Spin angular momentum eigenfunctions

In summary, proper spin eigenfunctions must be constructed from antisymmetric (i.e., determinental) wavefunctions as demonstrated above because the total S2 and total Sz remain valid symmetry operators for many-electron systems. Doing so results in the spin-adapted wavefunctions being expressed as combinations of determinants with coefficients determined via spin angular momentum techniques as demonstrated above. In... 

Electrons (and many other particles) have associated with them an intrinsic angular momentum that has come to be called spin . One of the greatest successes of relativistic quantum mechanics is that spin is seen to arise naturally within the relativistic formalism, and does not need to be added post facto as it is in non-relativistic treatments. As with orbital angular momentum, spin angular momentum has x, y, and z components, and the operators 5, Sy, and S, together with orthonormal eigenfunctions a and fi of electron spin, are defined from ... 

Physically, it means that it is possible to know simultaneously the square of the intensity of the spin angular momentum and its component along z. Since the spin wavefunctions are not eigenfunctions of the operators S or /, it is impossible to... 

The wavefunction must be an eigenfunction of the operators corresponding to the square of the total spin angular momentum (S2) and to its z component S2.f... 

The spin eigenfunction 0[w] associated with an axis u u = X, y, or z) has a zero projection of spin angular momentum into this axis. In terms of the usual electron spin functions a... 

Electrons and most other fundamental 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 eigenfunctions of the intrinsic spin angular momentum operator 3... 

UHF calculations consequently give different spatial orbitals for a- and j6-spins. This causes a problem in that UHF wavefunctions are often not eigenfunctions for the square of the total spin operator, S. The total spin operator S is the sum of the spin angular momentum operators of the constituent electrons, s, and is written as... 

The usual choice of the basis functions in the spin space is a and JS, These functions are adapted to the z axis in real space in the sense of being the eigenfunctions of the z component of the spin angular momentum operator ha /2, with eigenvalues +h/2 and - /2, respectively. The usual choice of spin functions similarly adapted to the orthogonal 7 and x axes in real space is (a i )/j2 and (a P)/j2 respectively. These are eigenfunctions of the spin angular momentum operators hGy/2 and fta /2, respectively. 

Dirac notation (p. 19) time evolution equation (p. 20) eigenfunction (p. 21) eigenvalue problem (p. 21) stationary state (p. 22) measurement (p. 22) mean value of an operator (p. 24) spin angular momentum (p. 25) spin coordinate (p. 26)... 

If all spins of a collection of particles, say electrons, are coupled to one total spin angular momentum S, the corresponding total spin eigenfunction psM is constructed. 

We convert this classical expression for the energy into a Hamiltonian operator by replacing the classical quantity f by the operator /. Thus, H = -yBI. Let Mj) denote the function that is simultaneously an eigenfunction of the operators P (for the square of the magnitude of the nuclear-spin angular momentum) and 1. We have... 

As discussed in the chapter on symmetry (chapter 6), neither orbital nor spin angular momentum provide good quantum numbers for the Dirac equation in a central field, and we must instead turn to eigenfunctions of the operators and with eigenvalues j j -1-1) and nij. For a one-electron wave function the angular momentum part can be expressed in a basis of coupled products of a spherical harmonic and a Pauli spinor Ti(mj)... 

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