2. Central Potential
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Potentials depending only on distance from origin: $V(\vec{r}) = V(r)$
Separation in Spherical Coordinates
For $V = V(r)$, assume:
$$\psi(r,\theta,\phi) = R(r)Y(\theta,\phi)$$
Separates into radial and angular parts
Angular Equation
The angular part satisfies:
$$\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial Y}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2 Y}{\partial\phi^2} = -\ell(\ell+1)Y$$
Solutions are spherical harmonics $Y_{\ell}^m(\theta,\phi)$
Radial Equation
The radial part satisfies:
$$-\frac{\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) + \left[V(r) + \frac{\hbar^2\ell(\ell+1)}{2mr^2}\right]R = ER$$
Centrifugal term acts like a repulsive potential
Effective Potential
$$V_{\text{eff}}(r) = V(r) + \frac{\hbar^2\ell(\ell+1)}{2mr^2}$$
The second term is the centrifugal barrier:
- Repulsive, goes as $1/r^2$
- Larger for higher angular momentum
- Prevents particle from reaching origin (except $\ell = 0$)
Reduced Radial Equation
Define $u(r) = rR(r)$:
$$-\frac{\hbar^2}{2m}\frac{d^2u}{dr^2} + V_{\text{eff}}(r)u = Eu$$
This looks like 1D Schrödinger equation!
Boundary conditions:
- $u(0) = 0$ (regularity at origin)
- $u(\infty) \to 0$ (bound states)
Quantum Numbers
Complete set for central potential:
$$|n,\ell,m\rangle$$
- $n$: Principal quantum number (energy)
- $\ell$: Orbital angular momentum quantum number, $0 \leq \ell < n$
- $m$: Magnetic quantum number, $-\ell \leq m \leq \ell$
Behavior at Origin
For small $r$, the centrifugal term dominates:
$$R(r) \sim r^\ell \quad \text{as } r \to 0$$
Implications:
- $\ell = 0$ (s-wave): finite at origin
- $\ell > 0$: vanishes at origin
- Higher $\ell$ means less penetration to small $r$
Asymptotic Behavior
Bound states ($E < 0$):
$$R(r) \sim e^{-\kappa r} \quad \text{as } r \to \infty$$
where $\kappa = \sqrt{-2mE}/\hbar$
Scattering states ($E > 0$):
$$R(r) \sim \sin(kr + \delta_\ell) \quad \text{as } r \to \infty$$
where $\delta_\ell$ is the phase shift
Conservation of Angular Momentum
For central potentials, $[\hat{H}, \hat{\vec{L}}] = 0$
- Angular momentum is conserved
- Energy eigenstates are also angular momentum eigenstates
- $\ell$ and $m$ are good quantum numbers
- Motion in fixed plane (classical analog)