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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 self-defense) 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 in-depth 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
15 References
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54

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