# Stationary States & Superposition > The time-dependent Schrödinger equation looks fearsome — a complex PDE — but for time-independent > potentials it surrenders to the oldest trick in mathematical physics: separation of variables. The > reward is the *time-independent* equation, whose solutions — stationary states — are the atoms of > quantum dynamics. Individually they do nothing at all, like a cat at 2 p.m.; superpose two of > them, and the relative phase makes everything move. ## Learning Objectives After this lesson you will be able to: 1. Carry out **separation of variables** on the Schrödinger equation and obtain the **time-independent Schrödinger equation** $\hat H\psi = E\psi$. 2. Prove that the separation constant $E$ must be **real** and must exceed $V_{\min}$. 3. Derive the three defining properties of **stationary states**: frozen $|\Psi|^2$, constant expectation values, and **definite energy**. 4. Expand an initial state in energy eigenstates via **orthonormality** and **Fourier's trick**, interpreting $|c_n|^2$ as energy-measurement probabilities. 5. Derive the **two-state sloshing** of the infinite square well and explain why all quantum dynamics is evolving **relative phase**. --- ## Intuition Faced with $i\hbar\,\partial_t\Psi = \hat H\Psi$, ask the physicist's first question: are there solutions with *trivial* time dependence? Yes — products $\psi(x)\,e^{-iEt/\hbar}$, whose time dependence is a pure phase. The Born rule sees only $|\Psi|^2$, so the phase cancels: the probability density is *frozen*. These are the **stationary states**, one per allowed energy. That sounds like a dead end — a theory of motion whose basic solutions never move. The escape is linearity: the general solution is a *superposition* of stationary states, each spinning its phase at its own rate $E_n/\hbar$. In a superposition the phases no longer cancel in $|\Psi|^2$; they beat against each other, and the beats are the dynamics. An electron sloshing in a well, a vibrating bond, a qubit mid-gate — all of it is relative phase between energy eigenstates. The strategy for every problem in Courses P.5–P.6 follows: solve the time-independent equation once for the catalog $\{\psi_n, E_n\}$, then assemble any motion from them. --- ## Theory ### Separation of variables Let $V = V(x)$ and try a product solution $\Psi(x,t) = \psi(x)\,\varphi(t)$. Insert into $i\hbar\,\partial_t\Psi = -\frac{\hbar^2}{2m}\partial_x^2\Psi + V\Psi$ and divide by $\psi\varphi$: $$i\hbar\,\frac{1}{\varphi}\frac{d\varphi}{dt} = -\frac{\hbar^2}{2m}\frac{1}{\psi}\frac{d^2\psi}{dx^2} + V(x).$$ The left side depends only on $t$, the right only on $x$; a function of $t$ can equal a function of $x$ only if both equal the same **constant**, $E$ (the name is not innocent). The PDE splits into two ODEs. **Time part.** $i\hbar\,\dot\varphi = E\varphi$, so $$\varphi(t) = e^{-iEt/\hbar}$$ (the integration constant is absorbed into $\psi$): every separable solution wears the same uniform, a phase rotating at $\omega = E/\hbar$ — Planck–Einstein again, now as an *output*. **Space part.** The remainder is the **time-independent Schrödinger equation** (TISE): $$\boxed{\;\hat H\psi = E\psi, \qquad \hat H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)\;}$$ — an *eigenvalue problem* for the **Hamiltonian** $\hat H$: allowed separation constants are eigenvalues, stationary profiles are eigenfunctions. For bound problems, the admissibility conditions of [P.4.1](01-schrodinger-equation-born-rule.md) (normalizability, continuity) select a *discrete* set $\{\psi_n, E_n\}$: quantization emerges from a boundary-value problem, not from an extra postulate. ### $E$ must be real, and $E > V_{\min}$ **Real.** If $E = E_0 + i\Gamma$ with $\Gamma \neq 0$, then $|\varphi(t)|^2 = e^{2\Gamma t/\hbar}$ and $\int|\Psi|^2dx = e^{2\Gamma t/\hbar}\int|\psi|^2dx$ grows or decays — contradicting conservation of probability ([P.4.1](01-schrodinger-equation-born-rule.md)). Normalizability for all time forces $E \in \mathbb{R}$. (Term 1 language: $\hat H$ is Hermitian, so its eigenvalues are real — [0.1.5](../../term-0-foundations/c01-linear-algebra/05-eigen-spectral-theorem.md).) **$E > V_{\min}$.** Rearrange the TISE: $$\frac{d^2\psi}{dx^2} = \frac{2m}{\hbar^2}\big(V(x) - E\big)\,\psi.$$ If $E < V_{\min} \le V(x)$ everywhere, then $V - E > 0$ everywhere, so **$\psi''$ has the same sign as $\psi$ at every point**: where $\psi>0$ it curves up, away from the axis; where $\psi<0$ it curves down, also away. Such a function bends away from zero and cannot decay to it at both ends — it blows up at $\pm\infty$ and is never normalizable. Physically: $\langle T\rangle \ge 0$, so you cannot have less energy than the least potential energy. ($E$ may still be negative if $V_{\min}<0$ — the bound states of wells, [P.5.2](../c05-bound-states/02-step-well-barrier.md).) The classical version of that floor is $T = \tfrac12 mv^2 \ge 0$. Reversing the motion cannot break it, and braking only walks the energy down toward zero, never past it: ```qm-demo kinetic-energy-floor ``` ### Properties of stationary states Let $\Psi_n = \psi_n(x)e^{-iE_nt/\hbar}$, $\psi_n$ normalized. Three properties, one line each: **1. Frozen density.** $|\Psi_n|^2 = \psi_n^*e^{+iE_nt/\hbar}\psi_ne^{-iE_nt/\hbar} = |\psi_n|^2$ — time-independent; hence *stationary*. **2. Every expectation value constant.** For any $\hat Q = Q(\hat x,\hat p)$ with no explicit $t$, the phase passes through $\hat Q$ and cancels: $$\langle Q\rangle = \int \psi_n^*e^{+iE_nt/\hbar}\,\hat Q\,\psi_ne^{-iE_nt/\hbar}dx = \int\psi_n^*\hat Q\,\psi_n\,dx.$$ Nothing measurable ever changes. **3. Definite energy.** Using $\hat H\psi_n = E_n\psi_n$: $\langle H\rangle = \int\psi_n^*\hat H\psi_ndx = E_n$, and $\langle H^2\rangle = \int\psi_n^*\hat H(\hat H\psi_n)dx = E_n^2$, so $$\sigma_H^2 = \langle H^2\rangle - \langle H\rangle^2 = 0:$$ an energy measurement returns $E_n$ with certainty. Stationary states are states of definite energy. > **Caution.** A single stationary state predicts **nothing moving**. Not "slowly" — every > probability and every expectation value of every observable is constant. If your system visibly > does anything, it is *not* in a stationary state. Motion — including every interference trick > quantum computing plays — lives in superpositions and their evolving relative phases. Stationary > states are the alphabet; dynamics is the poetry. ### Vocabulary: eigensolutions and degeneracy Fix the words now, because the rest of the term uses them constantly. A problem whose solutions exist only for a **specific set of allowed values** of some parameter — here the energy — is an **eigenvalue problem**; the allowed values are the **eigenvalues**, and the particular function attached to each is its **eigenfunction**. Together they form an **eigensolution**. Since our parameter is an energy, the $E_n$ are **eigenenergies** and the $\psi_n$ are **energy eigenfunctions**. "Stationary state", "energy eigenstate", and "eigensolution of $\hat H$" all name the same object. One eigenvalue need not have only one eigenfunction. When **two or more linearly independent eigenfunctions share a single eigenvalue**, the eigenvalue is called **degenerate**, and the number of independent eigenfunctions belonging to it is its **degeneracy**. Nothing forbids it, and symmetry tends to cause it: the 1D problems of Course P.5 are all *non*-degenerate (a second-order ODE with two decaying-tail boundary conditions has one solution per energy), but the moment a problem gains a symmetry direction, degeneracy appears — a particle in a *cube* has $E \propto n_x^2 + n_y^2 + n_z^2$, so $(2,1,1)$, $(1,2,1)$ and $(1,1,2)$ share one energy, a three-fold degeneracy. The hydrogen atom's $n^2$-fold degeneracy in $\ell$ and $m$ ([P.6.2](../c06-angular-momentum/02-radial-equation-hydrogen.md)) is the headline example, and lifting degeneracies with fields is how spectroscopy reads atoms apart ([P.6.3](../c06-angular-momentum/03-magnetic-fields-stern-gerlach-spin.md)). Degeneracy matters practically for the expansion below: within a degenerate set, *any* linear combination is still an eigenfunction of the same energy, so the basis is not unique — you may (and should) choose combinations adapted to another symmetry, exactly the freedom exploited when a degenerate subspace is diagonalized in [0.1.5](../../term-0-foundations/c01-linear-algebra/05-eigen-spectral-theorem.md). ### The general solution: superposition of stationary states Linearity lets us add separable solutions, and the spectral theorem guarantees *every* solution arises this way: $$\Psi(x,t) = \sum_n c_n\,\psi_n(x)\,e^{-iE_nt/\hbar}, \qquad c_n \in \mathbb{C}.$$ This is the master formula of wave mechanics: solve the TISE once, then evolution is "attach the phase wiggles." **Orthonormality and completeness.** The eigenfunctions can be chosen with $$\int_{-\infty}^{\infty}\psi_m^*\psi_n\,dx = \delta_{mn},$$ and they are **complete**: any admissible $f(x)$ expands as $f = \sum_nc_n\psi_n$. This is the spectral theorem of [0.1.5 Eigenvalues & the Spectral Theorem](../../term-0-foundations/c01-linear-algebra/05-eigen-spectral-theorem.md) promoted to infinite dimensions: $\hat H = \sum_n E_n\lvert\psi_n\rangle\langle\psi_n\rvert$. **Fourier's trick.** Multiply $\Psi(x,0) = \sum_nc_n\psi_n$ by $\psi_m^*$ and integrate; orthonormality collapses the sum: $$\int\psi_m^*\,\Psi(x,0)\,dx = \sum_n c_n\,\delta_{mn} = c_m \quad\Longrightarrow\quad c_n = \int_{-\infty}^{\infty}\psi_n^*(x)\,\Psi(x,0)\,dx.$$ Exactly the Fourier-coefficient recipe of [0.3.2](../../term-0-foundations/c03-fourier-methods/02-fourier-series-transforms.md) — a Fourier series *is* an eigenfunction expansion (for the infinite well, literally a sine series). **What the $c_n$ mean.** Insert the expansion into $\int|\Psi|^2dx$ and $\langle H\rangle$; the cross terms $c_m^*c_ne^{i(E_m-E_n)t/\hbar}\int\psi_m^*\psi_ndx$ die by orthonormality: $$1 = \sum_n|c_n|^2, \qquad \langle H\rangle = \sum_n|c_n|^2E_n,$$ both time-independent (energy is conserved). The reading, confirmed by [1.1.2](../../term-1-quantum-mechanics/c01-postulates/02-observables-measurement-postulate.md): **$|c_n|^2$ is the probability that an energy measurement returns $E_n$** — the Born rule in the energy basis, with $\sum_n|c_n|^2 = 1$ its consistency condition. ### Dynamics from interference: the sloshing well To watch phases *do* something, take the cleanest arena: the infinite square well on $[0,L]$, whose stationary states (derived in full in [P.5.2](../c05-bound-states/02-step-well-barrier.md)) are $$\psi_n(x) = \sqrt{\frac{2}{L}}\,\sin\frac{n\pi x}{L}, \qquad E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad n = 1,2,3,\dots$$ Prepare $\Psi = \frac{1}{\sqrt2}\big(\psi_1e^{-iE_1t/\hbar} + \psi_2e^{-iE_2t/\hbar}\big)$ and compute the density (the $\psi_n$ are real): $$|\Psi|^2 = \frac12\Big(\psi_1^2 + \psi_2^2 + \psi_1\psi_2\big(e^{i(E_1-E_2)t/\hbar} + e^{-i(E_1-E_2)t/\hbar}\big)\Big),$$ and since the bracket is $2\cos\big((E_2-E_1)t/\hbar\big)$, $$\boxed{\;|\Psi(x,t)|^2 = \frac12\big(\psi_1^2 + \psi_2^2\big) + \psi_1(x)\psi_2(x)\cos(\omega_{21}t), \qquad \omega_{21} \equiv \frac{E_2 - E_1}{\hbar}\;}$$ The cross term — pure interference — oscillates at the **Bohr frequency** $\omega_{21}$. Since $\psi_1\psi_2 > 0$ on the left half of the well and $< 0$ on the right, the probability lump **sloshes** left–right with period $T = 2\pi\hbar/(E_2-E_1)$. Note what mattered: not the absolute phases (a global $e^{i\gamma}$ changes nothing) but the **relative** phase $(E_2-E_1)t/\hbar$. *All quantum dynamics is relative phase in a superposition.* This is also the frequency an atom radiates: $\hbar\omega_{21} = E_2 - E_1$, the Bohr condition of [P.2.2](../c02-radiation-old-quantum/02-atomic-models-and-spectra.md). **Connections.** The phases assemble into the evolution operator $U(t) = e^{-i\hat Ht/\hbar} = \sum_ne^{-iE_nt/\hbar}\lvert\psi_n\rangle\langle\psi_n\rvert$ — exactly the unitary of [1.1.3 The Evolution Postulate](../../term-1-quantum-mechanics/c01-postulates/03-evolution-postulate.md); what we found by separating variables, Term 1 postulates directly. And a stationary state's $e^{-iE_nt/\hbar}$ alone is a *global* phase — unobservable; between two components it is a *relative* phase — the whole show: precisely the distinction of [1.1.1](../../term-1-quantum-mechanics/c01-postulates/01-state-postulate.md), now with a clock attached. --- ## Worked Examples ### Example 1 — Energy statistics of a three-term superposition A particle in the infinite well is prepared in $\Psi(x,0) = \frac{1}{\sqrt6}\psi_1 + \frac{i}{\sqrt3}\psi_2 + \frac{1}{\sqrt2}\psi_3$. Check normalization; find the energy-measurement outcomes, probabilities, and $\langle H\rangle$. $\sum|c_n|^2 = \frac16 + \frac13 + \frac12 = 1$. ✓ Outcomes are the eigenvalues present: $P(E_1) = \frac16$, $P(E_2) = \frac13$, $P(E_3) = \frac12$ (the $i$ in $c_2$ never enters — only $|c_2|^2$ does). With $E_n = n^2E_1$: $$\langle H\rangle = \tfrac16E_1 + \tfrac13(4E_1) + \tfrac12(9E_1) = 6\,E_1.$$ Note $6E_1$ is not an eigenvalue: the *average* need not be a possible *outcome*. And all these numbers are constants of the motion — measure at any $t$, same statistics. ### Example 2 — The sloshing amplitude $\langle x\rangle(t)$ For the two-state superposition above, compute $\langle x\rangle(t)$. $$\langle x\rangle = \frac12\int_0^L x\big(\psi_1^2 + \psi_2^2\big)dx + \cos(\omega_{21}t)\int_0^L x\,\psi_1\psi_2\,dx.$$ Each $|\psi_n|^2$ is symmetric about $L/2$, so the first term is $L/2$. For the cross integral, use $2\sin A\sin B = \cos(A-B) - \cos(A+B)$ and $\int_0^L x\cos\frac{n\pi x}{L}dx = \frac{L^2}{n^2\pi^2}\big((-1)^n - 1\big)$: $$\int_0^L x\,\frac2L\sin\frac{\pi x}{L}\sin\frac{2\pi x}{L}dx = \frac1L\left(-\frac{2L^2}{\pi^2} + \frac{2L^2}{9\pi^2}\right) = -\frac{16L}{9\pi^2}.$$ Therefore $$\langle x\rangle(t) = \frac L2 - \frac{16L}{9\pi^2}\cos(\omega_{21}t) \approx \frac L2 - 0.18\,L\cos(\omega_{21}t):$$ the mean position swings across the well at the Bohr frequency $\omega_{21} = 3\pi^2\hbar/2mL^2$. Each ingredient state alone predicts a frozen $\langle x\rangle = L/2$; the *superposition* moves. --- ## Hands-on (Python) ```python import numpy as np import matplotlib.pyplot as plt # --- Two-state sloshing in the infinite square well -------------------------- # Numerical values hbar = m = L = 1 (physics keeps ħ explicit; arrays don't care). hbar, m, L = 1.0, 1.0, 1.0 x = np.linspace(0, L, 800) psi_n = lambda n, x: np.sqrt(2/L) * np.sin(n*np.pi*x/L) # √(2/L) sin(nπx/L) E_n = lambda n: n**2 * np.pi**2 * hbar**2 / (2*m*L**2) # n²π²ħ²/2mL² w21 = (E_n(2) - E_n(1)) / hbar # Bohr frequency T = 2*np.pi / w21 # sloshing period Psi = lambda x, t: (psi_n(1,x)*np.exp(-1j*E_n(1)*t/hbar) + psi_n(2,x)*np.exp(-1j*E_n(2)*t/hbar)) / np.sqrt(2) for frac in [0, 0.25, 0.5, 0.75]: # snapshots across one period plt.plot(x, np.abs(Psi(x, frac*T))**2, label=f"t = {frac:.2f} T") plt.xlabel("x/L"); plt.ylabel(r"$|\Psi|^2$"); plt.legend(); plt.show() # Expected: lump on the left at t=0, symmetric at T/4 and 3T/4, on the right at T/2. t = np.linspace(0, 2*T, 400) x_mean = [np.trapz(x*np.abs(Psi(x, tt))**2, x) for tt in t] plt.plot(t/T, x_mean); plt.axhline(0.5, ls="--", c="gray") plt.xlabel("t/T"); plt.ylabel(r"$\langle x\rangle$"); plt.show() # Expected: (t) = 0.5 - (16/9π²)cos(ω21 t) ≈ 0.5 - 0.180 cos(ω21 t): # a clean cosine at exactly ω21 = (E2 - E1)/ħ. ``` ```python # --- Expand a triangular initial state in well eigenstates ------------------- tri = np.where(x < L/2, x, L - x) # triangle peaked at L/2 tri = tri / np.sqrt(np.trapz(tri**2, x)) # normalize N = 40 # c_n by numerical quadrature c = np.array([np.trapz(psi_n(n, x)*tri, x) for n in range(1, N+1)]) print(np.cumsum(np.abs(c)**2)[[0, 2, 9, 39]]) # Expected: ≈ [0.9855 0.9977 0.9998 1.0000] — Σ|c_n|² → 1 (completeness); # even n vanish by symmetry and |c_n|² ~ 1/n⁴, so convergence is fast. def Psi_rec(x, t): # reconstruct Σ c_n ψ_n e^{-iE_n t/ħ} return sum(c[n-1]*psi_n(n, x)*np.exp(-1j*E_n(n)*t/hbar) for n in range(1, N+1)) T1 = 2*np.pi*hbar/E_n(1) # fundamental period plt.plot(x, tri**2, "k--", label=r"exact $|\Psi(x,0)|^2$") for tt, lab in [(0, "t=0"), (0.1*T1, "t=0.1$T_1$"), (0.25*T1, "t=0.25$T_1$")]: plt.plot(x, np.abs(Psi_rec(x, tt))**2, label=lab) plt.xlabel("x/L"); plt.legend(); plt.show() # Expected: at t=0 the reconstruction overlays the triangle (dashed); later # snapshots show the peak melting into interference wiggles — dynamics generated # purely by the relative phases e^{-iE_n t/ħ}. ``` --- ## Exercises **E1 (easy).** Verify by substitution that $\Psi_n = \psi_n(x)e^{-iE_nt/\hbar}$ solves the time-dependent Schrödinger equation whenever $\hat H\psi_n = E_n\psi_n$.
Solution Left side: $i\hbar\,\partial_t\Psi_n = i\hbar(-iE_n/\hbar)\psi_ne^{-iE_nt/\hbar} = E_n\psi_ne^{-iE_nt/\hbar}$. Right side: $\hat H\Psi_n = (\hat H\psi_n)e^{-iE_nt/\hbar} = E_n\psi_ne^{-iE_nt/\hbar}$ (the phase is constant in $x$, so it passes through $\hat H$). Equal. ✓
**E2 (easy).** For $\Psi(x,0) = \frac{1}{\sqrt5}\psi_1 + \frac{2}{\sqrt5}\psi_4$ in the infinite well: which energies can be measured, with what probabilities? What is $\langle H\rangle$ in units of $E_1$? Do the answers change at later times?
Solution $E_1$ with $P = 1/5$; $E_4 = 16E_1$ with $P = 4/5$. $\langle H\rangle = \frac15E_1 + \frac45\cdot16E_1 = 13\,E_1$. No change: evolution sends $c_n \to c_ne^{-iE_nt/\hbar}$, so every $|c_n|^2$ is constant — energy statistics are conserved.
**E3 (medium).** Prove that eigenfunctions of $\hat H$ with distinct energies are orthogonal: $E_m \neq E_n \Rightarrow \int\psi_m^*\psi_n\,dx = 0$. (Hint: evaluate $\int\psi_m^*(\hat H\psi_n)dx - \int(\hat H\psi_m)^*\psi_ndx$ two ways.)
Solution Call the difference $D$. The $V$ terms cancel ($V$ real); the kinetic terms give $-\frac{\hbar^2}{2m}\int(\psi_m^*\psi_n'' - \psi_m''^*\psi_n)dx = -\frac{\hbar^2}{2m}\big[\psi_m^*\psi_n' - \psi_m'^*\psi_n\big]_{-\infty}^{\infty} = 0$ after two integrations by parts (boundary terms vanish for normalizable states). So $D = 0$. But evaluating directly with the eigenvalue equations (and real eigenvalues, proved in Theory): $D = (E_n - E_m)\int\psi_m^*\psi_ndx$. Since $E_m \neq E_n$, the integral vanishes — Hermiticity at work, as in [0.1.5](../../term-0-foundations/c01-linear-algebra/05-eigen-spectral-theorem.md).
**E4 (medium).** For $\Psi = \frac{1}{\sqrt2}\big(\psi_1e^{-iE_1t/\hbar} + e^{i\alpha}\psi_2e^{-iE_2t/\hbar}\big)$ with constant $\alpha$, show $|\Psi|^2$ has the same sloshing form as in Theory but time-shifted. What does this say about $\alpha$ versus a global phase?
Solution The cross term becomes $\psi_1\psi_2\cos(\omega_{21}t - \alpha)$: the relative phase $\alpha$ shifts the *starting position* of the slosh ($\alpha = \pi$ starts the lump on the right) but not the frequency. A global phase $e^{i\gamma}\Psi$ cancels in $|\Psi|^2$ entirely. Relative phase: physical initial condition. Global phase: nothing — the verdict of [1.1.1](../../term-1-quantum-mechanics/c01-postulates/01-state-postulate.md).
**E5 (hard).** Show that in the infinite well *any* state revives exactly: $\Psi(x, t + T_{\mathrm{rev}}) = \Psi(x,t)$ with $T_{\mathrm{rev}} = 4mL^2/\pi\hbar$. Why does no such universal revival occur for a generic potential?
Solution $E_n = n^2E_1$ with $E_1 = \pi^2\hbar^2/2mL^2$. After time $T$, the $n$-th term gains $e^{-in^2E_1T/\hbar}$. Choose $E_1T/\hbar = 2\pi$, i.e. $T_{\mathrm{rev}} = 2\pi\hbar/E_1 = 4mL^2/\pi\hbar$: then $e^{-2\pi in^2} = 1$ for every integer $n$, so every coefficient returns exactly (sub-multiples of $T_{\mathrm{rev}}$ give nontrivial global phases and fractional revivals). The magic: all $E_n$ are integer multiples of one quantum $E_1$, so all Bohr frequencies are commensurate. A generic potential has incommensurate level spacings — the relative phases never realign, and the state never exactly recurs.
--- ## Checkpoint 1. Walk through separation of variables: what plays the role of the separation constant, and which two ODEs result? 2. Why must $E$ be real, and why must $E > V_{\min}$? 3. Name and derive (one line each) the three defining properties of a stationary state. 4. Given $\Psi(x,0)$, how do you find the $c_n$, and what do $|c_n|^2$, $\sum|c_n|^2$, and $\sum|c_n|^2E_n$ mean? 5. Nothing about a stationary state changes in time — so where does quantum dynamics come from?
Answers 1. Try $\Psi = \psi(x)\varphi(t)$; dividing by $\psi\varphi$ separates $t$ from $x$, each side equaling a constant $E$. Results: $i\hbar\dot\varphi = E\varphi$ (so $\varphi = e^{-iEt/\hbar}$) and the TISE $\hat H\psi = E\psi$. 2. Complex $E$ makes $\int|\Psi|^2dx \propto e^{2\,\mathrm{Im}(E)t/\hbar}$, violating conservation of probability. If $E < V_{\min}$, $\psi''$ and $\psi$ share a sign everywhere, so $\psi$ curves away from the axis and cannot be normalizable. 3. (i) $|\Psi_n|^2 = |\psi_n|^2$ — the phases cancel; (ii) $\langle Q\rangle$ constant — the phase passes through $\hat Q$ and cancels; (iii) $\langle H\rangle = E_n$, $\langle H^2\rangle = E_n^2$, so $\sigma_H = 0$ — definite energy. 4. Fourier's trick: $c_n = \int\psi_n^*\Psi(x,0)dx$ (orthonormality projects each coefficient out). $|c_n|^2$ = probability of measuring $E_n$; $\sum|c_n|^2 = 1$ = normalization; $\sum|c_n|^2E_n = \langle H\rangle$ — all constant in time. 5. From superposition: components $c_n\psi_ne^{-iE_nt/\hbar}$ accumulate *relative* phases at the Bohr frequencies $(E_m - E_n)/\hbar$, which survive in the cross terms of $|\Psi|^2$. All quantum dynamics is evolving relative phase.
--- ## Further Reading - **[Gri]** Griffiths & Schroeter, §2.1 — stationary states; the source of this lesson's structure. - **[Gri]** Griffiths & Schroeter, §2.2 — the infinite square well and its superposition dynamics. - **[Sha]** Shankar, §4.3 & Ch. 5 — the energy basis and the propagator $U(t)$, bridging to Term 1. - **[ER]** Eisberg & Resnick, §5.5–5.7 — separation of variables and the meaning of eigenvalues. - **[Pre]** Preskill, *Ph219*, Ch. 2 — Hamiltonian evolution stated axiomatically, for contrast. --- ← Prev: [The Schrödinger Equation & the Born Rule](01-schrodinger-equation-born-rule.md) · Up: [Pre-Term](../README.md) · Next: [Wave Packets, Uncertainty & the Momentum Operator](03-matter-waves-uncertainty-momentum.md) →