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###### tags: `Quantum Computing`
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# Quantum Mechanics

## Eigen Vector and Eigen Value
>[color=#00EF9d] **$Mat * \vec{x} = \lambda * \vec{x}$**

推推看還會不會(sucessfuly proofed on 2021/2/7)

可藉由 $det(Mat - \lambda*I) = 0$ 算出$\lambda$
## Quantum Mechanics
### Definitions
- 正交基 **orthogonal** : **if true => inner product=0**
單範正交基 **Orthonormal** basis: e.g. Hermitian matrix
- 滿足 $Q^{-1}=Q^T$且複矩陣存在與實對稱矩陣相應的美好矩陣，稱為 **Hermitian matrix** or **Conjugate matrix**
1. 若 A 是 Hermitian 矩陣，對於任意向量 $\mathbf{x}\in\mathbb{C}^n，\mathbf{x}^{\ast}A\mathbf{x}$ 是實數
2. 特徵值皆為實數
3. 對應相異特徵值的特徵向量互為正交
4. 特徵值的代數重數等於幾何重數
- Particles are discrete
- Schrödinger equation: We can't know the speed and the position of a quatum object at the same time
- Colliding(碰撞) particles bounce off each other
Colliding waves interfere with each other
- Wave don't have a well defined position
- 
- A quantum state is represented by a ket $|\psi>$ in the Hilbert space
- Concept of Classical Physics v.s. Quantum Mechanics

|Classical|Quantum|
|-|-|
|Coordinate System|Hilbert Space|
|Classical Observables|Operator that acts on state **kets**|

### Broglie equation: $\lambda=\dfrac{h}{P}$
- P is momentum
- h=$6.63*10^{-34}$
- Ex: human walking through concrete wall
$\lambda=1.105*10^{-5}=\dfrac{6.63*10^{-34}}{60*velocity}$
=> if velocity=10^{-30}, human can walk through concrete walls

### Stern-Gerlach Experiment

- measuring electron spin
- After passing through a Stern-Gerlach apparatus(equipment), there are 2 values of elecron spin (discrete)
- The orientation(方向) of the experiment corresponds to **the basis** of measurement of electron spin

### Entanglement
- Two entangles quantum state will always be opposite
- More than two quantums can be entangled at the same time, but they can't be fully entangled
#### Application
Teleportation, Cryptography, coding, speedups

## Quantum Postulate

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