(igamma_{mu}partial^mu-m)psi=0 eching

$$( i \gamma _ { \mu } \partial ^ { \mu } - m ) \psi = 0$$

m uchun yechish

$$\left\{\begin{matrix}\\m=i\gamma _{μ}∂^{\mu }\text{, }&\text{unconditionally}\\m\in \mathrm{C}\text{, }&\psi =0\end{matrix}\right,$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

i\gamma _{μ}∂^{\mu }-m ga \psi ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$i\gamma _{μ}∂^{\mu }\psi -m\psi =0$$

Ikkala tarafdan i\gamma _{μ}∂^{\mu }\psi ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-m\psi =-i\gamma _{μ}∂^{\mu }\psi$$

Tenglama standart shaklda.

$$\left(-\psi \right)m=-i\gamma _{μ}\psi ∂^{\mu }$$

Ikki tarafini -\psi ga bo‘ling.

$$\frac{\left(-\psi \right)m}{-\psi }=-\frac{i\gamma _{μ}\psi ∂^{\mu }}{-\psi }$$

-\psi ga bo'lish -\psi ga ko'paytirishni bekor qiladi.

$$m=-\frac{i\gamma _{μ}\psi ∂^{\mu }}{-\psi }$$

-i\gamma _{μ}∂^{\mu }\psi ni -\psi ga bo'lish.

$$m=i\gamma _{μ}∂^{\mu }$$

γ_μ uchun yechish

$$\left\{\begin{matrix}\gamma _{μ}=-\frac{im}{∂^{\mu }}\text{, }&\mu =0\text{ or }∂\neq 0\\\gamma _{μ}\in \mathrm{C}\text{, }&\psi =0\text{ or }\left(m=0\text{ and }∂=0\text{ and }\mu \neq 0\right)\end{matrix}\right,$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

i\gamma _{μ}∂^{\mu }-m ga \psi ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$i\gamma _{μ}∂^{\mu }\psi -m\psi =0$$

m\psi ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$i\gamma _{μ}∂^{\mu }\psi =m\psi$$

Tenglama standart shaklda.

$$i\psi ∂^{\mu }\gamma _{μ}=m\psi$$

Ikki tarafini i∂^{\mu }\psi ga bo‘ling.

$$\frac{i\psi ∂^{\mu }\gamma _{μ}}{i\psi ∂^{\mu }}=\frac{m\psi }{i\psi ∂^{\mu }}$$

i∂^{\mu }\psi ga bo'lish i∂^{\mu }\psi ga ko'paytirishni bekor qiladi.

$$\gamma _{μ}=\frac{m\psi }{i\psi ∂^{\mu }}$$

m\psi ni i∂^{\mu }\psi ga bo'lish.

$$\gamma _{μ}=-\frac{im}{∂^{\mu }}$$