Chapter 2 Key Takeaways

The Three Pillars Introduced in This Chapter

Key Equations

Time-Dependent Schrödinger Equation (TDSE)

$$i\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi = \left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\right]\psi$$

• Governs all quantum evolution between measurements.
• First order in time: $\psi(x,0)$ determines $\psi(x,t)$ uniquely.
• Linear: superposition of solutions is a solution.
• Deterministic: the randomness is in measurement, not in evolution.

Time-Independent Schrödinger Equation (TISE)

$$\hat{H}\phi = E\phi \qquad \Longleftrightarrow \qquad -\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2} + V(x)\phi = E\phi$$

• An eigenvalue equation: $\phi$ is an eigenfunction, $E$ is an eigenvalue.
• Solutions exist only for specific values of $E$ (quantization).
• Valid when $V$ is time-independent.

Normalization

$$\int_{-\infty}^{\infty}|\psi(x,t)|^2\,dx = 1$$

Expectation Values

$$\langle A \rangle = \int_{-\infty}^{\infty}\psi^*\,\hat{A}\,\psi\,dx$$

Probability Current

$$j = \frac{\hbar}{m}\operatorname{Im}\left(\psi^*\frac{\partial\psi}{\partial x}\right)$$

Continuity equation: $\frac{\partial|\psi|^2}{\partial t} + \frac{\partial j}{\partial x} = 0$.

Stationary States

$$\psi(x,t) = \phi(x)\,e^{-iEt/\hbar}, \qquad |\psi|^2 = |\phi|^2 \text{ (time-independent)}$$

General Solution (Superposition)

$$\psi(x,t) = \sum_n c_n\,\phi_n(x)\,e^{-iE_n t/\hbar}, \qquad |c_n|^2 = \text{probability of measuring }E_n$$

Key Concepts

1. The wave function $\psi$ is a probability amplitude. It is complex-valued, and $|\psi|^2$ gives the probability density. $\psi$ itself is not directly observable.

2. The Born rule bridges theory and experiment. It is the only rule connecting the wave function to measurement outcomes.

3. The Schrödinger equation is deterministic. Quantum randomness enters only through the Born rule at the moment of measurement, not through the evolution of $\psi$.

4. Stationary states have definite energy. A measurement of energy in a stationary state always yields the eigenvalue $E$.

5. Superposition is genuinely quantum. $\psi = c_1\psi_1 + c_2\psi_2$ is not "either $\psi_1$ or $\psi_2$" — it is a new state with interference effects that have no classical analogue.

6. Quantization emerges from boundary conditions. The requirement that $\psi$ be normalizable restricts energy to discrete values for bound states.

7. Probability is conserved. The continuity equation ensures that if $\psi$ is normalized at $t = 0$, it remains normalized forever.

8. Physical wave functions must be well-behaved: continuous, single-valued, square-integrable, with continuous first derivative (where $V$ is finite).

Decision Framework: When to Use TDSE vs. TISE

Is the potential V time-independent?
│
├─ NO → Use the full TDSE: i ℏ ∂ψ/∂t = Ĥ(t) ψ
│        (Time-dependent perturbation theory, Ch 21)
│
└─ YES → Solve the TISE: Ĥφₙ = Eₙφₙ
          │
          ├─ Is the system in a single stationary state?
          │   └─ YES → ψ(x,t) = φₙ(x) e^{-iEₙt/ℏ}
          │             |ψ|² is time-independent
          │             Energy is definite: E = Eₙ
          │
          └─ Is the system in a superposition?
              └─ YES → ψ(x,t) = Σ cₙ φₙ(x) e^{-iEₙt/ℏ}
                        |ψ|² is time-DEPENDENT
                        ⟨E⟩ = Σ |cₙ|² Eₙ
                        Find cₙ from initial condition:
                        cₙ = ∫ φₙ*(x) ψ(x,0) dx

Common Mistakes to Avoid

What Comes Next