The Bloch Sphere and Spin in Quantum Mechanics

     It is not possible to simultaneously measure more than one component of a Pauli spin ½ vector operator, (σ_X, σ_Y ,σ_Z).  This is one of the conclusions from studying the Stern-Gerlach experiment and is, of course, due to the Heisenberg Uncertainty Principle.  Most commonly Heisenberg is discussed in terms of position and momentum, but using a spin ½ gives different insight into measurement, dispersion and the Principle.

     If, for example, a beam of particles with spin ½ is measured in the same direction as it is polarized, say the expectation value of σ_Z  along the laboratory ±Z axes, then measurements give with 100% certainty (no dispersion  ) outcomes of +1 or -1. (we won’t write ±½ ),

     If however an attempt is made to measure the expectation value of  σ_Z   along the ±X direction (by rotating the Stern-Gerlach magnetic field by 90 degrees), then

     This is a completely random outcome: zero lies half way between the two possible values, ±1 , with a dispersion equal to the maximum range of outcomes, ±1 .  In other words any attempt to measure a component of spin other than along its axis of quantization gives dispersion which increases to a maximum in the case that the spin component and the Stern-Gerlach filer are perpendicular.

     The expectation values are plotted in Figure 1,

     One basic reason that it is impossible to simultaneously measure spin in two directions lies in their states.  The Z and X states are superpositions of each other and this makes them complementary.  A common choice of representing the states of a spin ½ in the laboratory Z direction is,

Then the states that express the X components are

     That is the X states are superpositions of the Z states making it impossible to measure in the X states unless the filter angle is rotated to the X direction.  Then the X spin components can be measured with no dispersion, but the Z states then become superpositions of the X states and they cannot be measured.  Only two dispersion free states can be measured in the Stern-Gerlach experiment at any instant.

     Another way of saying this, although less usual, is it is impossible to have the Stern-Gerlach filter simultaneously pointing in two directions or more.  If a spin happened to have more than one axis of quantization we could never tell experimentally. We can only look in one direction.

     This complementary nature of states with non-commuting operators, (σ_X, σ_Y ,σ_Z), is the basis for the Copenhagen Interpretation of Quantum Mechanics (CI).  It states, basically, that if the Z states exist then the X do not, and vice versa. I would rather conclude that it is impossible to determine experimentally if spin has more than one axis of quantization.

     The CI does, however, refer to measurement and clearly it is true that it is impossible to simultaneously measure two observables whose operators do not commute (canonically conjugate operators, or states that are superpositions of each other).

     Let’s go back to the dispersion of spin states.  In the previous Blog the expectation value for measuring σ_Z in an arbitrary direction, θ,φ, depends upon the difference between two probability: one for the number of spins pointing up and the other for the number pointing down—the polarization,

where recall Malus’ Law gives the probabilities from the Stern-Gerlach experiment.  Likewise the dispersion can be obtained,

so that the expectation value with its dispersion displayed is

     The pair, (cosθ, sinθ) map out a circle shown in Figure 3 with the expectation values of  σ_Z  displayed along the ordinate as a function of the filter angle, θ, with the corresponding dispersion on the abscissa.

From this Figure 3 it is easy to read off both the expectation value of  <σ_Z>  and its dispersion when the field points in other directions. The arrow at the top pointing in the Z direction defines the Z states, |+>_0,0=Z   but as the Stern-Gerlach field is rotated, those states are no longer pure (sharp, dispersion free), while the states lying along the new Stern-Gerlach field direction are pure, denoted by  |+>_θ,φ .

This shows there is nothing special about the Z direction and measuring σ_θ,φ in the same direction of θ,φ gives pure states

Now, of course, the axes from Figure 3 must change to reflect the fact that now the |±>_θ,φ states are pure and dispersion free while the states |± >_Z  have dispersion.  So let us just decide that we know that when the field is pointing in some direction and passes through two antipodal points that those states are pure and all the other states on the circle are not pure.

     Hence the coordinate axes in Figures 1 and 3 are now redundant and not used in Figure 4.  Note, though, that putting the field in another direction simply defines another axis, (indeed another experiment) but those new pure states are indistinguishable from the original Z states so we can define a new Z direction. More commonly the old Z basis states  are retained, but then the expectation values and dispersions are more complicated functions, but mean the same thing.

     Suppose that there is no field present.  Then all the infinite number of states on the circle are pure, even though all those states are various superpositions of only two states, usually chosen as |± >_Z .

Also note that the states are always defined with a specific choice of phase, φ , and the phase does not come into the  Malus’ Law probabilities.  Hence there is azimuthal symmetry in the Bloch circle which makes it into the Bloch sphere,

Hence the Bloch Sphere represents all the possible pure spin states.  Pairs of pure state are antipodal points and are orthogonal.  The states are given by

for one choice of phase.  Other choices exist.

     The Bloch sphere nicely represents a spin ½ and is commonly used today to describe qubits.  It is a normal sphere (2-sphere) of unit radius.  It also forms a convex set of states with those on the surface being pure and those on the inside being mixed. The density operator can be written as,

Where the polarization vector, P , is also called the Bloch vector.  It points from the center of the Bloch sphere in any direction.  If the absolute value of the polarization is unity, |P| =1  , then it points to the surface of the Bloch sphere where the states are pure.  If  |P| <1 , then the Bloch vector lies within the sphere and represents a mixed state.

     Felix Bloch is credited as one of the discoverers of Nuclear Magnetic Resonance, (NMR) which is an invaluable tool in determining the structure of molecules, and has led to the important medical imaging technique of Magnetic Resonance Imaging (MRI). The Bloch sphere is extremely useful in understanding pulsed NMR since the states in one direction can be rotated to another direction.  As such the spin polarization vector, P ,  can be visualized as moving across the surface of the Bloch sphere.

     Although the Bloch sphere nicely represents a spin of ½ magnitude it is difficult to generalize to spin with greater magnitude.  This is primarily because spins > ½ have quadrupole , octapole and higher tensor symmetry, and these do not transform as vectors on the surface of a sphere.