Single Measurements of Quantum-mechanical Spin-1/2 Systems:
Exercises

Part of the Collection of OSP-based Programs and Curricular Materials for Advanced Physics

The curricular material uses the OSP Spins program to show the effect of measurement on an ensemble of spin-1/2 particles.  In the virtual experiment, a beam of spin-1/2 particles leaves an oven incident on one or multiple ideal analyzers (ideal Stern-Gerlach apparatus) the result of which is detected as a click on a detector.

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Spin-1/2 systems are perhaps the simplest, yet interesting, systems to explore the effect of quantum-mechanical measurement on states.   There are, in some sense, two different kinds of time evolution in quantum mechanics: predictable time evolution governed by the Schrödinger equation, which in one-dimensional position space is

[−(ħ2/2m)∂2/∂x2 + V(x)] ψ(x,t) = (∂/∂t) ψ(x,t) ,      

and the abrupt time evolution (collapse) of wave functions when something is measured.  Feynman once stated that "no one understands quantum mechanics,"  by this he most certainly meant to the collapse of the wave function due to measurement.  To complicate matters, multiple measurements in quantum mechanics need not yield the same results due to the uncertainty principle (such as the measurement of x then p then x again). 

For a spin-1/2 system, the 1/2 refers to the quantum number s.  Spin is spin angular momentum and is related to the intrinsic magnetic moment of a particle.  Unlike orbital angular momentum (l = 0, 1, 2,...), spin angular momentum can take on integer and half-integer values (s = 0, 1/2, 1, 3/2, 2,...).  Half-integer particles are called fermions (obeying Fermi-Dirac statistics) and integer spin particles are called bosons (obeying Bose-Einstein statistics).

 
© Mario Belloni and Wolfgang Christian (2006).