Quantum Mechanics for Mathematicians

u+(x) is determined by setting it to be

(

1

0

)

at the North pole, and defin-

ing it at other pointsxon the sphere by acting on it by the element Ω

which, acting on vectors by conjugation (as usual using the identification

of vectors and complex matrices), would take the North pole tox.

• With the specific choices made,u+(x) is discontinuous at the South pole,
  wherex 3 =−1, andφis not uniquely defined. For topological reasons,
  there cannot be a continuous choice ofu+(x) with unit length for all
  x. In applications one generally will be computing quantities that are
  independent of the specific choice ofu+(x), so the discontinuity (which is
  choice-dependent) should not cause problems.

One can similarly pick a solutionu−(x) to the equation

(σ·x)u−(x) =−u−(x)

for eigenvectors with eigenvalue−1, with a standard choice

u−(x) =

1

√

2(1 +x 3 )

(

−(x 1 −ix 2 )

1 +x 3

)

=

(

−e−iφsinθ 2

cosθ 2

)

For eachx,u+(x) andu−(x) satisfy

u+(x)†u−(x) = 0

so provide an orthonormal (for the Hermitian inner product) complex basis for
C^2.

Digression.The association of a different vector spaceC⊂ H=C^2 to each
pointxby taking the solutions to equation 7.5 is an example of something called
a “vector bundle” over the sphere ofx∈S^2. A specific choice for eachxof a
solutionu+(x)is called a “section” of the vector bundle. It can be thought of as
a sort of “twisted” complex-valued function on the sphere, taking values not in
the sameCfor eachxas would a usual function, but in copies ofCthat vary
withx.
These copies ofCmove around inC^2 in a topologically non-trivial way: they
cannot all be identified with each other in a continuous manner. The vector bun-
dle that appears here is perhaps the most fundamental example of a topologically
non-trivial vector bundle. A discontinuity such as that found in the section
u+of equation 7.6 is required because of this topological non-triviality. For a
non-trivial bundle like this one, there cannot be continuous non-zero sections.

While the Bloch sphere provides a simple geometrical interpretation of the
states of the two-state system, it should be noted that this association of points
on the sphere with states does not at all preserve the notion of inner product.
For example, the North and South poles of the sphere correspond to orthogonal
vectors inH, but of course (0, 0 ,1) and (0, 0 ,−1) are not at all orthogonal as
vectors inR^3.