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Schrodinger equation

きゅうり

Do you know Schrodinger equation?
\displaystyle \left(-\frac{\hbar^2}{2m}\nabla^2+V(r)\right)\psi(r,t)=i\hbar\frac{d}{dt}\psi(r,t)

It is one of the most important equations in quantum physics. It looks very difficult but the equation just shows energy of a particle. I will explain that at this article.

In classical physics(dynamics), the sum of kinetic energy and potential energy is total energy.
The total energy of a particle is shown by the following equation
\displaystyle E=K+V,
where K is kinetic energy, V is potential energy and E is total energy and constant value.

Assuming the mass of the particle is m and the momenta is P, the kinetic energy equals to \displaystyle p^2/2m. Substituting the equation into the equation of energy, you can obtain
\displaystyle E=\frac{p^2}{2m}+V.

This equation is similar to Schrodinger equation, isn't it?

In quantum physics, physical value is operator. The eigenstates are wavefunctions. we can observe eigenvalues of the operator by experiments.The operator of energy and momentum are
\displaystyle \hat{p}\rightarrow i\hbar\nabla,\hspace{3mm}\hat{E}\rightarrow i\hbar\frac{d}{dt}
respectively.
This replacement is called quantization.

Therefore the classical energy equation converts to
\displaystyle i\hbar\frac{d}{dt}\psi(r,t)=\left(-\frac{\hbar^2}{2m}\nabla^2+V(r)\right)\psi(r,t)

Schrodinger eq was derived!