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Undergraduate lecture notes in physics

Spin Operators

Siegfried Hess

Published Jan 1, 2015• 0 citations
Publisher
DOI
10.1007/978-3-319-12787-3_13

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arXiv:quant-ph/9906002•Jun 1, 1999•0 citations
This chapter demonstrates how the two kinds of spin can be represented in terms of spin up and spin down operators. The Pauli matrices have many interesting properties. The operators and representing the components of the electron spin can be written in terms of the spin raising and spin lowering operators. Considering the Pauli matrices representing the spin operators, this chapter helps to prove that the operators are Hermitian. This result is what the readers could expect as the operators represent a physical (measurable) quantity, the electron spin. It shows that the operator each has eigenvalues +1, -1. The chapter also shows that the operators obey the commutation relation. These commutation relations show that the three components of the spin cannot be measured simultaneously with the same precision. The result is a confirmation of the conservation of the total spin of the system that the magnitude of the total spin vector is constant.
Recently, we have shown how the interpretation of quantum mechanics due to Lande' can be used to derive from first principles generalized formulas for the operators and some eigenvectors for spin 1/2 Though we gave the operators for all the components of the spin, we did not give the eigenvectors of the operators for the x and y components of the spin. We now give these vectors. In addition, we present a new and simple method of deriving the operators for the x and y components of the spin as well as their vectors from those for the z component. We give a general proof that the operator for the square of the spin is the unit matrix multiplied by the value of the square of the spin.