Chapter 3 Key Takeaways

Essential Equations

Infinite Square Well ($0 \le x \le a$)

$$E_n = \frac{n^2\pi^2\hbar^2}{2ma^2}, \quad n = 1, 2, 3, \ldots$$

$$\psi_n(x) = \sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right)$$

• Orthonormality: $\int_0^a \psi_m^*(x)\psi_n(x)\,dx = \delta_{mn}$
• Completeness: any function $f(x)$ with $f(0) = f(a) = 0$ can be expanded as $f = \sum c_n\psi_n$
• Zero-point energy: $E_1 = \pi^2\hbar^2/(2ma^2) > 0$

Free Particle

$$\Psi_k(x,t) = Ae^{i(kx - \omega t)}, \quad E = \frac{\hbar^2 k^2}{2m}, \quad \omega = \frac{E}{\hbar}$$

• Plane waves are not normalizable — use wave packets
• Group velocity: $v_g = d\omega/dk = \hbar k/m = p/m$ (classical velocity)
• Phase velocity: $v_\phi = \omega/k = p/(2m) = v_g/2$
• Gaussian wave packet spreading: $\sigma(t) = \sigma_x\sqrt{1 + (t/\tau)^2}$, $\tau = 2m\sigma_x^2/\hbar$

Finite Square Well (centered, half-width $a$, depth $V_0$)

• Even bound states: $l\tan(la) = \kappa$
• Odd bound states: $-l\cot(la) = \kappa$
• Where $l = \sqrt{2m(E + V_0)}/\hbar$ and $\kappa = \sqrt{-2mE}/\hbar$
• Always at least one bound state in 1D
• Number of bound states $\approx 1 + \lfloor 2z_0/\pi\rfloor$, where $z_0 = a\sqrt{2mV_0}/\hbar$

Step Potential (height $V_0$ at $x = 0$)

$E > V_0$:

$$R = \left(\frac{k_1 - k_2}{k_1 + k_2}\right)^2, \quad T = \frac{4k_1 k_2}{(k_1 + k_2)^2}, \quad R + T = 1$$

where $k_1 = \sqrt{2mE}/\hbar$, $k_2 = \sqrt{2m(E-V_0)}/\hbar$.

$E < V_0$: $R = 1$ (total reflection), penetration depth $\delta = 1/\kappa = \hbar/\sqrt{2m(V_0 - E)}$.

Tunneling Through a Rectangular Barrier (height $V_0$, width $d$)

Exact:

$$T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa d)}{4E(V_0 - E)}}$$

Thick-barrier approximation ($\kappa d \gg 1$):

$$T \approx \frac{16E(V_0 - E)}{V_0^2}\,e^{-2\kappa d}, \quad \kappa = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$$

Finite Difference Method

Second derivative: $\psi''(x_i) \approx (\psi_{i+1} - 2\psi_i + \psi_{i-1})/(\Delta x)^2$

Hamiltonian matrix elements:

$$H_{ii} = \frac{\hbar^2}{m(\Delta x)^2} + V_i, \quad H_{i,i\pm 1} = -\frac{\hbar^2}{2m(\Delta x)^2}$$

Solve $\mathbf{H}\boldsymbol{\psi} = E\boldsymbol{\psi}$ using standard eigenvalue solvers.

Decision Framework: Which Potential Model to Use

Key Conceptual Points

1. Quantization arises from boundary conditions. It is not assumed — it emerges from requiring the wavefunction to be physically sensible (continuous, normalizable, single-valued).

2. Confinement → discrete spectrum. No confinement → continuous spectrum. The infinite well has only discrete energies; the free particle has only continuous energies; the finite well has both.

3. Zero-point energy is real. A confined quantum particle is never at rest. This is a direct consequence of the uncertainty principle.

4. Plane waves are idealizations. Physical particles are described by wave packets — superpositions of plane waves with both position and momentum spread.

5. Wave packets spread. A free Gaussian wave packet broadens over time because different momentum components travel at different speeds (dispersion).

6. Quantum tunneling is real. Particles pass through classically forbidden barriers. The probability is $T \propto e^{-2\kappa d}$ — exponentially sensitive to barrier width, height, and particle mass.

7. Partial reflection has no classical analogue. A quantum particle encountering a step potential where $E > V_0$ has a nonzero probability of reflection, even though classically it would always pass through.

8. Numerical methods are not a last resort — they are the primary tool for realistic problems. The finite difference method converts the TISE into a matrix eigenvalue problem solvable on any laptop.

Common Mistakes to Avoid

Looking Ahead