QuantumPhysics.dvi

get, up to an immaterial overall phase,

|n〉= cosθ eiψ|+〉+ sinθe−iψ|−〉 (16.2)

so that

Pn=

∣∣

∣acosθe−iψ+bsinθ eiψ

∣∣

∣

2

(16.3)

Demanding that the probability be independent ofnis tantamount to demanding that it be inde-
pendent ofθandψ. Evaluating it forα= 0,π/2 gives|a|=|b|, so that we may seta=|a|eiαand
b=|a|eiβ, so that

Pn=|a|^2

∣∣

∣cosθ+ sinθ e^2 iψ+iβ−iα

∣∣

∣

2

=|a|^2 (16.4)

which requires|a|^2 cos(2ψ+β−α) = 0, for allψ. This is possible only if|a|= 0, but then we
would have|a|^2 +|b|^2 = 0, which contradicts the normalization of the state. We conclude that an
unpolarized beam cannot be described mathematically by a state in Hilbert space.

16.2 The Density Operator

A new mathematical tool is needed to describe the particles in both polarized, unpolarized and
partially polarized beams. This formalism was introduced by John von Neumann in 1927, and
the key object is referred to traditionally referred to as thedensity operator, but it is actually
more accurate to designate it as thestate operator, since this object will characterize a state of a
quantum system.

Apure ensemble, by definition, is a collection of physical systems such thatevery member of
the ensemble is characterized by the same element|ψ〉in Hilbert space.

Amixed ensemble, by definition, is a collection of physical systems such thata fraction of the
members is characterized by a pure state|ψ 1 〉, another fraction of the members is characterized
by a pure state|ψ 2 〉, and so on. We shall assume that each pure state ket|ψi〉is normalized.
Thus, a mixed ensemble is characterized by a numberNof pure states,|ψi〉fori= 1,···,N. Each
pure state|ψi〉enters into the mixed state with apopulation fractionwi≥0, which quantitatively
indicates the proportion of state|ψi〉in the mixture. The population fractions are normalized by

∑N

i=1

wi= 1 (16.5)

A mixture is anincoherent superposition of pure stateswhich means that all relative phase infor-
mation of the pure states must be lost in the mixture. This is achieved by superimposing, not the
states|ψi〉in Hilbert space, but rather the projection operators|ψi〉〈ψi|associated with each pure
state. Thedensity operatorρfor an ensemble ofNpure states|ψi〉incoherently superimposed
with population fractionswiis defined by

ρ=

∑N

i=1

|ψi〉wi〈ψi| (16.6)