測試章節

• KaTeX – The fastest math typesetting library for the web

Here is an inline example, $\pi (\theta )$,

an equation,

$$\nabla f(x)\in {\mathbb{R}}^n,$$

and a regular $ symbol.

Another common application of classical superposition is finding the total magnitude and direction of quantities such as force, electric field, magnetic field, etc. For example, to calculate the total electric force ${\vec{F}}_{\text{total}}$ on a charge $q_2$ produced by other charges $q_1$ and $q_3$, one would sum the forces produced by each individual charge: ${\vec{F}}_{\text{total}}={\vec{F}}_{12}+{\vec{F}}_{32}$. The challenge here is that forces are vectors, so vector addition is needed, as shown in Figure.

經典疊加的另一個常見應用是求合力，電場，磁場等量的正比。例如，計算由其他電荷 $q_1$ 和 $q_3$ 產生的電荷 $q_2$ 上的總力 ${\vec{F}}_{\text{total}}$，總和為每個單獨電荷產生的力：${\vec{F}}_{\text{total}}={\vec{F}}_{12}+{\vec{F}}_{32}$。這裡的挑戰是力量是向量，因此需要向量加法。

Dirac bra-ket notation

In order to work with qubits, it is useful to know how one can express quantum mechanical states with mathematical formulas.

Dirac or bra-ket notation is commonly used in quantum mechanics and quantum computing. The state of a qubit is enclosed in the right half of an angled bracket, called the ket A qubit, $|\Psi ⟩$, could be in a $|0⟩$ or $|1⟩$ state which is a superposition of both $|0⟩$ and $|1⟩$. This is written as

$$|\Psi ⟩=\alpha |0⟩+\beta |1⟩$$

with $\alpha$ and $\beta$ called the amplitudes of the states. Amplitudes are generally complex numbers (a special type of number used in mathematics and physics). However, to understand the meaning of amplitudes, we can just imagine the amplitudes as being ordinary (real) numbers. Amplitudes allow us to mathematically represent all of the possible superpositions.

為了使用量子位，知道一個人如何表達量子力學狀態是很有用的與數學公式。狄拉克（Dirac）或“布雷克”（bra-ket）表示法通常在量子力學中使用和量子計算。量子位的狀態包含在尖括號的右半部分，叫做 ket。量子位 $|\Psi ⟩$ 可能處於 $|0⟩$ 或 $|1⟩$ 狀態，這是兩個 $|0⟩$ 的疊加和 $|1⟩$ 。這寫成

$$|\Psi ⟩=\alpha |0⟩+\beta |1⟩$$

其中α和β稱為狀態的振幅。振幅通常是複數（數學和物理中使用的特殊數字類型）。但是，要了解其含義在振幅方面，我們只能將振幅想像為普通（實數）數。振幅使我們能夠數學上表示所有可能的疊加