**Do you like math?**

**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}

*A*

⋅ =
• Eigenvalues and eigenkets

,...

, ,

*a* *a* *a*

′ ′′ ′′′
*A a* ′ = *a a* ′ ′ , *A a* ′′ = *a a* ′′ ′′

**Bra Space** **Bra Space**

• Dual Correspondence (DC)

### α α

↔^{DC}

,...

, ,...

,

*a* *a* *a*

*a*

′ ′′ ↔^{DC}′ ′′

### β α

### β

### α + ↔

^{DC}

### +

### 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* ⋅ =

*X* *Y*

*Y*

*X* + = +

### (

*Y*

*Z*

### ) (

*X*

*Y*

### )

*Z*

*X*+ + = + +

### ( ) ^{α} ^{⋅}

^{X}^{=} ^{α}

^{X}**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* =

**Operator Multiplication I** **Operator Multiplication I**

*YX* *XY* ≠

### ( ) ( ) ^{YZ} ^{XY} ^{Z} ^{XYZ}

^{YZ}

^{XY}

^{Z}

^{XYZ}

*X* = =

### (

^{Y}### ) ^{( )}

^{XY}

^{XY}### (

^{X}### )

^{Y}^{( )}

^{XY}

^{XY}*X* α = α = α ,** ** β = β = β

**Operator Multiplication II** **Operator Multiplication II**

### ( ^{Y} ) ( ^{Y}

^{Y}

^{Y}

^{†}

### ) ^{X}

^{X}

^{†}

^{Y}

^{Y}

^{†}

^{X}

^{X}

^{†}

*X*

*XY* α = α ↔ α = α

### ( ) ^{XY}

^{XY}

^{†}^{=} ^{Y}

^{Y}

^{†}^{X}

^{X}

^{†}• Outer product

### ( ) ( ) ^{β} ^{⋅} ^{α} ^{=} ^{β} ^{α}

**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*

**Eigenvalues of Hermitian** **Operator**

**Eigenvalues of Hermitian** **Operator**

**Theorem. **

**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.*

**Eigenkets as base Kets** **Eigenkets as base Kets**

*a* *c*

*a*

*a*

### ′

### = ∑

′ ′

### α

### α

*a* *c*

_{a}_{′}

### = ′

### α

### α *a* *a*

*a*

′

=

### ∑

′′

**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

**Projection Operator** **Projection Operator**

*a* *a*

*A*

_{a}_{′}

### ≡ ′ ′

*I*

*a*

*a*

### =

### ∑ Λ

′

′

### ( ^{a} ′ ^{a} ′ ) ⋅ α = ^{a} ′ ^{a} ′ α = ^{c}

^{a}

^{a}

^{a}

^{a}

^{c}

*a*

_{′}

^{a} ′

^{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 measurement*

*a′*

### α

*A measurement*

*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.,*

### [ ]

^{A}^{,}

^{B}^{=} ^{0}

• *Observables A and B are incompatible when*

### [ ]

^{A}^{,}

^{B}^{≠} ^{0}

**Uncertainty Principle** **Uncertainty Principle**

### ( ) ( )

^{2}

^{2}

### [ ] ^{,}

^{2}

### 4

### 1 *A* *B* *B*

*A* Δ ≥

### Δ

where

*A* *A*

*A* = − Δ

### ( ) ^{Δ} ^{A}

^{A}

^{2}

^{=} ( ^{A}

^{A}

^{2}

^{−} ^{2} ^{A} ^{A} ^{+} ^{A}

^{A}

^{A}

^{A}

^{2}

### ) ^{=} ^{A}

^{A}

^{2}

^{−} ^{A}

^{A}

^{2}

• 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 2

### ≥ Δ Δ

### ≥ Δ

### Δ *p* *x*

*i* *p*

*x*

### , *i*

### Χ Ρ = − ∇

### [ ] ^{Χ,} ^{Ρ} ^{=} ^{i}

^{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*

^{†}: Creation operator

### [ ] ^{=} ^{⎛} ^{1} ^{⎞} ^{(} ^{−} ^{[ ] [ ]} ^{+} ^{)} ^{=}

### (

^{†}

### ) (

^{†}

### )

2

2 , *m* *a* *a*

*i*
*p*
*a*

*m* *a*

*x* = + = ω − +

ω

**Number Operator I** **Number Operator I**

## •

Number Operator*a* *a* *N* =

^{†}

### ( ^{+}

^{1}

_{2}

### )

### = *N*

*H* ω

*n* *n* *n*

*N*

=
### ( ) ^{n} ^{n}

^{n}

^{n}

*n*

*H*

= + ^{1}

_{2}

### ω

**Number Operator II** **Number Operator II**

### [ ^{N a} ^{,} ] ^{=} ^{⎡} _{⎣} ^{a a a}

^{N a}

^{a a a}

^{†}

^{,} ^{⎤} _{⎦} ^{=} ^{a a a}

^{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*

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*

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

### = *n* *n*

*E* _{n} ω

_{n}