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Classroom In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. "Classroom" is equally a forum for raising broader ·issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. Kovid Goyal Department of Physics Matrix Magic: Spin Half Systems St. Xavier's College Mahapalika Marg Mumbai 400001, India. Keywords. Quantum mechanics . Figure 1. The Stern-Gerlach filter for spin one-half particles. A beam of spin half particles entering a Stern-Gerlach (S-G )filter gets split into two beams. The upper with spin + 1/2 and the lower with spin -1/2. Either of the two beams can be blocked, so that the device acts as a filter and the output contains particles of only one type. The arrow in the upper right corner indicates the direction of increasing magnetic field gradient and defines an axis for the device. The problem is to carry out a quantum mechanical analysis of this device. Specifically, to calculate what the probabilities are for a particle in a known initial state to be in the upper or lower beam, and how these probabilities change when the device is rotated. The Solution The device can be analysed quantum mechanically, by considering a single particle at a time. First, we identify ( y ~ -76-----------------------------~-----------R-ES-O-N-A-N-C-E--I-F-eb-r-ua-r-y-2-0-0-3 CLASSROOM each spin state (+ or -) with a vector in a Hilbert space. Then, the device with one of the beams blocked, is a projective Von Neumann measurement on the Hilbert space. We further recognise that the Hilbert space is of dimension two since a particular particle can be in at most two mutually orthogonal spin states. The Feynman lectures, Vol. 3, Ch.6. This identification of a single spin half particle as the quantum system of interest, with the device as a projective measurement immediately leads to the conclusion that every Stern-Gerlach filter defines its own basis for the Hilbert space of the particle. Two different S-G filters can differ only in their orientations with respect to some fixed co-ordinate system. Thus, a rotation of one filter into another corresponds to a unitary transformation on the Hilbert space of the particle. This greatly clarifies the problem. The question of finding the probability of a particle that exits from one filter being in the upper or lower beam of the next simply reduces to finding a unitary transformation (a rotation) that relates the two filters. Indeed, this is so simple that it can be accomplished for spin half particles simply from the properties of space and the axioms of quantum mechanics 1 . Let 1+ Z) or 1- Z) denote the spin states of a particle that has passed through the upper or lower beam respectively of a S-G filter with its axis along the Z-direction. Then, for a rotation R( B) about the X -axis: I+Z') cos ~I+Z) - I-Z') -iSin~I+Z) + cos~I-Z), 2 isin~I-Z) 2 (1) (2) where Z' is obtained by rotating Z by B about the Xaxis. -E-S-O-N-A-N-C-E-I-F-e-br-Ua-r-Y-2-0-0-3---------'~-----------------------------n R CLASSROOM For a rotation R( 8) about the Y-axis: I+Z') I-Z') cos ~I+Z) 8 + sin~I-Z) 8 - sin 21+Z ) + cos 21-Z), (3) (4) where Z' is obtained by rotating Z by 8 about the Yaxis. For a rotation R( 8) about the Z-axis: I+Z') I-Z') e- iOj2 1+Z) eiOj2 1_Z), (5) (6) where Z' is obtained by rotating Z by 8 about the Zaxis. Thus, we have obtained the necessary transformation laws that govern the rotation of Stern-Gerlach filters for spin half particles. However, these laws are still in abstract form and cannot be used conveniently for actual calculation. In order to perform actual calculations, we must select a particular representation for the Hilbert space of a spin half particle. A convenient choice is the space of 2 x 1 column matrices over the field C (Complex numbers). Then we make the convenient identification: I+Z) (7) I-Z) (8) Substituting these into the transformation laws, with a little algebra the following matrix relations can be obtained: I+Z') I-Z') R(8)I+Z) R(8)I-Z), (9) (10) --------~-------RESONANCE I February 2003 78 CLASSROOM where the kets are now column matrices and R(B} is a 2 x 2 unitary matrix that represents a rotation of the filter by B about a particular axis. The following unitary matrices were obtained: R(B z ) = ( e-iO/2 0 R(B ) = (COSB/2 -i sin B/2 x R( B ) = 0) eifJ/2 (11) -isinB/2) cos B/2 (12) (COS B/2 - sin B/2) (13) sin B/2 cos B/2 where R( Bi ) denotes a rotation of B about the i axis. Y To demonstrate the power of this formulation, a quick example: Suppose the output from a S-G filter A in which the lower beam is blocked is fed into another filter B, which is different from A by a rotation of a about the X -axis. We wish to calculate the probability that the particle will emerge in the upper beam of B. Then, = [( (+BI+A) = (R(ax) + AI+A) (14) _~oS::~~2 -~:~:/~2) x (~)] t x (~) (15) (cos cr/2 i sincr/2) x (~) (16) = = cos a/2. (17) Thus the required probability is cos 2 a. This method can be easily extended. For compound rotations, simply multiply the corresponding elementary rotation matrices. For a series of n filters, multiply the corresponding probability amplitudes from last to first. This illustrates the power of formulating a problem precisely. It also demonstrates the great utility of the matrix method of doing quantum mechanics. -Fe-b-rU-ar-Y-2-0-0-3--------~--------------------------7-9 -RE-S-O-NA-N-C-E--'