Applications de mecanique quantique by Aslangul C.

By Aslangul C.

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P + βmc2 ] . 189) Calculons d’abord le commutateur de L avec la partie “cin´etique” de HD . 190) puisque p agit de la mˆeme fa¸con sur chaque composante (p Ψµ = −i ∇ Ψµ ). p] . 192) u d’o` u: [L, HD] = i c (α × p) . p ] 0 . 195) u d’o` u: [Σ, HD] = 2 ic 0 p×σ p×σ 0 ≡ 2 ic p × α . 196), le commutateur de J avec le Hamiltonien de Dirac est : [J, HD] ≡ [L + 2 Σ, HD ] = i c (α × p + p × α) = 0 . 197) Ce r´esultat permet bien d’interpr´eter 2 Σ comme l’expression du moment cin´etique intrins`eque de spin de l’´electron : ajout´e vectoriellement au moment cin´etique orbital, le moment cin´etique r´esultant J est une constante du mouvement.

219)   Ψ−, −1 Comme il se doit, les deux ´etats |Ψε, ±1 et |Ψ−ε, ±1 sont orthogonaux. Pour chaque ´energie, l’observable Σz peut prendre les deux valeurs ±1 ; la valeur propre +1 repr´esente un ´etat o` u la composante Sz du spin est dans le mˆeme sens que l’impulsion k, −1 ´etant associ´e `a l’´etat o` u le spin est “en arri`ere” vis-`a-vis de l’impulsion. Au total, on imagine une rotation autour de la direction de propagation qui est tantˆ ot dans un sens, tantˆ ot dans l’autre : c’est pourquoi on parle d’´etats `a h´elicit´e positive et n´egative, respectivement.

A. 4. INTERACTION SPIN-ORBITE ET STRUCTURE FINE o` u En=1 ∼ e2 a0 est l’ordre de grandeur de l’´energie fondamentale de l’atome d’hydrog`ene. 79). 152)). En tout cas, combin´es avec la r`egle de Bohr, ils donnent lieu a` l’apparition d’une structure dans les raies spectrales : avec une r´esolution m´ediocre, on ne voit qu’une seule raie. En regardant mieux, on constate que chaque raie est le plus souvent constitu´ee de plusieurs raies tr`es voisines : c’est ce que l’on appelle la structure fine46 .

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