PowerPoint Template

Basic Mathematics for Quantum Mechanics

(Sakurai, Modern Quantum Mechanics)

이 병 호

서울대 전기공학부

byoungho@snu.ac.kr

Kets and Bras

β α

bracket

α β

• Hilbert space

P. A. M. Dirac

Ket Space

γ β

α + =

c

c α = α

Operator and Eigenkets

( )

^{α Aα}

A⋅ =

• Eigenvalues and eigenkets

,...

,

, a a

a′ ′′ ′′′

A a′ = a a′ ′ , A a′′ = a a′′ ′′

Bra Space

• Dual Correspondence (DC)

α α ↔^DC

,...

, ,...

, a a a

a′ ′′ ↔^DC ′ ′′

β α

β

α ^{+ ↔DC +}

β α

β

α _{β α β}

α + c ↔ c^∗ + c^∗

c

Inner Product

( ) ( )

^{β α}

α

β = ⋅

Bra (c) ket

• Postulates

≥ 0 α α

= α β ∗

α β

where the equaility sign holds only if is a null ket.^α

Orthogonality & Normalization

• Orthogonality

= 0 β α

• Normalization

α α α α

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

= ⎛ 1

~

~ 1

~α = α

Operators

( )

^{α X α}

X ⋅ =

X Y

Y

X + = +

(

Y Z

) (

X Y

)

Z X + + = + +

( )

^{α ⋅ X = α X}

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 =

Operator Multiplication I

YX XY ≠

( ) ( )

^{YZ XY Z XYZ}

X = =

(

^Y

) ( )

^{XY XY}

(

^X

)

^Y

( )

^{XY XY}

X α = α = α , β = β = β

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

• 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

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′′ = ′′^∗ ′′

(

^{a′ − a ′′∗}

)

^{a ′′ a′ = 0}

Proof.

Eigenkets as base Kets

a c

a

a

′

= ∑

′ ′

α

α

a c

_a′

= ′

α

α a a

a

′

=

∑

′

′

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

Projection Operator

a a

A

_a′

≡ ′ ′

I

a

a

=

∑ Λ

′

′

( a ′ a ′ ) ⋅ α = a ′ a ′ α = c

a_′

a ′

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

∑

∑

′ ′ ′ = ′ ′ ′

=

a a

a a a a

c α

α

a′ ^A measurement a′

α ^A measurement ^a′

for 2

y

Probabilit a′ = a′ α

Expectation value

α α A A =

obtaining for

y

probabilit123

a a'

a a

α a a

a a A a a

A =

∑∑

′′ ′′ ′ ′ =

∑

′ ′

′

′ ′′

α α

measured value a'

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}

Uncertainty Principle

( ) ( )

^{2 2}

[ ]

^{, 2}

4

1 A B

B

A Δ ≥

Δ

where

A A

A = − Δ

( )

^{ΔA 2 =}

(

^{A2 −2A A + A 2}

)

^{= A2 − A 2}

• The uncertainly relation can be obtained from the Schwarz inequality in a straight forward manner.

Example of Uncertainty Relation

( ) ( )

2

4

1

²

2 2

h

h

≥ Δ

Δ

≥ Δ

Δ p x

i p

x

, i

Χ Ρ = − ∇h

[ ]

^{Χ,Ρ = i}_h

Schrödinger’s Cat

Newton Highlight - 양자론, 뉴턴코리아, 2006.

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

h

h , 2

2

†

a : Annihilation operator a^† : Creation operator

[ ] ( [ ] [ ]

^{, ,}

)

¹

2

, ^† 1 ⎟ − + =

⎠

⎜ ⎞

⎝

= ⎛ i x p i p x a

a h

(

^†

) (

^†

)

2

2 , m a a

i p a

m a

x = + = ω − +

ω ^h

h

Number Operator I

•

Number Operator

a a N = ^†

(

^{+ 1}₂

)

= N

H _hω

n n n

N =

( )

^{n n}

n

H = + ¹_{2 h}ω

( )

^{+ 1}_{2 h}^ω

= n E_n

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

−1

= c n 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

hω

2 1 0 = E

( ) + 12 h , ( = 0 , 1 , 2 , 3 , K )

= n n

E

_n

ω

Energies of a Harmonic Oscillator

Who Made the Law?

Is There God?

Richard Dawkins, The God Delusion (2006)

Francis S. Collins, The Language of God (2006)

Alister McGrath and Joanna C. McGrath, The Dawkins Delusion? (2007)

Timothy Keller, The Reason for God (2008)

Ian G. Barbour, When Science Meets Religion (2000)