Spin - Spin Michael Fowler 11/26/06 Introduction The Stern...

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Spin Michael Fowler 11/26/06 Introduction The Stern Gerlach experiment for the simplest possible atom, hydrogen in its ground state, demonstrated unambiguously that the component of the magnetic moment of the atom along the z -axis could only have two values. It had been well established by this time that the magnetic moment vector was along the same axis as the angular momentum. This is obviously true for the Bohr model of hydrogen, where the circulating electron is equivalent to a ring current, generating a magnetic dipole. The problem is, though, that a magnetic moment generated in this way by orbital angular momentum will have a minimum of three possible values of its z -component: the lowest nonzero orbital angular momentum is 1, l = with allowed values of the z -component ,1 , 0 , mm =− = 1 . Recall, however, that in our derivation of allowed angular momentum eigenvalues from very general properties of rotation operators, we found that although for any system the allowed values of m form a ladder with spacing we could not rule out half-integral m values. The lowest such case, would in fact have just two allowed m values: However, this cannot be any kind of orbital angular momentum because the z -component of the orbital wave function , = 1/2, l = 1/2, 1/2. m ψ has a factor , i e ϕ ± and therefore picks up a factor -1 on rotating through 2, π meaning is not single-valued, which doesn’t make sense for a Schrödinger wave function. Yet the experimental result is clear. Therefore, this must be a new kind of non-orbital angular momentum. It is called “spin”, the simple picture being that just as the Earth has orbital angular momentum in its yearly circle around the sun, and also spin angular momentum from its daily turning, the electron has an analogous spin. But the analogy has obvious limitations: the Earth’s spin is after all made up of material orbiting around the axis through the poles, the electron’s spin cannot similarly be imagined as arising from a rotating body, since orbital angular momenta always come in integral multiples of . = Fortunately, this lack of a simple quasi-mechanical picture underlying electron spin doesn’t prevent us from using the general angular momentum machinery previously developed, which followed just from analyzing the effect of spatial rotation on a quantum mechanical system. Recall this led to the spacing = of the ladder of eigenvalues, and to values of the matrix elements of angular momentum components J i between the eigenkets ,: j m enough information to construct matrix representations of the rotation operators for a system of given angular momentum. As an example, for the orbital angular momentum 1 jl = = state, we constructed the matrix representation of an arbitrary rotation operator 33 × iJ e θ G G = in the space with orthonormal basis 1,1 , 1, 0 1 , 1, (in the , lm notation). The spin 1/2 js = = case can be handled in exactly the same way.
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2 Spinors, Spin Operators, Pauli Matrices The Hilbert space of angular momentum states for spin one-half is two dimensional. Various
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Spin - Spin Michael Fowler 11/26/06 Introduction The Stern...

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