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American Journal of Physics

Stern–Gerlach measurements with arbitrary spin: Quantum combs

Kurt Schönhammer

Published Jan 1, 2000• 4 citations
Publisher

Abstract

As in quantum mechanics the operators of different Cartesian components of the angular momentum do not commute these observables cannot simultaneously have sharp values. In this paper we present a simple method to calculate the probability distributions for the measurements of angular momentum components other than ĵz in eigenstates of ĵz with an arbitrary value of j, where ℏ2j(j+1) is the eigenvalue of ĵ2. Surprisingly this method is not found in textbooks, where usually the case j=12 is presented in discussions of Stern–Gerlach experiments. For larger values of j, the probability distributions have a comb-like structure. A detailed discussion of the quasiclassical limit j≫1 is presented for the measurement of ĵx. It yields rather surprising results.

DOI
10.1119/1.19372

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As in quantum mechanics the operators of different Cartesian components of the angular momentum do not commute these observables cannot simultaneously have sharp values. In this paper we present a simple method to calculate the probability distributions for the measurements of angular momentum components other than ĵz in eigenstates of ĵz with an arbitrary value of j, where ℏ2j(j+1) is the eigenvalue of ĵ2. Surprisingly this method is not found in textbooks, where usually the case j=12 is presented in discussions of Stern–Gerlach experiments. For larger values of j, the probability distributions have a comb-like structure. A detailed discussion of the quasiclassical limit j≫1 is presented for the measurement of ĵx. It yields rather surprising results.
We analyze a coincidence Stern-Gerlach measurement often discussed in connection with the derivation and illustration of Bell's theorem. The treatment is based on our recent analysis of the original Stern-Gerlach experiment (PCCP, 14, 1677‐1684 (2012)), where it is concluded that it is necessary to include a spin relaxation process to account for the experimental observations. We consider two limiting cases of a coincidence measurement using both an analytical and a numerical description. In on limit relaxation effects are neglected. In this case the correlation between the two spins present in the initial state is conserved during the passage through the magnets. However, at exit the z coordinate along the magnetic field gradient is randomly distributed between the two extreme values. In the other limit T2 relaxation is assumed to be fast relative to the time of flight through the magnet. In this case the z coordinate takes one of two possible values as observed in the original Stern‐Gerlach experiment. Due to the presence of a relaxation process involving transfer of angular momentum between particle and magnet the initially entangled spin state changes character leading to a loss of correlation between the two spins. In the original derivations of Bell's theorem based on a coincidence Stern‐Gerlach setup one assumes both a perfect correlation between the spins and only two possible values for the z‐coordinate on exit. According to the present calculations one can satisfy either of these conditions but not both simultaneously.
Coherent states provide an appealing method to reconstruct efficiently the pure state of a quantum mechanical spin s. A Stern-Gerlach apparatus is used to measure (4s+1) expectations of projection operators on appropriate coherent states in the unknown state. These measurements are compatible with a finite number of states which can be distinguished, in the generic case, by measuring one more probability. In addition, the present technique shows that the zeroes of a Husimi distribution do have an operational meaning: they can be identified directly by measurements with a Stern-Gerlach apparatus. This result comes down to saying that it is possible to resolve experimentally structures in quantum phase-space which are smaller than ¯h. To determine the unknown state of a quantum system poses a problem which is interesting from a theoretical, an experimental, and a conceptual point of view. To set up a successful approach to state reconstruction (see [1] and [2] for reviews) means to identify a collection of observables, a quorum, such that their expectation values contain
Abstract We analyze idealized sequential Stern–Gerlach (SG) experiments with higher spin particles. This analysis serves at least two purposes: the widely discussed spin-1/2 case leads to some misunderstandings since the probabilities are always evenly distributed for the sequential orthogonal magnets, which does not generalize to higher spins. A detailed discussion of the higher spin case, as is done here, is highly useful. Secondly, the Wigner rotation matrices for generic spins become conceptually more transparent with this physical example. We also give compact formulas for the probabilities in terms of the angle between the sequential SG apparatus for generic spins. We work out the spin-1/2, spin-1 cases explicitly. Since there are some confusing issues regarding the actual experiment, we also compile a ‘facts and fiction’ section on the SG experiments.
This chapter presents key results on angular momentum in quantum mechanics, beginning with a theorem on the simultaneous diagonalization of commuting operators. The eigenvalues and eigenstates of both the total angular momentum operator and its z-component are derived. For a quantum rotator in the p-state with zero magnetic quantum number, it is shown that the most probable trajectory lies in a plane containing the z-axis, and the angular momentum vector is perpendicular to it. The procedure for angular momentum addition is explained in detail, including the use of Clebsch–Gordan coefficients to perform the explicit decomposition of tensor products of irreducible representations. The Bloch sphere representation of qubit spin states is introduced. The chapter then provides an experimental background to the discovery of spin, introduces the Pauli matrices, and demonstrates how they act on spin states. It concludes with the formulation of the Pauli equation, which incorporates spin into the framework of nonrelativistic quantum mechanics. Metadata/Keywords: angular momentum, simultaneous diagonalization, magnetic quantum number, Clebsch–Gordan coefficients, spin, Pauli matrices, Bloch sphere, spin states, tensor product, Pauli equation
Coherent states provide an appealing method to reconstruct efficiently the pure state of a quantum mechanical spin s . A Stern-Gerlach apparatus is used to measure (4 s +1) expectations of projection operators on appropriate coherent states in the unknown state. These measurements are compatible with a finite number of states which can be distinguished, in the generic case, by measuring one more probability. In addition, the present technique shows that the zeros of a Husimi distribution do have an operational meaning: they can be identified directly by measurements with a Stern-Gerlach apparatus. This result comes down to saying that it is possible to resolve experimentally structures in quantum phase space which are smaller than .