1.24: Free Particles
- Page ID
- 50125
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In free space, the potential, V, is constant everywhere. For simplicity we will set V = 0.
Next we solve Equation (1.23.5) with V = 0.
\[ -\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\psi(x)=E\psi(x) \nonumber \]
Rearranging slightly gives the second order differential equation in slightly clearer form
\[ \frac{d^{2}\psi}{dx^{2}} = -\frac{2mE}{\hbar^{2}}\psi \nonumber \]
A general solution is
\[ \psi(x)=\psi(0)\text{exp}[ikx] \nonumber \]
where
\[ k=\sqrt{\frac{2mE}{\hbar^{2}}} \nonumber \]
Inserting the time dependence (see Equation (1.23.7)) gives
\[ \psi(x,t)=\psi(0)\text{exp}[i(kx-\omega t)] \nonumber \]
where
\[ \omega = \frac{E}{\hbar}=\frac{\hbar k^{2}}{2m} \nonumber \]
Thus, as expected the solution in free space is a plane wave.