#The Schrodinger Equation

Consider a particle constrained on the x-axis, with mass $m$, subjected to a force $F(x,t)$.

– In Classical mechanics, we determined $x(t)$, the position. Then we find $v = \frac{dx}{dt}$ the velocity, and momentum $p = mv$. Finally, we determine the kinetic energy $T = \frac{1}{2}mv^2$… or any other dynamical quantity of interest.

– From there we find $x(t)$ using $F=ma$, for conservative systems.

Luckily, conservative systems are the only ones which occur at microscopic scales. We know that $$F = – \frac{\partial V}{\partial x }$$ and thus $$m \frac{d^2x}{dt^2} = – \frac{\partial V}{\partial x}$$ which, along with initial conditions $(x,t)$, we determine $x(t)$.

In Quantum Mechanics, the name of the game is different. We wish to find the particle’s *wave function*, $\psi ( x,t)$, which we do by solving the Schrodinger equation:

\begin{equation}

i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar ^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V\psi

\end{equation}

-more notes to be written soon.

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