adotar dois pontos de vista clara- mente distintos. O primeiro, chamado depassivo, consiste em efetuar a transforma¸cão nos eixos coordenados, mantendo fixos cada ponto P do espa¸co e as grandezas f´ısicas a ele associadas; isto é, mantendo fixosr_S e r_M no exemplo da Fig. 8.1. O segundo ponto de vista, chamado deativo, consiste em manter fixos os eixos e efetuar a transforma¸cão sobre o aparato, representado pelos vetoresr_S e r_M. Os dois pontos de vista são equivalentes:

efetuar uma transforma¸cão nos eixos coordenados ou a transforma¸cãoinversa nos vetores corresponde, essencialmente, à mesma coisa. De agora em diante, pensaremos sempre no ponto de vista ativo, a não ser quando especificado ao contrário.

Uma transforma¸cão de coordenadas é tal que as ‘novas’ coordenadas,r⁰, se relacionam com as antigas,r, através de

r⁰ =Tr, (8.2.1)

onde T é a matriz de transforma¸cão. Dependendo de sua forma particular, T pode representar, por exemplo, uma transla¸cão, uma rota¸cão, uma reflexão, etc., ou, ainda, uma combina¸cão de mais de uma destas transforma¸cões. Na maioria dos casos estaremos interessados em transforma¸cõesortogonais, isto é, as que são descritas por matrizes reais ortogonais:

T ¹ = ˜T =T^†, (8.2.2)

onde ˜ e^† representam ‘transposta’ e ‘hermitiana conjugada’, respectivamente.

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8.2. TRANSFORMATIONS 139

Figure 8.2: YYY

Podemos pensar num conjunto de transforma¸cões que atuam nasfun¸cões, ao invés de nas coordenadas; este conjunto pode ser tomado como isomorfo ao anterior. Chamemos de P_T o operador que corresponde a T, e adotemos a conven¸cão, devida a Wigner, de que a seguinte rela¸cão seja satisfeita:

P_T f(Tr) =f(r); (8.2.3)

ou, equivalentemente,

PTf(r) =f(T ¹r). (8.2.4)

Assim, P_T modifica a forma funcional de f(r) de modo a compensar a mudan¸ca de coordenadas efetuada porT. A Fig. 8.2 ilustra isto no caso simples da fun¸cãof(x) =x, em que a transforma¸cão corresponde à opera¸cão de transla¸cão por um valor constante, a: x⁰ =T(x) =x+a.

8.2.2 Transformation of Quantum States

Em Mecânica Clássica, as variáveis dinâmicas (e.g., posi¸cões e momentos) têm uma rela¸cão bem direta com os estados dinâmicos, no sentido de que ao transformarmos as variáveis, a transforma¸cão nos estados está automaticamente especificada; isto é, o espa¸co tri-dimensional é o cenário das transforma¸cões. Já em Mecânica Quântica devemos ser mais cuidadosos ao especificar a ‘transforma¸cão no sistema f´ısico’, pois a rela¸cão entre variáveis e estados dinâmicos é menos direta.

Com efeito, seja| i um estado qualquer do sistema que, sob uma transforma¸c˜aoT,

é levado em | ⁰i=T [| i ];T estabelece, portanto, uma rela¸cão bijetora entre vetores no espa¸co de Hilbert. Considere agora dois estados quaisquer |ui e |vi, e seus respectivos transformados |u⁰i e |v⁰i. Por hipótese, a transforma¸cão conserva as propriedades f´ısicas dos estados dinâmicos. Assim, com o sistema transformado no estado |u⁰i, a probabilidade de que uma medida dê um resultado correspondendo a|v⁰i deve ser igual

`

a probabilidade de que a mesma medida em|ui dˆe um resultado correspondendo a |vi. Isto ´e,

|hu⁰|v⁰i|² =|hu|vi|², 8 |ui,|vi, (8.2.5)

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de modo que a correspondência estabelecida por T preserva os módulos dos produtos escalares, o que é menos restritivo do que conserva¸cão dos produtos escalares apenas.

Evidentemente, qualquer transforma¸cão unitária é, de acordo com esta defini¸cão, uma transforma¸cão de simetria. Cabe, no entanto a pergunta inversa: qual será a transforma¸cão mais geral que satisfaz à rela¸cão (8.2.5)? A resposta corresponde ao Teorema de Wigner:¹ Toda transforma¸cão de simetria é implementável por uma isometria linear ou anti-linear. Considere então os kets |↵i,| i,| i=c₁|↵₁i+c₂|↵₂i, e seus respectivos transformados, |↵⁰i=T|↵i,| ⁰i=T| i, | ⁰i=T| i; note que admitimos que a transforma¸cão T seja efetuada nos kets pelo operador T. O Teorema de Wigner garante que só há duas escolhas para as fases dos kets transformados:

(I)

(h↵| i=h↵⁰| ⁰i

| ⁰i=c₁|↵⁰₁i+c₂|↵⁰₂i, (8.2.6) ou

(II)

(h↵| i=h↵⁰| ⁰i^⇤=h ⁰|↵⁰i

| ⁰i=c^⇤₁|↵⁰₁i+c^⇤₂|↵⁰₂i. (8.2.7) No primeiro caso (I), a transforma¸cão é unitária (e linear), enquanto que, no segundo (II), é anti-unitária (e anti-linear). Se as fases do produto escalar tivessem significado f´ısico, então só o caso (I) deveria ser considerado. Em Mecânica Quântica, (II) é uma possibilidade que não pode ser descartada; veremos mais tarde que ela descreve o com- portamento de um sistema sob inversão temporal.

Vejamos agora algumas opera¸c˜oes alg´ebricas com operadores anti-lineares que atuam no espa¸co de Hilbert:

(i) Multiplica¸c˜ao por uma constante. SeA´e um operador anti-linear ecuma constante complexa, (8.2.6) implica em

cA=Ac^⇤ 6=Ac. (8.2.8)

(ii) Produtos de operadores anti-lineares. SejamA₁ eA₂ dois operadores anti-lineares.

Então, o produto A₁A₂ é um operador linear; isto é,

(A₁A₂)|ui=A₁(A₂|ui). (8.2.9) Esta última propriedade pode ser utilizada para mostrar um resultado muito importante: toda transforma¸cão que depende de um conjunto cont´ınuo de parâmetros e que satisfaz a propriedade de grupo

T_bT_a=T_ba (8.2.10)

é unitária. Em rota¸cões, por exemplo, ae b são ângulos, em transla¸cões são vetores de transla¸cão, etc. Para demonstrar isto, note que a opera¸cão a transforma o estado | i

1Para uma demonstra¸c˜ao do teorema veja, por exemplo, Gottfried (1966), Cap. V, Se¸c˜ao 27.

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8.2. TRANSFORMATIONS 141 em | ai e este em | bai por meio deb. Por outro lado, a transforma¸c˜ao ba leva| i em

| bai diretamente. Se a transforma¸c˜ao ´e cont´ınua, podemos escrever

Tc =T_c/2T_c/2, (8.2.11)

isto é, podemos dividir a opera¸cão numa sucessão de duas opera¸cões iguais. Logo, seT_c/2 fôr anti-unitário, T_c é unitário, o que é uma contadi¸cão. Portanto, T_c é necessariamente unitário.

E interessante notar que tanto para operadores unit´´ arios quanto anti-unit´arios, temos

T T^†=T^†T = 1. (8.2.12)

8.2.3 Transformation of Observables

Examinemos agora as transforma¸cões nos aparelhos de medida, representados por ob- serváveis. SejamQum observável eQe=T [Q] sua transformada pela opera¸cão T. Fisi- camente,Qerepresenta a medida efetuada com o aparato tendo sofrido a transforma¸cão;

veja a Fig. 8.1. Logo, o valor esperado das medidas de Q, efetuadas quando o sistema está no estado | i, é igual ao valor esperado das medidas de Qe quando o sistema está no estado | ⁰i=T[| i]; isto é,

h |Q| i=h ⁰|Qe| ⁰i, (8.2.13) ou, ainda, com | i=T| ⁰i,

h |T^†QTe | i=h |Q| i, (8.2.14) de modo que

T^†QTe =Q ) Qe=T QT^†. (8.2.15) Note que uma transforma¸cão unitária não preserva, necessariamente, regras de comuta¸cão. Por exemplo, sejamAeB operadores tais que [A, B] = , onde é um número complexo. É fácil ver que [A⁰, B⁰] = ^⇤ seA⁰=T AT^†, comT anti-linear.

Por outro lado, vejamos o que ocorre com o espectro de um observável quando ele é submetido a transforma¸cões unitárias. Considere a equa¸cão de autovalores,

A|ai=a|ai. (8.2.16)

Atuando com T nesta equa¸c˜ao obtemos

T A|ai ⌘T AT^†T|ai=A⁰|a⁰i=a|a⁰i, (8.2.17) ondeA⁰ =T AT^† e cada |aié levado em |a⁰i=T|ai. Com isto estabelecemos o seguinte teorema: Observáveis unitariamente equivalentes têm o mesmo espectro.

A igualdade descrita por (8.2.14) expressa a situa¸c˜ao em que tanto o aparelho de medida quanto o sistema f´ısico sofrem a mesma transforma¸c˜ao. Podemos nos perguntar,

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por outro lado, qual o resultado da medida de Q quando somente o sistema sofre a transforma¸c˜ao. A resposta ´e obtida calculando-se

h ⁰|Q| ⁰i=h |T^†QT| i=h |Q⁰| i, (8.2.18) ondeQ⁰ ´e o resultado da opera¸c˜ao inversa atuando emQ,

Q⁰ =T^†QT, (8.2.19)

como ilustrado na Fig. 8.1.

Estabelecemos, portanto, uma equivalência entre transformar o sistema, deixando fixo o medidor, e efetuar a transforma¸cão inversa no medidor, deixando fixo o sistema f´ısico. Isto é análogo ao que ocorre no espa¸co tri-dimensional, onde transformar um vetor mantendo fixo o sistema de coordenadas equivale a efetuar a transforma¸cão inversa no sistema de coordenadas, mantendo fixo o vetor.

8.2.4 Symmetries, Conservation Laws, and Degeneracies

Let us now assume the transformation (8.2.19) leaves invariant some operatorQ,

Q=T^†QT. (8.2.20)

Acting on the left-hand-side withT yields

T Q=QT, (8.2.21)

that is, invariance of the operator under a transformationT implies in their commutation,

[Q, T] = 0. (8.2.22)

Consider now symmetry operations which depend on a continuous parameter; when the parameter is infinitesimal, the operator must di↵er infinitesimally from the identity transformation,

T =1+ T. (8.2.23)

Unitarity implies

T T^†=1+ T + T^†+O⇥ ( T)²⇤

, (8.2.24)

Therefore, to this order,

T = T^†, (8.2.25)

that is, T is anti-Hermitian, and can be written as

T = i F, with F Hermitian. (8.2.26)

As inspired by the time evolution operator (see both Chapter 4, and in what follows), we may now assume a generic infinitesimal parameter,", to factor out from F, and, for dimensional purposes, we also introduce the factor 1/~,

T =1 i"

~G, (8.2.27)

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8.2. TRANSFORMATIONS 143 whereGis a Hermitian operator, called thegenerator of the transformation. Notice that the product "G must have units of angular momentum, which allows us to anticipate that if " = dt, then G = H; if " = dx, G = P_x, or, "G = da·P; if "= d✓ (an angle around an axis ˆn), G= ˆn·J (J is the angular momentum operator), and so forth; we will discuss each of these in the sections below. The invariance condition (8.2.22) now becomes the commutation betweenQand the infinitesimal generator,

[Q, G] = 0. (8.2.28)

Especially important are transformations leaving the Hamiltonian invariant,

[H, T] = 0, (8.2.29)

or, in terms of the infinitesimal generator,

[H, G] = 0. (8.2.30)

Indeed, if T (or G) does not have an explicit time dependence, Eq. (8.2.29) implies, through the Heisenberg equation of motion, thatT (orG) is a constant of motion,

dT_H

dt = dG_H

dt = 0. (8.2.31)

We now discuss the e↵ect of a symmetry operation T, on the eigenstates, |ni, of a Hamiltonian, H, which is invariant uponT. By virtue of Eq. (8.2.29), we have

, then all kets T( )|ni are degenerate in energy.

8.2.5 Definition of a Group

In order to explore the symmetry properties in a systematic way, it is useful to define a group. Consider a set of elements,a, b, . . . for which a multiplication is defined. We say these elements form a groupG, if the following properties are satisfied:

• The product of any two elements ofG,c=ab, is also in G;

• There is an identity element inG,I, such thatIa=a;

• Each elementahas an inverse, a ¹, also in G, and such thataa ¹ =I.

As we will see, the symmetry operations on a physical system will form a group.

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8.3 Time Displacements

Já examinamos a evolu¸cão temporal de um sistema no Cap. 4, mas é ilustrativo retomar a discussão, encarando-a como uma transforma¸cão de simetria. Mostraremos inicial- mente que a unitariedade da transforma¸cão determina, necessariamente, que o gerador da transforma¸cão seja a Hamiltoniana do sistema; a partir da´ı chega-se à equa¸cão de Schrödinger.

Considere um sistema isolado, isto é, aquele que não interage com qualquer agente, a não ser que uma medida seja efetuada. Neste caso, com o sistema tendo sido preparado em t0, o operador de evolu¸cão temporal depende apenas da diferen¸ca t t0; isto é, U(t, t₀)⌘U(t t₀). Por depender de um parâmetro que pode variar continuamente,U

é unitário, e satisfaz às propriedades de grupo:

U(t t₀) =U(t t⁰)U(t⁰ t₀). (8.3.1) Fazendot=t₀ obtemos o elemento identidade do grupo

U(0) =1, (8.3.2)

o que define a opera¸c˜ao inversa

U ¹(t t⁰) =U(t⁰ t) =U^†(t t⁰). (8.3.3) The arguments leading to Eq. (8.2.27) can now be specialised to infinitesimal time displacements, so thatT now becomes, for an isolated system,

U( t) =1 i

~ H t. (8.3.4)

whereHis a Hermitian operator with dimension of energy, and by analogy with Classical Mechanics, it must be the Hamiltonian of the system; therefore, H is the generator of infinitesimal time displacements.

Vejamos agora que a Equa¸cão de Schrödinger é recuperada como conseqüência da unitariedade e da propriedade de grupo da evolu¸cão temporal. De fato, usando (8.3.1) obtemos

U(t+dt) U(t) = [U(dt) 1]U(t) = i

~HU(t) dt, (8.3.5) ou

i~@

@tU(t) =HU(t), (8.3.6)

que é equivalente à Equa¸cão de Schrödinger.

Integrando (8.3.6), com a condi¸c˜ao inicial (8.3.2), chegamos a

U(t) =e ^iHt/^~. (8.3.7)

Assim, se| ié autoestado deH com energia E, sua evolu¸cão temporal é dada por

| ⁰i=e ^iEt/^~| i, (8.3.8)

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8.4. SPATIAL TRANSLATIONS 145 ou seja, por diferirem apenas por uma fase, | ⁰i e | i representam o mesmo estado.

Evidentemente isto não é verdade para um estado arbitrário.

Finalizando, considere um sistema que não esteja isolado entre observa¸cões, como no caso de um feixe atômico que passe por um campo eletromagnético alternado. Os resultados desta se¸cão podem ser generalizados para este caso, com modifica¸cões perti- nentes, tais como o fato de queU agora depende tanto do instante inicial de prepara¸cão quanto do instante da medida: U =U(t, t₀). A discussão segue de modo análogo, com H=H(t), o que traz a necessidade de introdu¸cão do operador de ordenamento temporal;

veja a Se¸c˜ao 4.3.

8.4 Spatial Translations

Considere dois observadores cujos eixos coordenados estejam rigidamente deslocados um em rela¸cão ao outro. O primeiro tem um arsenal completo de aparelhos de medida com os quais prepara os estados|ui,|vi, etc. A origem do segundo observador está deslocada de a com rela¸cão ao primeiro; o segundo observador também se supre de um conjunto completo de estados {|u,ai}, que têm, de seu ponto de vista, as mesmas propriedades que os{|ui} do primeiro observador. Devemos ter, pois,

|hu|vi|=|hu,a|v,ai| (8.4.1) e, de acordo com o Teorema de Wigner podemos escolher as fases de |ui, |u,ai, etc., de modo que estes dois conjuntos de kets sejam relacionados por um operador unitário ou anti-unitário. Porém, como as transla¸cões dependem de um parâmetro cont´ınuo, a transforma¸cão deve ser unitária.

Devido à homogeneidade espacial, as transla¸cões no espa¸co tri-dimensional formam um grupo parametrizado pelos deslocamentos. Denotando um elemento deste grupo por G(a), as rela¸cões de grupo são então

G(a)G(a⁰) =G(a+a⁰), (8.4.2)

Identidade: G(0), (8.4.3)

G ¹(a) =G( a). (8.4.4)

As opera¸cões G(a) atuam no espa¸co tri-dimensional, e a cada uma destas deve cor- responder um operador unitário no espa¸co de Hilbert, que chamaremos de U(a). A correspondência G$U deve preservar as rela¸cões de grupo:

U(a)U(a⁰) =U(a+a⁰), (8.4.5)

U(0) = 1, (8.4.6)

U(a)U( a) =U( a)U(a) = 1, (8.4.7) ou

U( a) =U ¹(a) =U^†(a). (8.4.8)

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Como no caso de deslocamento temporal, consideremos transla¸c˜oes infinitesimais:

U( a) = 1 + U. (8.4.9)

Novamente, a propriedade de grupo e unitariedade implicam em U = a·F, com F sendo um operador anti-hermitiano independente dea. Podemos escrever, ent˜ao,

U( a) = 1 i

~ a·P, (8.4.10)

onde o operador hermitiano Pé o momento linear total do sistema, e é o gerador das transla¸cões infinitesimais. Apesar de ser bastante intuitivo que P seja, de fato, um operador vetorial, a demonstra¸cão rigorosa fica adiada até o estudo de rota¸cões.

Vejamos agora as rela¸cões de comuta¸cão satisfeitas pelas componentes de P. Ao transladarmos um objeto de uma distância a_y ao longo da dire¸cão y e, em seguida, de ax ao longo de x, devemos chegar ao mesmo resultado que se realizássemos estes deslocamentos na ordem inversa. Por conseguinte, [G(a_x,0,0),G(0, a_y,0)] = 0, de modo que os geradores infinitesimais devem também comutar:

[P_x, P_y] = 0, etc. (8.4.11)

Um grupo cujos geradores comutam é chamado de grupo Abeliano. Como conseqüência desta propriedade, podemos integrar (8.4.10) para obter o operadorU no caso de transla¸cões finitas, de modo análogo ao feito na se¸cão anterior. De fato,

U(a+ a) U(a) = [U( a) 1]U(a) = i

~ a·PU(a), (8.4.12) e, lembrando que o lado esquerdo pode ser escrito como a·rU(a), a Eq. (8.4.12) fica

rU(a) = i

~PU(a). (8.4.13)

A integra¸c˜ao ´e, portanto, imediata,

U(a) =e ^iP^·^a/^~. (8.4.14)

O mesmo resultado pode ser obtido através da fórmulae^x = limn!1(1 +x/n)ⁿ, já que todas as componentes dePcomutam entre si, podendo ser tratadas como números.

Os kets associados ao observador transladado s˜ao dados, portanto, por

Logo, os autovetores de P são invariantes por transla¸cões. Note também que, como o grupo é Abeliano,U(a) já é diagonal na base que diagonaliza P.

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8.4. SPATIAL TRANSLATIONS 147 Suponha agora que o primeiro observador me¸ca o valor médio de um observável A no estado|u,aipreparado pelo observador deslocado; isto é, no que se refere ao primeiro observador, o sistema é transladado de a. Ele mede, pois,

hu,a|A|u,ai=hu|eîP^·â/^~Ae îP^·â/^~|ui. (8.4.16) Se aé infinitesimal, temos

hu, a|A|u, ai=hu|A+ i

~ a·[P, A]|ui

=hu|A|ui+ i

~ a·hu|[P, A]|ui. (8.4.17) Então, se [P, A] = 0 o resultado de uma medida da propriedadeAnão depende da escolha da origem do sistema de coordenadas: um observável é translacionalmente invariante se comuta com P; este resultado pode, evidentemente, ser adaptado se A comutar com apenas algumas das componentes deP.

Let us consider, in particular, systems for which [H,P] = 0. The consequences are:

• The Hamiltonian is invariant under translations.

• There are states which are simultaneously invariant under time and spatial displacements.

• In the Heisenberg picture,P_His a constant of motion.

• For P in the Schr¨odinger picture, and if | (t)i is a solution to the Schr¨odinger equation, one has

h (t)|P| (t)i=h (0)|e^~ⁱ^H^tPe ^~ⁱ^H^t| (0)i

=h (0)|P| (0)i, (8.4.18) so thatP is a constant of motion.

We have therefore concluded the quantum mechanical version of the fact that if a system is invariant under translations, linear momentum is conserved.

We now determine how the eigenstates of the operator R transform under translations. First, we note that since [R, g(P)] = i~rPg(P), where g(P) is a di↵erentiable function of the operador P, we have

[R, U(a)] =i~rPU(a)

=aU(a). (8.4.19)

With|r⁰₀i=U(a)|r0i, we let R act on the left, R|r⁰₀i=R[U(a)|r0i]

=UR|r₀i+aU|r₀i

= (r₀+a) [U(a)|r₀i], (8.4.20)

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that is, the displaced ket has eigenvaluer+a.

We can now examine the e↵ect of translation on the wave function,

0(r)⌘ hr| ⁰i=hr|U(a)| i

=hr a| i, (8.4.21)

or,

0(r) = (r a), (8.4.22)

which is in agreement with Eq. (8.2.4).

The translated position operator is transformed according to Eq. (8.2.19) which, with the aid of (8.4.19), leads to

R⁰ =U^†(a)RU(a) =R+a. (8.4.23) Indeed, if we now calculate the e↵ect of the transformed position operator onto the fixed ket|r₀i, we get

R⁰ |r₀i= (r₀+a)|r₀i, (8.4.24) illustrating that it is equivalent to transform the ket with U or transform the operator withU^†RU.

Devemos ter em mente que o operadorPnão foi especificado, nesta se¸cão, em detalhe suficiente que permitisse sua constru¸cão expl´ıcita. Esta é uma questão que deve ser atacada separadamente para cada sistema f´ısico. Tudo o que podemos garantir até o momento é que deve ser sempre poss´ıvel construir um operadorPpara qualquer sistema realista, já que todos os sistemas podem ser deslocados espacialmente.

8.5 Rotations and Angular Momentum

Refs: G§32-36; S Ch. 3; MS Ch. XIII; MZ Ch. 16; B Ch. 17.

8.5.1 Rotations in 3D

Let us consider rotations of vectors,V!V⁰, in the three-dimensional space; we denote by Rany rotation. As a special case, we first examine rotations by an angle ↵ around thezaxis; see Fig. 8.3. The transformation of coordinates can be expressed as follows,

V_i ^R!V_i⁰ =X

j

R_ijV_j, (8.5.1)

where

V⁰ = 0

@V_x⁰ V_y⁰ V_z⁰

1

A, V= 0

@Vx

V_y V_z

1

A, (8.5.2)

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8.5. ROTATIONS AND ANGULAR MOMENTUM 149

Figure 8.3: Rotation by↵ about theOz axis.

and R_ij are the elements of the rotation matrix, Rz(↵) =

0

@

cos↵ sin↵ 0 sin↵ cos↵ 0

0 0 1

1

A, (8.5.3)

Note that the rotation matrix is orthogonal,

R^T_z(↵) =Rz( ↵) =R_z¹(↵), (8.5.4) which guarantees that the norms of vectors are preserved under the rotation.

Had the rotations been around the x and y axes, the matrices would have been Rx(↵) =

0

@1 0 0 0 cos↵ sin↵ 0 sin↵ cos↵

1

A, and Ry(↵) = 0

@ cos↵ 0 sin↵

0 1 0

sin↵ 0 cos↵ 1

A, (8.5.5) respectively.

It is instructive to consider infinitesimal rotations,↵!"⌧1, and keep terms up to O("²). We get

Rx(") = 0

@1 0 0 0 1 "²/2 "

0 " 1 "²/2 1

A, and Ry(") = 0

@1 "²/2 0 "

0 1 0

" 0 1 "²/2 1

A. (8.5.6) We note that their ‘commutator’ is

[Rx("),Ry(")] =Rz("²) 1, (8.5.7) from which we conclude: (1) rotations in ordinary three-dimensional space are not com- mutative; (2) the commutation of the matrices for x and y rotations is expressed in terms of the matrix forz rotation; this will be crucial in establishing the commutation relations between angular momentum from purely geometrical arguments.

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8.5.2 Rotation of states

To each rotation,R, in three-dimensional space we associate a rotation in Hilbert space, D(R), with the following properties:

unitarity: D^†(R ¹) =D(R) (8.5.8)

group: D(R2)D(R1) =D(R2R1) (8.5.9)

identity: D(O) =1 (8.5.10)

inverse: D(R)D(R ¹) =D(R ¹)D(R) =1 (8.5.11) associativity: D(R3)[D(R2)D(R1)] = [D(R3)D(R2)]D(R1), (8.5.12) and, since it depends on a continuous parameter, Wigner’s theorem guarantees it is linear, not anti-linear. These rotations therefore form a group.

A physical system, represented by a state|ai(astands for a set of quantum numbers defining the state), undergoes a rotation,R, after which it is represented by the rotated ket,

|a;Ri=D(R)|ai, (8.5.13)

We start with a sub-group of the rotations, namely, that of rotations about the z- axis; they are parametrized by a single angle,D(R)! D_z(↵), and their properties are special cases of the general ones, e.g.,

D_z(↵₂)D_z(↵₁) =D_z(↵₁+↵₂) (8.5.14)

D_z(↵) =D_z^†( ↵), (8.5.15)

and so forth.

According to our general analyses of Sec. 8.2.4, Eq. (8.2.27) allows us to write for infinitesimal rotations (↵!"⌧1),

D_z(")'1 i

~"J_z, (8.5.16)

whereJ_z is the generator of rotations about thez-axis.

We can convince ourselves thatJ_z is an angular momentum operator, by considering spinless particles, and examine the e↵ect such rotation has on the wave function,

D_z(") (x, y, z)' (x+"y, "x+y, z) ' (x, y, z) +"(y@x x@y )

=

✓

1 i"L_z

~

◆

(x, y, z), (8.5.17)

where the first equality is the Wigner prescription, Eq. (8.2.4), and the second is the result of expanding near (x, y, z).

For a rotation about an arbitrary direction, ˆu, Eq. (8.5.16) generalizes to D_u_ˆ(")'1 i

~"L·u.ˆ (8.5.18)

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8.5. ROTATIONS AND ANGULAR MOMENTUM 151 For N particles, L should clearly stand for the total angular momentum. By the same token, if one also includes the spin degrees of freedom, the infinitesimal rotation operator must be given by

D_u_ˆ(")'1 i

~"J·u,ˆ (8.5.19)

which, for a system without classical analogue, should be taken as defining the total angular momentum operator. Equation (8.5.19) therefore establishes that the total angular momentum operator is the generator of rotations, as expected from the analogy with Classical Mechanics.

For finite rotations, Eq. (8.5.19) can be iterated (or integrated) as we did for translations in Sec. 8.4, with the identification "⌘↵/N,N 1,

D_u_ˆ(↵) = lim

N!1

 1 i

✓J·uˆ

~

◆ ↵ N

N

= exp

✓ i

~↵J·uˆ

◆

. (8.5.20)

At this point we should stress the important di↵erences between the groups of translations and of rotations. First, the angle ↵ has restrictions about its domain, namely,

↵ 2 [0,2⇡), unlike the translations, which are not subject to similar restrictions. A group with restrictions about the domain of its parameters is known as acompact group, which, in turn, is connected to the fact that the spectrum of their generators is discrete.

In the present case, the spectrum of angular momentum operators is discrete, while that of linear momentum (the generator of space translations) is not.

The second di↵erence is the fact that the di↵erent components of the generator of the rotation group (the total angular momentum operator) do not commute with each other, again as opposed to what happens to the generator of space translations. One therefore says that the rotation group isnon-abelian, while the translation group isabelian. Note, however, that if one considers solely rotations about a fixed axis, the subgroupisabelian.

The parametrization of rotations discussed so far is through an axis, specified, for instance, by the angles✓ and ', relative to fixed cartesian directions, and by the angle of rotation↵. An alternative parametrization is through the Euler angles, ( ,✓, ), as in Classical Mechanics:

• We first rotate the cartesian axes by aroundOZ, so that (X, Y, Z)!(X⁰, Y⁰, Z);

see Fig. 8.4(a).

• Then we rotate the axes by ✓around OX⁰, so that (X⁰, Y⁰, Z)!(X⁰, Y⁰⁰, Z⁰); see Fig. 8.4(b).

• And, finally, we rotate the axes by aroundOZ⁰, so that (X⁰, Y⁰⁰, Z⁰)!(X⁰⁰, Y⁰⁰⁰, Z⁰);

see Fig. 8.4(c).

In Quantum Mechanics texts, one often denotes these angles as ( ,✓, )!(↵, , ), so that the operation is referred to as

R(↵, , ) =RZ⁰( )RX⁰( )RZ(↵). (8.5.21)

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Figure 8.4: Definition of an Euler rotation: (a) Rotation of about Z; (b) Rotation of

✓about X⁰; (c) Rotation of about Z⁰. [Image from http://zone.ni.com/reference/en- XX/help/371361H-01/gmath/3d cartesian coordinate rotation euler/]

As it stands, this formulation is not very convenient for operators, since the combined rotation is expressed in terms of the rotated axes, instead of fixed ones. Nonetheless, it can be shown (see Problem 4) that the operator performing the Euler rotation can be expressed as

D(↵ ) =e ^i↵J^z^/~e ^{i J}^y^/~e ^{i J}^z^/~, (8.5.22) where it should be noted that all axes are the original (i.e., fixed) ones.

8.5.3 Rotations of Observables

As discussed in Sec. 8.2.3, one can rotate the observables instead of the states. More specifically, ifD(R) rotates the kets, then an observableQ transforming as

Q⁰ =D^†(R)QD(R) (8.5.23)

produces the same net e↵ect.

Some operators have specific transformation properties under rotations:

• Scalar operators,S, are invariant under rotations, that is,

S⁰ =D^†(R)SD(R) =S, 8R. (8.5.24) Acting with D(R) on the right yields

[D(R), S] = 0. (8.5.25)

Operators such as P²,R²,J² =J·J, (J is an angular momentum operator) are scalars.

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• Vector operators, V, are such that, on the one hand, its components follow the transformation law in ordinary space, as given by Eq. (8.5.1); on the other hand, since each component is an operator, it must also rotate according to Eq. (8.5.23).

We can therefore write,

V_i⁰ =D^†(R)V_iD(R) =X

j

R_ijV_j. (8.5.26)

Let us consider, for simplicity, an infinitesimal rotation about Oz; then, with the rotation operator being given by Eq. (8.5.16), we may write,

V_x⁰ '⇣ 1+i"

~J_z⌘ V_x⇣

1 i"

~J_z⌘

=V_x+i"

~[J_z, V_x] +O("²) (8.5.27)

=V_x "V_y+O("²), (8.5.28)

where the last equality follows from the corresponding one in Eq. (8.5.26), withR_ij being given by Eq. (8.5.3), but keeping terms only up to linear in↵=". Comparing the terms linear in"in Eqs. (8.5.27) and (8.5.28) leads to

[J_z, V_x] =i~V_y. (8.5.29) If one now considers infinitesimal rotations about theOxandOyaxis, it is a simple matter to generalise the above result to

[J_i, V_j] =i~"_ijkV_k. (8.5.30) Two remarks about this important result are in order. First, this equation can be used as a definition of vector operators; that is, the cartesian components of any vector operator must satisfy this algebra. And, secondly, since the angular momentum operator is itself a vector operator, it does satisfy this algebra; we have therefore established that the angular momentum algebra, which appeared in Eq. (7.2.8) as a plausible extension of Eq. (7.2.4), is indeed a manifestation of its vector character.

• Tensor operators are natural generalisations of vector operators, and play a very important role in, e.g., determining selection rules. If one sticks to cartesian components, the discussion of tensor operators can be somewhat cumbersome; simplicity arises if one uses spherical tensor coordinates instead. However, in order to introduce this system of coordinates, one must first analyse representations of the rotation operators, which we now discuss.

8.5.4 Representations of the Rotation Operator

We will now obtain representations for the rotation operator. For simplicity, we first consider the case of spinless particles, and the spin will be included subsequently.

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8.5.4.1 Spinless Particles

Similarly to any general ket, under a rotation operationR, the eigenstates ofL² andL_z transform according to

|`m;Ri=D(R)|`mi. (8.5.31) In order to discuss representations for D(R), we first note that since L² commutes with all components ofL one has

[L², D(R)] = [L²,e ^i↵L·^n/~^ˆ ] = 0, (8.5.32) where, for simplicity, we consider rotations of an angle↵ about a given axis, ˆn. Taking the matrix element between arbitrary orbital angular momentum states yields

h`⁰m⁰|[D(R), L²]|`mi=~²[`(`+ 1) `⁰(`⁰+ 1)] h`⁰m⁰|D(R)|`mi= 0, (8.5.33) so that

h`⁰m⁰|D(R)|`mi= 0, if `6=`⁰. (8.5.34) This reflects the fact thata rotation can change the system’s orientation (m), without altering its angular momentum (`).

SinceD(R) does not connect states with di↵erent angular momenta, there is a completeness relation for each`,

X` m⁰= `

|`m⁰ih`m⁰|=1, (8.5.35) where, in this context, 1 is to be understood as the identity matrix in the (2`+ 1)- dimensional subspace.

This result is similar to what we had already established in Chapter 7 for the components ofJ: D(R) is represented by matrices with a block structure, whose elements are defined by

D_m^(`)₀_m(R)⌘ h`m⁰|D(R)|`mi, (8.5.36) where the index (`) emphasizes the block-structure of this representation. Since each

`-block cannot be subject to further reductions in size, it is said to form an irreducible representation for the rotation matrix. One should also have in mind that these matrix elements are determined solely by the properties of the angular momentum operators;

that is, they are the same, irrespective of what the dynamics of the system is.

The interpretation of these matrix elements is immediate. Suppose a system is prepared in a state of angular momentum`, with a projectionmalong a given direction ˆ

u, which is at an angle ✓ with ˆz; see Fig. 8.5. If we measure L_z, the probability of obtaining the valuem⁰~ is then given by

h`m⁰|`m;Ri ² = h`m⁰|D(R)|`mi ² = D^(`)_m₀_m(R) ², (8.5.37) whereRis the rotation by an angle ✓that takes ˆu to ˆzaround ˆn, as shown in Fig. 8.5.

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Figure 8.5: Rotation of ...

The matrices D^(`) share the same group properties as the rotation operators, discussed in Sec. 8.5.2; that is, it is easy to see that

D_mm^(`) ₀(R1R2) =X

m⁰⁰

D^(`)_mm₀₀(R1)D^(`)_m₀₀_m₀(R2), (8.5.38) and that

D^(`)_m₀_m(R) =D_mm^(`)^⇤₀(R ¹). (8.5.39) Also, since the ‘no rotation’ operation O=RR ¹ must be represented by the identity matrix, we also have, as a special case of Eq. (8.5.38),

X

m⁰⁰

D^(`)_mm₀₀(R)D^(`)_m₀₀_m₀(R ¹) = _mm0. (8.5.40) With Eqs. (8.5.39) and (8.5.40), the unitarity relations are expressed as

X

m⁰⁰

D_mm^(`) ₀₀(R)D_m^(`)₀_m^⇤₀₀(R) =X

m⁰⁰

D^(`)_m₀₀^⇤_m(R)D^(`)_m₀₀_m₀(R) = _mm0. (8.5.41) The transformation of spherical harmonics under rotations plays an important role in the theory of representations of the rotation group. In order to discuss this, we first insert Eq. (8.5.35) into Eq. (8.5.31), and use (8.5.36) to get

|`m;Ri= X` m⁰= `

D^(`)_m₀_m(R)|`m⁰i. (8.5.42) This should be compared with Eq. (8.5.31): now the rotated state is expressed as a linear combination of allm-states within the same`-subspace, with the coefficients being the

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matrix elements of the rotation operator. In view of this, one says that a simultaneous eigenstate ofL² andL_z transforms according to the`-th irreducible representation of the rotation group.

Now we show that the spherical harmonics transform in the same way. Recalling Eq. (7.4.18), we project Eq. (8.5.42) onto|nˆi, to obtain

hnˆ|`m;Ri=X

m⁰

hnˆ|`m⁰iD_m^(`)₀_m(R) =X

m⁰

Y_`^m⁰(✓,')D^(`)_m₀_m(R). (8.5.43) One should note, however, that hnˆ|`m;Ri is not a spherical harmonic in (✓,'); one obvious manifestation of this is the fact that, for a general rotation, a ' dependence crops up even ifm= 0. Indeed, since

hnˆ|`m;Ri=hnˆ|D(R)|`mi=hR ¹nˆ|`mi, (8.5.44) where R ¹nˆ denotes the inverse rotation in the usual 3D space, (✓,') ! (✓⁰,'⁰), we have

hnˆ|`m;Ri ⌘Y_`^m(R ¹n),ˆ (8.5.45) so that

Y_`^m(R ¹ˆn) =X

m⁰

D_m^(`)₀_m(R)Y_`^m⁰(ˆn), (8.5.46) thus completing the proof that the spherical harmonics also transform according to the

`-th irreducible representation of the rotation group.

The inverse of Eq. (8.5.46), in the sense that it is the vector nˆ who undergoes the rotationnˆ!Rn, isˆ

Y_`^m(Rn) =ˆ X

m⁰

D^(`)_m₀_m(R ¹)Y_`^m⁰(ˆn) =X

m⁰

D^(`)_mm^⇤₀(R)Y_`^m⁰(ˆn), (8.5.47) where we have used Eq. (8.5.39). Equation (8.5.47) will be used below to establish the transformation properties of tensor operators; see Sec. 8.5.6.

8.5.4.2 Spin-1/2 Particles

Without orbital motion, the rotation operator for a spin-1/2 particle is simply

D^(1/2)(R) = e ^i↵ⁿ^ˆ^·^S/^~ = e ^i↵ⁿ^ˆ^· ^/2, (8.5.48) where are the Pauli matrices, whose properties have been explored in detail in Chapters 5 and 7. The rotation operatorD^(1/2)(R) is therefore represented by the 2⇥2 matrix

D^(1/2)(↵n) =ˆ

✓cos^↵₂ in_zsin^↵₂ ( in_x n_y) sin^↵₂ ( inx+ny) sin^↵₂ cos^↵₂ +inzsin^↵₂

◆

, (8.5.49)

wheren_x,n_y, and n_z are the cartesian components of ˆn.

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8.5. ROTATIONS AND ANGULAR MOMENTUM 157 One should note that if↵= 2⇡, the rotation matrix becomes 1, meaning that only after a rotation by 2⇥2⇡ the identity operation is recovered; this double-valuedness of rotations occurs for all half-integer j. There is no inconsistency with this, since the isomorphism between rotations in three-dimensions and those in the Hilbert space is only validfor infinitesimal rotations. Further, consider two operators D⁰ and D⁰⁰ which are associated with the same three-dimensional rotation, but di↵er by an angle of 2⇡.

Then, we may have, say, | ⁰i =D⁰| i = | i, while | ⁰⁰i=D⁰⁰| i =| i; however, no measurable e↵ect is observed on an observable A, sinceD⁰AD^0†=D⁰⁰AD^00†.

When orbital motion is taken into account, the Hilbert space for a spin-1/2 particle is the direct product of the space associated with the three classical degrees of freedom, Er, and the two-dimensional space corresponding to the spin, Es. Thus, a state | i of this spin-1/2 particle in the space E =Er⌦Es can be represented by an object, called the two-component spinor,

[ ](r)⌘

✓

+(r) (r)

◆

, with _"(r)⌘ hr"| i, (8.5.50)

where the two spin projections, "and #, are denoted by "= + and , respectively.

The rotation of a state | i 2E is then carried out by

| ⁰i=D(R)| i, (8.5.51)

with

D(R) = e î↵J^·^n/^ˆ ^~ =D^(r)(R)⌦D^(1/2)(R) = e î↵L^·^ˆ^n/^~e î↵S^·^ˆ^n/^~, (8.5.52) since the generator of rotations is now the total angular momentumJ=L+S; the last equality follows from the fact that [Lµ, S⌫] = 0, µ,⌫ = x, y, z. A pictorial view of the di↵erence between applying the rotation simultaneously in Er⌦Es and in Er alone is sketched in Fig. 8.6.

Let us now derive the transformation law for spinors. We start by projecting Eq. (8.5.51) onto the mixed coordinate-spin representation {|r"i}, to obtain

hr"| ⁰i= ⁰_"(r) =hr"|D(R)| i=X

"⁰

Z

d³r⁰hr"|D(R)|r⁰"⁰ihr⁰"⁰| i, (8.5.53) where the completeness relation

X

"⁰

Z

d³r⁰|r⁰"⁰ihr⁰"⁰|= 1 (8.5.54) inEr⌦Es has been used. Since

hr"|D(R)|r⁰"⁰i = hr|D^(r)(R)|r⁰ih"|D^(1/2)(R)|"⁰i=

= hR ¹r|r⁰iD_""^(1/2)₀ (R) =

= (r⁰ R ¹r)D_""^(1/2)₀ (R), (8.5.55)

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Figure 8.6: Pictorial representation of the e↵ect of rotating a state withL+S(left) and withL alone (right).

Eq. (8.5.53) becomes

0"(r) =X

"⁰

D_""^(1/2)₀ (R) _"0(R ¹r), (8.5.56)

which is a generalization of the Wigner prescription, Eq. (8.2.4),

0(r) = (R ¹r), (8.5.57)

which also appeared in the context of rotation of spinless particles, Eq. (8.5.44).

The above result can be easily generalized to a particle with arbitrary spin-s:

µ0(r) = Xs µ⁰= s

D_µµ^(s)₀(R) _µ0(R ¹r). (8.5.58) Therefore, under rotations di↵erent spinor components combine to yield the transformed component.

We see that the rotation matricesD^(j)(R) determine how kets, spinors, observables, etc., transform. Therefore, in order to exploit the symmetry properties in the most effi- cient manner, we must be able to calculate these matrices in a systematic way. Indeed, similarly to the problem of addition of angular momentum, givenD^(j¹⁾(R) andD^(j²⁾(R), one can construct the direct product matrices with the aid of the Clebsch-Gordan coefficients; see, e.g., Merzbacher,§16.7.

8.5.5 Rotational Invariance and Conservation of Angular Momentum Since a finite rotation can be expressed as a product of infinitesimal rotations,

D("n)ˆ '1 i

~"nˆ·J, (8.5.59)

with"⌧1, the invariance of any quantity under rotation is intimately connected to the properties of angular momentum.

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8.5. ROTATIONS AND ANGULAR MOMENTUM 159 Therefore, a state | iis invariant under rotations if

D("n)ˆ | i=| i ()J| i= 0, (8.5.60) that is, if the application of any component of Jvanishes identically; alternatively, this condition can be expressed as

J²| i= 0. (8.5.61)

A well known example is provided by the spherically symmetric`= 0 states (also known ass-states) of central-force problems, such as the Hydrogen atom, or three-dimensional harmonic oscillator.

As we saw in Sec. 8.5.3, an observableS which is invariant under rotations is called a scalar; for the case of infinitesimal rotations, this condition implies thatSmust commute with the components of the total angular momentum operator,

[J, S] = 0. (8.5.62)

Let us now stress some consequences of dealing with a system described by a scalar Hamiltonian,

[D(R),H] = 0 (8.5.63)

(a) Two states related by a rotation at a given instant, and satisfying the Schr¨odinger equation, remain solutions at all subsequent instants of time, if the Hamiltonian is invariant under that rotation.

Let | (t₀)i and | ⁰(t₀)i ⌘ D(R)| (t₀)i be two states connected by a rotation at t=t0. If| (t0)i is a solution of the Schr¨odinger equation, then

D(R)[i~@t H]| (t)i= 0

= [i~@_t H]D(R)| (t)i

= [i~@_t H]| ⁰(t)i, (8.5.64) where the second line follows from [D(R),H] = 0.

That is, sinceHgenerates time displacements, the equations of motion are invariant under rotation. By the same token, one cannot generate states with di↵erent energies simply by rotating a given eigenstate: the energy subspaces are globally invariant upon rotations.

(b) If the Hamiltonian is invariant by rotations, the angular momentum is a constant of motion.

Indeed,

i~dJ_H

dt =U^†(t)[J,H]U(t) = 0)Jis conserved. (8.5.65) Some simple examples are

(i) a spinless particle in a central potential,V(R) =V(R), for whichLis constant.

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(ii) a Hydrogen atom in a uniform magnetic field (neglecting the electron spin).

With

p!p e

cA, withA= 1

2H⇥r= H

2 ˆz⇥r, (8.5.66) The Hamiltonian now reads,

H= ~²

2mr² e² r +1

2m!²r²sin²✓+~!L_z, (8.5.67) with ! =eH/2mc. The last term shows that when H 6= 0 (! 6= 0), H only commutes with Lz, meaning that it is only invariant under rotations around theOz axis.

(c) Labelling of states.

Further,

[J,H] = 0 =)[J²,H] = [J_z,H] = 0. (8.5.68) This means that the eigenstates of H can be labelled as |kjmi (where k denotes all other quantum numbers apart fromj and m), since hkjm|H|k⁰j⁰m⁰i= 0, unless j=j⁰ and m=m⁰; clearly, additional selection rules apply to k and k⁰, depending on their specific nature. In other words, when expressed in the basis of simultaneous eigenstates of J² and J_z, the Hamiltonian will be block-diagonal: the size of each block depends onk, but each block is associated with a single pair (j, m).

(d) Rotational degeneracy

Yet another property of scalar Hamiltonians is the rotational degeneracy of their spectra: the energy depends onj, but not onm, so that the degeneracy of each level is at least 2j+ 1. To see how this comes about, recall that

[J,H] = 0 =)[J₊,H] = [J ,H] = 0. (8.5.69) Therefore,

E6=E(m). (8.5.71)

As an example, consider the spectrum of the Hydrogen atom, ignoring the electron spin: for a given principal quantum number,n= 1,2, . . .,`= 0,1, . . . , n 1, and the energy levels depend solely onn, being given byE_n= 13.6 eV/n². Thus, for a given nthe degeneracy isgn⌘n², as a result of combining two distinct degeneracies: one is the so-called ‘accidental’ degeneracy, which manifests itself by states with di↵erent

`(but the samen) having the same energy; the other is the rotational (or, to oppose the former, ‘essential’) degeneracy discussed above. The ‘accidental’ one is actually caused by an additional symmetry, which is not built into the {|n`mi} basis. This example also illustrates that the degeneracy can indeed be higher than 2`+ 1.

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8.5. ROTATIONS AND ANGULAR MOMENTUM 161 The relation between symmetry and degeneracy can be summarized as follows. If the Hamiltonian is invariant under a symmetry operationU, and if| iis an eigenstate ofH with eigenvalue E, then | ⁰i=U| i is either a di↵erent eigenstate, though degenerate with the first, or | ⁰i =| i, apart from a phase. Further, the Hamiltonian eigenstates are not necessarily invariant under the same symmetry operation which leaves H invariant, such as when| ⁰i 6=| i; in this case, the state has a lower symmetry than the Hamiltonian itself, and, especially in the context of many-body physics, such state is said to have broken symmetry. Again, the H atom provides an example: H is rotationally invariant, but onlys-states are spherically symmetric.

8.5.6 Tensor Operators and the Wigner-Eckart theorem

Refs.: G§36; S§3.10; Baym Ch. 17.

8.5.6.1 Motivation

We have already established that a scalar observable, S, is invariant under rotations.

Since angular momentum states, {|jmi}, form a very useful basis, and have definite transformation properties under rotations, one expects that general selection rules for matrix elements of S can be derived.

Indeed, since [D(R), S] = 0, one has

[S, J²] = [S, J_z] = [S, J_±] = 0, (8.5.72) which imply

hj⁰m⁰|S|jmi= 0, ifj6=j⁰ orm6=m⁰ (8.5.73) This selection rule can be interpreted as due to geometrical constraints, in the sense that a spherically symmetric operator carries no angular momentum, so that when it acts upon a state |jmi, the magnitudeJ² (meaning j) is preserved, so thatS|jmi only overlaps with |j⁰m⁰i if j⁰ =j. By the same token, it does not alter the orientation (m) of the state it acts upon, so the outcome has no superposition with |jm⁰i, if m⁰ 6= m.

On the other hand, the specific nature ofS should a↵ect other quantum numbers, apart from (j, m).

The existence of selection rules like these provide important tools to extract information about the interactions at play in a given physical system, without their detailed knowledge. A systematic way of achieving this goal is to set up a classification of observables into scalars, vectors, and tensors, and exploit their transformation properties under rotations. As we will see, although selection rules can be drawn for the cartesian components of vector operators, a unified framework of tensor operators is readily established in terms of theirspherical components,Tq^(k), wherekis the rank of the tensor, andqlabels the component; see below. In terms of this representation, somegeometrical selection rules will emerge in a simple way, with the aid of the Wigner-Eckart theorem.

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8.5.6.2 Definitions

We have seen that the cartesian components of a vector operator, V, have a simple transformation rule under rotation:

V_i⁰=D^†(R)V_iD(R) =X

j

R_ijV_j, (8.5.74)

whereRis the matrix that performs the corresponding rotation of usual vectors in three- dimensional space. Likewise, we have seen that Eq. (8.5.74) leads to specific commutation relations with the cartesian components of the angular momentum operator,

[J_i, V_j] =i~✏_ijkV_k, (8.5.75) which can also be used as adefinitionof vector operators. Further, one should also note that the first equality in Eq. (8.5.74) also applies to any function of the vector operator, V:

f⁰(V) =D^†(R)f(V)D(R). (8.5.76) In classical Physics, a cartesian tensor of rank k is an object with 3^k components, whose transformation rules under rotations is an immediate generalization of those for vectors,

T_ijl...= X

i⁰,j⁰,l⁰,...

R_ii0R_jj0R_ll0T_i0j⁰l⁰...; (8.5.77) where the R_ii0 are the elements of the cartesian transformation matrix, and k is the number of indices: scalars correspond tok= 0, vectors tok= 1, and so forth. The fact that one has to perform products ofktransformation matrices makes it hard to extract general information from these rotations.

O exemplo mais simples de um tensor cartesiano de ordem 2 ´e uma di´adica, formada a partir de dois vetoresU eV, da seguinte forma:

T_ij ⌘U_iV_j. (8.5.78)

O leitor deve verificar que, de fato, as nove componentesTij se transformam sob rota¸c˜oes de acordo com a Eq. (8.5.77).

Further, we may foresee that cartesian components are not universally convenient to fully explore their behaviour under rotations. Indeed, cartesian tensors such as (8.5.77) arereducible; that is, they can be decomposed in objects which, as expressed in (8.5.77), follow di↵erent rotation rules. Por exemplo, a di´adica (8.5.78) pode ser escrita da seguinte forma:

U_iV_j = U·V

3 ^ij+(UiVj UjVi)

2 +

✓UiVj+UjVi

2

U·V 3 ^ij

◆

. (8.5.79) O primeiro termo do lado direito é T, o tra¸co do tensor, que, sendo um escalar, é invariante sob rota¸cões. O segundo termo pode ser escrito como um produto vetorial,

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8.5. ROTATIONS AND ANGULAR MOMENTUM 163 A_k ⌘✏_ijk(U⇥V)_k, de modo que se transforma como um vetor. O terceiro termo pode ser escrito como um tensor simétrico de tra¸co nulo, que tem apenas cinco componentes independentes (dos três elementos da diagonal, apenas dois são independentes). Logo, estes três objetos se transformam de acordo com as representa¸cões irredut´ıveis`= 0, 1 e 2, respectivamente; suas respectivas multiplicidades (2`+ 1) somam 3² = 9, que é o número de componentes do tensor.

A combination of cartesian components in some form of spherical coordinates should be preferred instead. In order to pursue along this line, we start by recalling that the

`= 1 spherical harmonics can be written in terms of cartesian coordinates as Y₁⁰=

r 3

4⇡cos✓= r 3

4⇡

z

r, (8.5.80)

Y₁^±¹ =⌥ r 3

8⇡sin✓e^±ⁱ =⌥ r 3

4⇡

x±iy

p2r , (8.5.81)

since x = rsin✓cos , y = rsin✓sin , and z = rcos✓. From now on, we may take r= 1, so that the tensor spherical coordinatesof the vector rare defined as

r_q⌘ r4⇡

3 Y₁^q(ˆn), q = 0,±1, (8.5.82) or, explicitly, as

r_±₁=⌥x±iy

p2 , and r₀=z. (8.5.83)

The transformation analogous to Eq. (8.5.74) is therefore the one that takes nˆ ! nˆ⁰ = Rˆn, or, to be more precise,Y₁^q(ˆn)!Y₁^q(Rn), as given byˆ

Y_`^m(ˆn⁰) = hnˆ⁰|`mi=D^†(R)Y_`^m(ˆn)D(R)

= hnˆ|D^†(R)|`mi=X

m⁰

hnˆ|`m⁰ih`m⁰|D^†(R)|`mi

= X

m⁰

Y_`^m⁰(ˆn)D_mm^(`)^⇤0(R), (8.5.84) where we have used ` instead of 1, in order to highlight (once again) that all spherical harmonics transform according to the `-th irreducible representation of the rotation group, instead of according to the rotation matrix, as in the second equality of Eq.

(8.5.74). We can then write

r⁰_q=X

q⁰

D_qq⁽¹⁾0^⇤(R)r_q0, (8.5.85) where we used the dummy indices q, q⁰ = `, `+ 1, . . . ,`, instead ofm, m⁰.

The above procedure can be extended along two directions. First, we replace the cartesian coordinates of the position vector by a generic vector operator,Vin Eq. (8.5.83):

r_±₁ !V_±₁ ⌘ ⌥V_x±iV_y

p2 and r₀ !V₀ =V_z. (8.5.86)