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이 병 호

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bracket

α β

• Hilbert space

P. A. M. Dirac

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A

⋅ =

• Eigenvalues and eigenkets

,...

, ,

′ ′′ ′′′

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Bra SpaceBra Space

• Dual Correspondence (DC)

DC

,...

, ,...

,

′ ′′ ↔DC ′ ′′

DC

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= ⋅

Bra (c) ket

• Postulates

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• Orthogonality

= 0 β α

• Normalization

α α α α

⎟⎟

⎜⎜

= ⎛ 1

~

~

~

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Y Z

X Y

Z X + + = + +

X

=α

X

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• X is called Hermitian adjoint of X if

• An operator X is said to be Hermitian if

DC

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Y

XY XY

X

Y

( )

XY XY

X α = α = α , β = β = β

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• Outer product

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Associative AxiomAssociative Axiom

• Important Examples

β α

γ = β

α γ

X

X

β X α

= β X

α = α X β

bra ket bra ket

X

• For a Hermitian X, we have

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Theorem.

The eigenvalues of a Hermitian operator A are real; the eigenkets of A corresponding to different eigenvalues are orthogonal.

′′ = ′′ ′′

Proof.

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a

a

a

a

=

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a

2

a a

2

2

a a

a

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a

a

a

a

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a a

a a a a

c

A measurement

A measurement

a′

for 2

y

Probabilit a′ = a

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obtaining for

y

probabilit a' a

a a

′′

α α

measured value a'

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Compatibility & IncompatibilityCompatibility & Incompatibility

Observables A and B are compatible when the corresponding operators commute, i.e.,

A

B

=0

Observables A and B are incompatible when

A

B

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2 2

2

where

2

2

2

2

−A

2

The uncertainly relation can be obtained from the Schwarz inequality in a

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2

2 2

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2 2

2

a : Annihilation operatora

: Creation operator

2

2 , m a a

i p a

m a

x = + = ω − +

ω

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Number Operator

12

=

= + 12

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2

2

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