Angular momentum half-integral

Note the possibility of half-integral (as well as integral) values for the quantum number j. For orbital angular momentum, only integral quantum numbers occur, but for spin angular momentum, we can have half-integral values. 

Schrodinger s equation has solutions characterized by three quantum numbers only, whereas electron spin appears naturally as a solution of Dirac s relativistic equation. As a consequence it is often stated that spin is a relativistic effect. However, the fact that half-integral angular momentum states, predicted by the ladder-operator method, are compatible with non-relativistic systems, refutes this conclusion. The non-appearance of electron... 

It was shown in Section 1.7 that when the operators Px, PY, Pz °t>ey general angular-momentum commutation relations, as in (5.41), then the eigenvalues of P2 and Pz are J(J+ )h2 and Mh, respectively, where M ranges from — J to J, and J is integral or half-integral. However, we exclude the half-integral values of the rotational quantum number, since these occur only when spin is involved. 

When placed in a static magnetic field of flux density B0, a nucleus may undergo nuclear magnetic resonance (NMR) [1-5] if it possesses an angular momentum p. This angular momentum is referred to as nuclear spin. The component of p in the direction of B0 (Fig. 1.1), denoted as p0, can only take on values which are half-integral or integral multiples m of hj2 n ... 

NUCLEAR SPIN. The intrinsic angular momentum of the atomic nucleus due to rotation about its own axis, It is usually designated I and has the magnitude, JI 1 + 1 )h/2iz /(/z/2jt), where I is the nuclear spin quantum number which has different (integral or half-integral) values... 

However, all three particles in this equation are fermions with intrinsic spins S = jh. Therefore, we cannot balance the angular momentum in the reaction as written. The spins of the proton and the electron can be coupled to 0 or 1 ft and can also have relative angular momenta with any integral value from the emission process. This simple spin algebra will never yield the half-integral value on the left-hand side of the equation. Another fermion must be present among the products. 

Vector representations correspond to integral values of the angular momentum quantum numberj and therefore to systems with an even number of electrons. Spinor representations correspond to systems with half-integral j and therefore to systems with an odd number of electrons. Note that T is the complex conjugate of T. 

We recall from section 5.2.4 that, from the general theory of angular momentum, j can take half-integral (more strictly half-odd) values as well as integral ones. The particular case of j = 1 /2 deserves special mention because of its importance in the discussion of electron or proton spin. For j = 1/2, there are two possible states l/2, 1/2) and 11/2, -1/2) which are often denoted a) and /3) respectively. The spin operators which define these states are particularly simple. For example,... 

Thus, we obtain a special class of so-called diagonal representations of so(4) characterized by j0 = 0 (or jl = j2). This is analogous to the situation in angular momentum theory where only integral values of orbital angular momentum are possible. The half-odd-integral values are ruled out because of the particular realization of L in terms of coordinates and momenta. 

We know from basic quantum mechanics that angular momentum is always quantized in half-integral or integral multiples of h, where h is Planck s constant divided by 2tt. For the electron spin, the multiple (or spin quantum number) is Y2, but the value for nuclear spin differs from one nuclide to another as a result of interactions among the protons and neutrons in the nucleus. If we use the symbol I to denote this nuclear spin quantum number (or, more commonly, just nuclear spin), we can write for the maximum observable component of angular momentum... 

TTje nuclei of certain isotopes possess what can be thought of as a mechanical spin, or angular momentum. In the quantum mechanical description this is characterized by the spin number, l, which has integral or half-integral values. The spin number is related to the mass and atomic number, as summarized in Table 7-1. 

In a free multielectron atom or ion, the spin and orbital angular moments of the electrons couple to give a total angular momentum represented in the Russell-Saunders scheme by the quantum number J. Since J arises from vectorial addition of L (the total orbital quantum number) and 5 (total spin quantum number), it may take integral (or half-integral... 

Many nuclei possess spin angular momentum, analogous to that of the electron. The nuclear spin, designated /, has an integral or half-integral value 0, 1.1,... 

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