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Do you like math?

Do you like math?

Here follows some elegant description of QM...

이 병 호

(2)

Kets and Bras Kets and Bras

β α

bracket

α β

• Hilbert space

P. A. M. Dirac

(3)

Ket Space Ket Space

γ β

α + =

c

c α = α

(4)

Operator and Eigenkets Operator and Eigenkets

( ) α A α

A

⋅ =

• Eigenvalues and eigenkets

,...

, ,

a a a

′ ′′ ′′′

A a ′ = a a ′ ′ , A a ′′ = a a ′′ ′′

(5)

Bra Space Bra Space

• Dual Correspondence (DC)

α α

DC

,...

, ,...

,

a a a

a

′ ′′ ↔DC ′ ′′

β α

β

α + ↔

DC

+

β α

β

α + ↔

+

(6)

Inner Product Inner Product

( ) ( ) β α

α

β

= ⋅

Bra (c) ket

• Postulates

≥ 0 α α

= α β

α

β

(7)

Orthogonality & Normalization Orthogonality & Normalization

• Orthogonality

= 0 β α

• Normalization

α α α α

⎟⎟

⎜⎜

= ⎛ 1

~

~

~

α

α

(8)

Operators Operators

( ) α X α

X ⋅ =

X Y

Y

X + = +

(

Y Z

) (

X Y

)

Z X + + = + +

( ) α

X

= α

X

(9)

Hermitian Adjoint & Hermitian Operator Hermitian Adjoint & Hermitian Operator

• X is called Hermitian adjoint of X if

• An operator X is said to be Hermitian if

X

X α ↔

DC

α

X

X =

(10)

Operator Multiplication I Operator Multiplication I

YX XY

( ) ( ) YZ XY Z XYZ

X = =

(

Y

) ( )

XY XY

(

X

)

Y

( )

XY XY

X α = α = α , β = β = β

(11)

Operator Multiplication II Operator Multiplication II

( Y ) ( Y

) X

Y

X

X

XY α = α ↔ α = α

( ) XY

= Y

X

• Outer product

( ) ( ) β α = β α

(12)

Associative Axiom Associative Axiom

• Important Examples

(

β α

)

γ = β

(

α γ

)

: If

X

= β α , then

X

= α β 예

( ) (

β X α

) (

= β X

) ( )

α = α X β

bra ket bra ket

( ) = { ( ) }

=

= β α α β α β

α

β X X X

X

• For a Hermitian X, we have

= α β

α

β X X

(13)

Eigenvalues of Hermitian Operator

Eigenvalues of Hermitian Operator

Theorem.

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

a a a

A ′ = ′ ′ a a

A

a

′′ = ′′ ′′

( )

Proof.

(14)

Eigenkets as base Kets Eigenkets as base Kets

a c

a

a

= ∑

α

α

a c

a

= ′

α

α a a

a

=

(15)

Completeness Relation Completeness Relation

I a

a

a

′ =

∑ ′

2

= ′

⎟ ⎠

⎜ ⎞

⎛ ′ ′ ⋅

=

a a

a a

a α α

α α

α

2

1

2

= ∑ ′ =

a a

a

a

c α

(16)

Projection Operator Projection Operator

a a

A

a

≡ ′ ′

I

a

a

=

∑ Λ

( aa ′ ) ⋅ α = aa ′ α = c

a

a

(17)

Measurement I Measurement I

• For an observable, there corresponds a Hermitian operator.

If we measure the observable, only eigenvalues of the Hermitian operator can be measured. A measurement

always causes the system to jump into an eigenstate of the

Hermitian operator.

(18)

Measurement II Measurement II

′ =

′ ′

=

a a

a a a a

c

α

α

a′

A measurement

a′

α

A measurement

a′

for 2

y

Probabilit a′ = a

α

(19)

Expectation value Expectation value

α α A A =

obtaining for

y

probabilit a' a

a a

α a a

a a A a a

A = ∑∑ ′′ ′′ ′ ′ = ∑ ′ ′

′′

α α

measured value a'

(20)

Compatibility & Incompatibility Compatibility & 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

0

(21)

Uncertainty Principle Uncertainty Principle

( ) ( )

2 2

[ ] ,

2

4

1 A B B

A Δ ≥

Δ

where

A A

A = − Δ

( ) Δ A

2

= ( A

2

2 A A + A

2

) = A

2

A

2

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

(22)

Example of Uncertainty Relation Example of Uncertainty Relation

( ) ( )

2

4

1

2

2 2

≥ Δ Δ

≥ Δ

Δ p x

i p

x

, i

Χ Ρ = − ∇

[ ] Χ, Ρ = i

(23)

Schrödinger’s Cat

Schrödinger’s Cat

(24)

Harmonic Oscillator Harmonic Oscillator

2 2

2 2

2

m x

m

H = p + ω

⎟ ⎠

⎜ ⎞

⎝ ⎛ −

⎟ =

⎜ ⎞

⎝ ⎛ +

= ω

ω ω

ω

m x ip a m

m x ip a m

2 2 ,

a : Annihilation operator a

: Creation operator

[ ] = 1 ( [ ] [ ] + ) =

(

) (

)

2

2 , m a a

i p a

m a

x = + = ω − +

ω

(25)

Number Operator I Number Operator I

Number Operator

a a N =

( +

12

)

= N

H ω

n n n

N

=

( ) n n

n

H

= + 12

ω

(26)

Number Operator II Number Operator II

[ N a , ] = a a a

, = a a a

[ ] , + a a a

, = − a

N a , a

⎡ ⎤ =

⎣ ⎦

[ ]

( , ) ( 1 )

Na n = N a + aN n = na n

( ) ( )

, 1

Na n = ⎡ ⎣ N a ⎤ ⎦ + a N n = n + a n

(27)

Annihilation and Creation Operators Annihilation and Creation Operators

− 1

= n c n

a

a n c

2

a

n =

c

2

n =

− 1

= n n n

a

1

n = n + 1 n +

a

(

) ( ) 0

=

= n N n n a a n

n

(28)

( ) + 1 2 , ( = 0 , 1 , 2 , 3 , )

= n n

E n ω

Energies of a Harmonic Oscillator

Energies of a Harmonic Oscillator

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