
Tutorial
Quantum Notation
The allusion to vectors representing quantum states can be made precise.
Using a notation introduced by Dirac, it is common to represent quantum
states as abstract vectors such as V> (vertical photon polarization)
or H> (horizontal photon polarization) in Hilbert space. These
objects are known as "kets," (the right portion of the word "bracket").
For our purposes we need not be concerned with the formal properties
of this vector space, and will merely concern ourselves with some of
the basic results.
One of the important properties of vectors is their length, or more generally
the length of the component of one vector along another, which is known as the
scalar product of the two vectors. In quantum notation, it is useful to introduce
the complex conjugate vectors, <V and <H, known as "bras" (from
the left hand portion of the word bracket) to define the scalar product using
the relations:
<
VV> = <HH> = 1, <VH> = <HV> = 0
The length of a quantum state vector is related to the probability of
an experimental outcome or measurement, which mathematically is represented
by the idea of projection: the component of a vector along another vector. For
example, a measurement of "V" polarization would be represented by
the projection operator, P(V) = V><V. Thus, a measurement for "V" polarization
on a V> photon state produces the state V>, corresponding to the 100%
probability that this state has the "V" polarization on measurement.
Conversely, there is no component of an H> state along the vector V> which
corresponds to the experimental result that an "H" photon will never
pass a test for "V"ness.

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