1.21: Schrödinger’s Wave Equation

The energy of our electron can be broken into two parts, kinetic and potential. We could write this as

\[ \text{total energy} = \text{kinetic energy} + \text{potential energy} \nonumber \]

Now kinetic energy is related to momentum by

\[ \text{kinetic energy} = \frac{1}{2}mv^{2} = \frac{p^{2}}{2m} \nonumber \]

Thus, using our operators, we could write

\[ \hat{E}\psi(x,t)=\frac{\hat{p}^{2}}{2m} \psi(x,t)+\hat{V} \psi(x,t) \nonumber \]

Where \(\hat{V}\) is the potential energy operator.

\[ \hat{V}=V(x,t) \nonumber \]

We can rewrite Equation (1.21.3) in even simpler form by defining the so called Hamiltonian operator

\[ \hat{H}=\frac{\hat{p}^{2}}{2m}+\hat{V} \nonumber \]

Now,

\[ \hat{E}|\psi\rangle = \hat{H}|\psi\rangle \nonumber \]

Or we could rewrite the expression as

\[ i\hbar \frac{d}{dt}\psi(x,t)=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}} \psi(x,t) +V(x,t)\psi(x,t) \nonumber \]

All these equations are statements of Schrödinger's wave equation. We can employ whatever form is most convenient.

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