In this special case, the eigenvalues of the Hamiltonian areh 0 ±h 3.
In the physical realization of this system by a spin^12 particle (ignoring its
spatial motion), the Hamiltonian is given by
H=ge
4 mc(B 1 σ 1 +B 2 σ 2 +B 3 σ 3 ) (3.6)where theBjare the components of the magnetic field, and the physical con-
stants are the gyromagnetic ratio (g), the electric charge (e), the mass (m) and
the speed of light (c). By computingU(t) above, we have solved the problem
of finding the time evolution of such a system, settinghj = 4 gemcBj. For the
special case of a magnetic field in the 3-direction (B 1 =B 2 = 0), we see that
the two different states with well-defined energy (|+^12 〉and|−^12 〉, recall that
the energy is the eigenvalue of the Hamiltonian) will have an energy difference
between them of
2 h 3 =
ge
2 mcB 3
This is known as the Zeeman effect and is readily visible in the spectra of atoms
subjected to a magnetic field. We will consider this example in more detail in
chapter 7, seeing how the group of rotations ofR^3 enters into the story. Much
later, in chapter 45, we will derive the Hamiltonian 3.6 from general principles
of how electromagnetic fields couple to spin^12 particles.
3.4 For further reading
Many quantum mechanics textbooks now begin with the two-state system, giv-
ing a much more detailed treatment than the one given here, including much
more about the physical interpretation of such systems (see for example [96]).
Volume III of Feynman’sLectures on Physics[25] is a quantum mechanics text
with much of the first half devoted to two-state systems. The field of “Quantum
Information Theory” gives a perspective on quantum theory that puts such sys-
tems (in this context called the “qubit”) front and center. One possible reference
for this material is John Preskill’s notes on quantum computation [69].