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Quantum Spin Systems and Their Local and Long-Time Properties Carolee Wheeler Faculty Advisor: Robert Sims

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Quantum Spin Systems and Their Local and Long-Time Properties Carolee Wheeler Faculty Advisor: Robert Sims URA Project Proposal Quantum Spin Systems and Their Local and Long-Time Properties Carolee Wheeler Faculty Advisor: Robert Sims In quantum mechanics, spin is an important concept having to do with atomic nuclei, hadrons, and elementary particles. Spin may be thought of as a measure of a particle’s rotation about its axis. However, spin differs from orbital angular momentum in the sense that the particles may carry integer or half-integer quantum numbers, i.e. 0, 1/2, 1, 3/2, 2, etc., whereas orbital angular momentum may only take integer quantum numbers. Furthermore, the spin of a charged elementary particle is related to a magnetic dipole moment. All quantum mechanic particles have an inherent spin. This is due to the fact that elementary particles (such as photons, electrons, or quarks) cannot be divided into smaller entities. In other words, they cannot be viewed as particles that are made up of individual, smaller particles that rotate around a common center. Thus, the spin that elementary particles carry is an intrinsic property [1]. An important characteristic of spin in quantum mechanics is that it is quantized. The magnitude of spin takes values S = h s(s + )1 , with h being the reduced Planck’s constant and s being the spin quantum number (a non- negative integer or half-integer). Spin may also be viewed in composite particles, and is calculated by summing the spins of the constituent particles. In the case of atoms and molecules, spin is the sum of the spins of unpaired electrons [1]. Particles with spin can possess a magnetic moment. The magnetic moment of an elementary particle is defined as q µ = g S , 2m where q is the charge of the particle, m is the mass, S is the spin value and g is a dimensionless quantity known as the g-factor (The g-factor is 1 for wholly orbital rotations.) [1]. From a mathematical standpoint, it is interesting to note the fact that spin has commutation properties similar to those of orbital angular momentum: h [Si , S j ] = i ε ijk S k 2 where ε ijk is the Levi-Civita symbol. Moreover, the eigenvectors of S and S z (expressed as kets in the total S basis) are: S 2 s, m = h 2 s(s + )1 s, m h S z s,m = m s, m . Acting on these eigenvectors, the spin raising and lowering operators produce: h S ± s,m = s(s + )1 − m(m + )1 s,m ±1 , where S± = S x ± iS y [1]. The quantum mechanical operators associated with spin observables are: - 1 - h S = σ x 2 x h S = σ y 2 y h S = σ z 2 z And in the case of spin-1/2, σx, σy, and σz are known as the Pauli matrices: 0 1 σ x = 1 0 0 − i σ y = i 0 1 0 σ z = 0 −1 Each of the Pauli matrices has two eigenvalues: +1 and -1. The related normalized eigenvectors are: The most common description of a particle’s spin involves a set of complex numbers that relate to amplitudes of a given value of projection of its angular momentum on a set axis. When measuring the electron spin on the x,y, or z axis, one can only yield an eigenvalue of the spin operator (S x, h h Sy, S z) on that axis, and - . With respect to spin, the quantum state of the particle can be 2 2 represented with a two-component spinor [1]: In 2002, Shruti Kapoor published an essay regarding symmetry in quantum spin systems. In her paper, Kapoor discusses a principle known as Bell’s inequality, which involves some locality properties of spin systems. For the purposes of her discussion, she considers a spin ½ system and three independent directions in which to measure spin. She assumes perfect correlation and ultimately examines the eight cases of possible spin orientations of the electron and positron: - 2 - The Possible Spin Orientations Bell’s work shows In her paper, Kapoor remarks that this work is a direct result from Einstein’s locality principle: “The real factual situation of the system S2 is independent of what is done on system S1, which is spatially separated from the former" [2]. In another paper, Heinz Horner, from the Institut f¨ur Theoretische Physik, presents his results dealing with low frequency and low temperature properties of a spin-boson system. His setup, based on a Liouville space formulation, allows him to examine these properties in terms of bath propagators and static linear and nonlinear susceptibilities. In Section 3 of his paper, he discusses the long time behavior of transverse and longitudinal correlation and response functions, including the Shiba relation and fluctuation dissipation theorems. He proposes that “the leading contribution for long time behavior is given by the lowest order projector which can be inserted.” Following his analysis, he concludes that the leading contribution may be expressed in terms of nonlinear susceptibilities and weighted bath propagators [3]. My research for the fall 2009 semester will involve studying these kinds of quantum spin systems. My work will require analyzing matrices that have to do with these systems, including finding eigenvalues and eigenvectors. We will particularly study the specific properties of the spin systems, both local and long-time. We will deal heavily with the Pauli matrices, defined above, and will work to discover any interesting properties within the systems. - 3 - References [1] “Spin.” Wikipedia. <http://en.wikipedia.org/wiki/Quantum_spin> (9 May 2009). [2] Kapoor, Shruti. “Symmetry in Quantum Spin Systems.” (18 Sep 2002), <http://sps.nus.edu.sg/~shrutika/essay.pdf> (12 May 2009). [3] Horner, Heinz. “Low frequency, low temperature properties of the spin-boson problem,” 15 Sep 2000, <http://arxiv.org/PS_cache/cond-mat/pdf/0007/0007122v2.pdf> (12 May 2009). - 4 - .
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