Spin Precession of Quantum-mechanical Spin-1/2 Systems:
Exercises

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

OSP Spins is an interactive computer program that simulates Stern-Gerlach-type measurements on spin-1/2 and spin-1 particles. The original Java Version was written by David McIntyre and was based on the original Macintosh version by Daniel Schroeder and Thomas Moore [Am. J. Phys. 61, 798 (1993)].  Within the context of the OSP project, the Java program has been extended allowing for "experimental" parameters to be saved in XML files.

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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).

The spin operators, Sx, Sy, and Sz represent the effect of a measurement of the spin in one of those three directions, while S2 describes the measurement of the spin squared of a particle.  It turns out that these four operators can be written in terms of Pauli 2 x 2 matrices

 

as Si = ħ/2 σi, where specifically

The spin operators also obey the following commutation relations

and [Si, S2] = 0, based on the commutation relations of the Pauli matrices. 

Since the z component of a spin-1/2 particle can take on just two values ms = +1/2 or -1/2, we often call these states "spin up" or "spin down" but we are actually referring just to the z component of spin.  Since there are two values, we can write these states as two-component column vectors which are eigenstates of Sz:

It is also of interest to write down the eigenstates of the other spin operators.  In particular, we find that the eigenstates of Sx are:

or that

We also have that

and finally

The form of these states should convince you that eigenstates of one component of spin, will not be eigenstates of the other components of spin.

 

© Mario Belloni and Wolfgang Christian (2006).