ai ai ai 
A=
i
38
14.5 Quantum measurement: General principles
In this section, A will denote a complete nondegenerate observable with
eigenvalues ai and eigenkets ai .
According to quantum measurement theory, the measurement of an ob
servable A of a ket ψ with respect to the basis {ai } produces the eigenvalue
ai with probability
ai  ψ 2
P rob (Value ai is observed) =
and forces the state of the quantum system to become the corresponding
eigenket ai .
Since quantum measurement is such a hotly debated topic among physi
cists, we (in selfdefense) quote P.A.M. Dirac[51]:
“A measurement always causes the (quantum mechanical) system
to jump into an eigenstate of the dynamical variable that is being
measured.”
Thus, the above mentioned measurement of observable A of ket ψ can
be diagrammatically represented as follows:
Meas. of A Meas. of A
ψ = ai ai  ψ aj aj aj
=’ ˜ =’
P rob = aj  ψ 2
P rob = 1
i
The observable
ai ai 
is frequently called a selective measurement operator (or a ¬ltration)
for ai . As mentioned earlier, it has two eigenvalues 0 and 1. 1 is a nonde
generate eigenvalue with eigenket ai , and 0 is a degenerate eigenvalue with
eigenkets {aj }j=i .
Thus,
ai ai 
ψ 1 · ai = ai ,
=’
P rob = ai  ψ 2
39
but for j = i,
ai ai
ψ 0 · aj = 0
=’
2
P rob = aj  ψ
14.6 Polarized light: Part III. Three examples of quan
tum measurement
We can now apply the above general principles of quantum measurement to
polarized light. Three examples are given below:9
Example 7
V
Vertical
Polaroid 1
P rob =
Rt. Circularly 2
¬lter =’
polarized photon
=’
 + i ” )
1
√
= (
2 =’
1
P rob = N
Measurement op. 2
 
Example 8 A vertically polarized ¬lter followed by a horizontally polarized
¬lter.
9
The last two examples can easily be veri¬ed experimentally with at most three pair
of polarized sunglasses.
40
Vert. Horiz.
Entangled polar. polar.
photon Vert.
¬lter ¬lter
2
polar. P
P rob = ±
± + β ” =’ . photon =’
=’

Normalized so that
± 2+ β 2=1
  ” ”
Example 9 But if we insert a diagonally polarized ¬lter (by 45o o¬ the ver
tical) between the two polarized ¬lters in the above example, we have:
2 1
± 1 1
2
= √ ( = √ (
 +  + ” )
)
2 2
’
’
    ”
where the input to the ¬rst ¬lter is ±  + β ” .
14.7 A Rosetta stone for Dirac notation: Part III. Ex
pected values
The average value (expected value) of a measurement of an observable
A on a state ± is:
A = ± A ±
For, since
ai ai  = 1 ,
i
41
we have
A = ± A ± = ± ai ai A aj aj  ± = ±  ai ai  A
i j i,j
But on the other hand,
ai A aj = aj ai  aj = aiδij
Thus,
±  ai ai ai  ± = ai  ± 2
A= ai
i i
Hence, we have the standard expected value formula,
aiP rob (Observing aj on input ± )
A=
i
14.8 Dynamics of closed quantum systems: Unitary
transformations, the Hamiltonian, and Schr¨dinger
o
equation
An operator U on a Hilbert space H is unitary if
U † = U ’1 .
Unitary operators are of central importance in quantum mechanics for many
reason. We list below only two:
• Closed quantum mechanical systems transform only via unitary trans
formations
• Unitary transformations preserve quantum probabilities
42
Let ψ(t) denote the state of a closed quantum mechanical system S as
a function of time t. Then the dynamical behavior of S is determined by the
Schr¨dinger equation
o
‚ i
ψ(t) = ’ H ψ(t) ,
‚t
and boundary conditions, where denotes Planck™s constant and H denotes
an observable of S called the Hamiltonian. The Hamiltonian is the quan
tum mechanical analog of the Hamiltonian classical mechanics. In classical
physics, it is the total energy of the system.
14.9 There is much more to quantum mechanics
There is much more to quantum mechanics. For more indepth overviews,
there are many outstanding books. Among such books are [65], [104], [51],
[95], [99], and many more. Some excellent insights into this subject are also
given in chapter 2 of [97].
43
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