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Do you like math?

Here follows some elegant description of QM...

이 병 호

Kets and Bras Kets and Bras

β α

bracket

α β

• Hilbert space

P. A. M. Dirac

Ket Space Ket Space

γ β

α + =

c

c α = α

Operator and Eigenkets Operator and Eigenkets

( ) ^{α A α}

A

• Eigenvalues and eigenkets

,...

, ,

a a a

A a ′ = a a ′ ′ , A a ′′ = a a ′′ ′′

Bra Space Bra Space

• Dual Correspondence (DC)

α α

,...

, ,...

,

a a a

a

β α

β

α + ↔

+

β α

β

α + ↔

+

Inner Product Inner Product

( ) ( ) ^{β α}

α

β

Bra (c) ket

• Postulates

≥ 0 α α

= α β

α

β

Orthogonality & Normalization Orthogonality & Normalization

• Orthogonality

= 0 β α

• Normalization

α α α α

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

= ⎛ 1

~

~

~

α

α

Operators Operators

( ) ^{α X α}

X ⋅ =

X Y

Y

X + = +

(

) (

)

( ) ^{α ⋅}

^{= α}

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 ↔

α

X

X =

Operator Multiplication I Operator Multiplication I

YX XY ≠

( ) ( ) ^{YZ XY Z XYZ}

X = =

(

) ^{( )}

(

)

^{( )}

X α = α = α , β = β = β

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}

Associative Axiom Associative Axiom

• Important Examples

(

)

(

)

: If

= β α , then

= α β 예

( ) (

) (

) ( )

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

Eigenvalues of Hermitian Operator

Eigenvalues of Hermitian Operator

Theorem.

a a a

A ′ = ′ ′ a a

A

a

( )

Proof.

Eigenkets as base Kets Eigenkets as base Kets

a c

a

a

′

= ∑

′ ′

α

α

a c

= ′

α

α a a

a

′

=

∑

′

Completeness Relation Completeness Relation

I a

a

a

′ =

∑ ′

′

∑

∑

= ′

⎟ ⎠

⎜ ⎞

⎝

⎛ ′ ′ ⋅

⋅

=

a a

a a

a α α

α α

α

2

1

2

= ∑ ′ =

∑

a

a

c α

Projection Operator Projection Operator

a a

A

≡ ′ ′

I

a

a

=

∑ Λ

′

′

( ^a ′ ^a ′ ) ⋅ α = ^a ′ ^a ′ α = ^c

^a ′

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.

Measurement II Measurement II

∑

∑

′ =

′ ′

=

a a

a a a a

c

α

α

a′

a′

α

a′

for 2

y

Probabilit a′ = a′

α

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'

Compatibility & Incompatibility Compatibility & Incompatibility

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

[ ]

^,

^{= 0}

• Observables A and B are incompatible when

[ ]

^,

^{≠ 0}

Uncertainty Principle Uncertainty Principle

( ) ( )

[ ] ^,

4

1 A B B

A Δ ≥

Δ

where

A A

A = − Δ

( ) ^{Δ A}

⁼ ( ^A

^{− 2 A A + A}

) ^{= A}

^{− A}

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

Example of Uncertainty Relation Example of Uncertainty Relation

( ) ( )

2

4

1

2 2

≥ Δ Δ

≥ Δ

Δ p x

i p

x

, i

Χ Ρ = − ∇

[ ] ^{Χ, Ρ = i}

Schrödinger’s Cat

Schrödinger’s Cat

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

[ ] ^{= ⎛ 1 ⎞ ( − [ ] [ ] + ) =}

(

) (

)

2

2 , m a a

i p a

m a

x = + = ω − +

ω

Number Operator I Number Operator I

•

a a N =

( ⁺

)

= N

H ω

n n n

N

( ) ^{n n}

n

H

ω

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 = n − a n

( ) ^{( )}

† † † †

, 1

Na n = ⎡ ⎣ N a ⎤ ⎦ + a N n = n + a n

Annihilation and Creation Operators Annihilation and Creation Operators

− 1

= n c n

a

†

a n c

a

n =

c

n =

− 1

= n n n

a

1

†

n = n + 1 n +

a

(

) ^⋅ ( ) ^{≥ 0}

=

= n N n n a a n

n

( ) ^{+ 1} ₂ ^{, ( = 0 , 1 , 2 , 3 , … )}

= n n

E _n ω

Energies of a Harmonic Oscillator

Energies of a Harmonic Oscillator