1. Free Particle
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The simplest quantum system: a particle with no potential energy ($V(x) = 0$).
Time-Independent Schrödinger Equation
For a free particle in 1D:
$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$$
This simplifies to:
$$\frac{d^2\psi}{dx^2} = -k^2\psi, \quad k = \frac{\sqrt{2mE}}{\hbar}$$
General Solutions
Two equivalent forms:
$$\psi(x) = Ae^{ikx} + Be^{-ikx} \quad \text{(plane waves)}$$
or
$$\psi(x) = C\sin(kx) + D\cos(kx) \quad \text{(standing waves)}$$
These represent momentum eigenstates $p = \pm\hbar k$
Time Evolution
Full time-dependent solution:
$$\Psi(x,t) = Ae^{i(kx - \omega t)} + Be^{-i(kx + \omega t)}$$
where $\omega = E/\hbar = \hbar k^2/(2m)$
Interpretation:
- First term: wave traveling to the right
- Second term: wave traveling to the left
Wave Packets
Plane waves are not normalizable. Physically realizable states are wave packets:
$$\Psi(x,t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi(k)e^{i(kx - \omega(k)t)}dk$$
where $\phi(k)$ is the momentum-space wave function
Group and Phase Velocity
Phase velocity: velocity of individual wave crests
$$v_{\text{phase}} = \frac{\omega}{k} = \frac{\hbar k}{2m}$$
Group velocity: velocity of wave packet (= particle velocity)
$$v_{\text{group}} = \frac{d\omega}{dk} = \frac{\hbar k}{m} = \frac{p}{m}$$
The group velocity matches classical velocity!
Dispersion
For a free particle, $\omega \propto k^2$, so:
$$\frac{d^2\omega}{dk^2} = \frac{\hbar}{m} \neq 0$$
Wave packets spread out over time due to dispersion:
- Different frequency components travel at different speeds
- Uncertainty in position increases with time
- Momentum uncertainty remains constant
Key Properties
- Energy spectrum: Continuous, $E \geq 0$
- Momentum: $p = \pm\hbar k$ (positive or negative)
- Degeneracy: Each energy corresponds to two momentum states
- Probability current: Constant in time for plane waves