Angular momentum particle with spin

The total angular momentum of a particle with spin, is... 

The electron is a charged particle with angular momentum (orbital and spin) and, as such, it possesses a magnetic moment, fig, given by... 

Fig. 1.11. Diagram of the spin angular momentum vector for a particle with spin quantum number s = j. The only measurable quantities are the spin length V (7+T)/l = and the projection of the spin on the quantization axis (chosen as coincident...

Fig. 12.10. Classical and quantum tops in space, (a) The space is isotropic and therefore the classical top ftteserves its angular momentum i.e.. its axis does not move with respect to distant stars and the top rotates about its axis with a cxxistant speed. This behavior is used in the gyroscopes that help to orient a spaceship with respect to distant stars, (b) The same tc in a homogeneous vector field. The space is no longer isotropic, and therefore the total angular momentum is no longer preserved. The projection of the total momentum on the field direction is still preserved. This is achieved by the precession of the top axis about the direction of the field, (c) A quantum top i.e., an elementary particle with spin quantum number / = in the magnetic field. The projection /- of its spin I is quantized /- =mjH with mj = —, + and, therefore, we have two energy eigenstates that correspond to two precession cones, directed up and down.

Fig. 1.11. Diagram of the spin angular momentum vector for a particle with spin quantum number r = 5 The only measurable...

The separated 3D wave equation can no longer describe any of these fundamental rotations. For this reason, the discovery of electron spin necessitated its introduction into quantum theory as another ad hoc postulate. The anomalous consequence is the unphysical situation of a point particle with spin. Only part of the spin function survives solution of the 3D wave equation, in the form of a complex variable interpreted as orbital angular momentum. In the so-called / -state, it has the peculiar property of non-zero orbital angular momentum with zero component in the direction of an applied magnetic field. This is the price to pay for elimination of spherical rotation. 

The phenomenon of NMR is based on the property of nucleus to possess intrinsic angular momentum. It is often called spin angular momentum or simply spin. Spin is a form of the angular momentum that is not due to the rotation of the particle, but is an intrinsic property of the particle itself. The concept of spin is difficult to visualize, because it is a quantum mechanical property with no analogue in the macroscopic world. 

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

The state of a particle with zero spin s = 0) may be represented by a state function (r, t) of the spatial coordinates r and the time t. However, the state of a particle having spin 5 (5 7 0) must also depend on some spin variable. We select for this spin variable the component of the spin angular momentum along the z-axis and use the quantum number ms to designate the state. Thus, for a particle in a specific spin state, the state function is denoted by (r, ms, t), where ms has only the (2s + 1) possible values —sh, (—s + )h,... 

The second term s may be called the operator for spin angular momentum of the photon. However, the separation of the angular momentum of the photon into an orbital and a spin part has restricted physical meaning. Firstly, the usual definition of spin as the angular momentum of a particle at rest is inapplicable to the photon since its rest mass is zero. More importantly, it will be seen that states with definite values of orbital and spin angular momenta do not satisfy the condition of transversality. 

Spin I> 0 nuclei possess a magnetic dipole or dipole moment, n, which arises from a spinning, charged particle. Nuclei that have a nonzero spin will also have a magnetic moment, and the direction of that magnetic moment is collinear with the angular momentum vector associated with the nucleus. This can be expressed as... 

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