The Schrödinger Equation (Time-Dependent and Time-Independent)

Chapter 2 said that a particle's state is a wave function and that measurements are statistical. It did not say how the state changes with time. That is the job of the Schrödinger equation, the dynamical law of non-relativistic quantum mechanics. Like Newton's second law it is a postulate — not derived from anything more basic — justified because its predictions match experiment to extraordinary precision. This chapter states it, extracts the time-independent Schrödinger equation by separation of variables, explains why its solutions ("stationary states") are the backbone of every bound-state calculation in Unit D, and treats the one unbound case that is fully solvable by hand: the free particle and its wave packets.

The organising idea is that the Hamiltonian operator $\hat H$ does two things at once. It generates time evolution, through $i\hbar\,\partial_t\Psi = \hat H\Psi$; and its eigenvalue problem $\hat H\psi = E\psi$ picks out the special states of definite energy, in terms of which every other state is expanded. Solving a quantum system means finding the spectrum and eigenfunctions of its $\hat H$; time evolution is then automatic, each eigenstate merely acquiring a phase $e^{-iEt/\hbar}$.

How to read this chapter. Two equations do all the work. The TDSE $i\hbar\,\partial_t\Psi = -\dfrac{\hbar^2}{2m}\partial_x^2\Psi + V\Psi$ governs how any state evolves. Separation of variables reduces it, for time-independent $V$, to the TISE $-\dfrac{\hbar^2}{2m}\psi'' + V\psi = E\psi$, an eigenvalue problem whose bound-state solutions exist only for discrete $E$. Chapters 6–8 are three worked instances (well, oscillator, angular momentum); this chapter is the machinery they share.

1. The Time-Dependent Schrödinger Equation

For a particle of mass $m$ in a potential $V(x,t)$, the wave function obeys

The right-hand side defines the Hamiltonian operator $\hat H = \hat p^2/2m + V(\hat x)$, the quantum version of the classical energy $p^2/2m + V$. Key features:

• It is first order in time: $\Psi(x,0)$ everywhere determines $\Psi(x,t)$ for all later $t$. The evolution is deterministic — the randomness in quantum mechanics is entirely in measurement, not in the dynamics.
• It is linear: if $\Psi_1,\Psi_2$ are solutions so is $c_1\Psi_1 + c_2\Psi_2$. This is what makes superposition consistent.
• The explicit $i$ forces $\Psi$ to be complex and makes the evolution unitary (norm-preserving), which is why $\int|\Psi|^2\,dx$ stays equal to $1$.

The equation cannot be derived from classical mechanics. It can be motivated: for a free particle, demanding that $e^{i(kx-\omega t)}$ be a solution with $E = \hbar\omega$ and $p = \hbar k$ related by $E = p^2/2m$ forces exactly the free ($V=0$) form. Adding $V$ is the natural guess, and it works.

2. Separation of Variables and the Time-Independent Equation

When $V = V(x)$ does not depend on time, look for product solutions $\Psi(x,t) = \psi(x)\,\varphi(t)$. Substituting and dividing by $\psi\varphi$:

The left side depends only on $t$, the right only on $x$; both must equal a constant, which has the dimensions of energy and is called $E$. This splits the problem into

The TISE is an eigenvalue equation $\hat H\psi = E\psi$: find the functions $\psi$ and numbers $E$ for which applying $\hat H$ just rescales $\psi$. The full separated solution is

3. Stationary States and Why They Are Called That

A separated solution $\Psi_E = \psi_E(x)e^{-iEt/\hbar}$ is a stationary state. Despite the oscillating phase, everything measurable about it is time-independent:

• $|\Psi_E|^2 = |\psi_E|^2$ — the probability density does not move.
• $\langle Q\rangle = \int\psi_E^*\,\hat Q\,\psi_E\,dx$ for any operator $\hat Q$ with no explicit time dependence — the phases $e^{\pm iEt/\hbar}$ cancel.
• The energy is sharp: $\langle H\rangle = E$ and $\langle H^2\rangle = E^2$, so $\Delta E = 0$. A measurement of energy in a stationary state returns $E$ with certainty.
• The probability current is constant in time (and, in 1D, constant in $x$).

"Stationary" refers to the observables, not the wave function. These states are the quantum analogue of the normal modes of a vibrating string: each oscillates at its own frequency $\omega = E/\hbar$ and nothing about its shape changes.

4. The General Solution as a Superposition

The TISE typically has many solutions $\psi_n(x)$ with energies $E_n$ (discrete for bound states; see Section 5). Because the TDSE is linear, the general solution is a superposition of stationary states:

The constants $c_n$ are fixed by the initial condition: at $t = 0$, $\Psi(x,0) = \sum_n c_n\psi_n(x)$, so by orthonormality of the $\psi_n$,

This is the entire method. To evolve any state: (1) expand the initial $\Psi(x,0)$ in energy eigenstates to get the $c_n$; (2) attach the phase factor $e^{-iE_n t/\hbar}$ to each term; (3) resum. The hard part is always step (1) — finding the spectrum and the $c_n$. Once done, time evolution is trivial.

Worked example — a two-term superposition in a box. Take $\Psi(x,0) = \tfrac{1}{\sqrt2}[\psi_1(x) + \psi_2(x)]$ in an infinite well. Then $\Psi(x,t) = \tfrac{1}{\sqrt2}[\psi_1 e^{-iE_1 t/\hbar} + \psi_2 e^{-iE_2 t/\hbar}]$, and

The probability density sloshes from side to side at the angular frequency $\omega_{21} = (E_2 - E_1)/\hbar$ — a genuinely time-dependent state, unlike either eigenstate alone. Its energy is not sharp: a measurement gives $E_1$ or $E_2$, each with probability $\tfrac12$.

5. Energy Quantisation for Bound States

For a potential that confines the particle (e.g. $V\to\infty$ as $|x|\to\infty$, or a well deep enough to trap it), the requirement that $\psi$ be normalisable — $\psi\to 0$ at infinity, $\psi$ and $\psi'$ continuous — cannot be met for arbitrary $E$. Generic $E$ produces a solution that diverges at large $|x|$; only a discrete set $E_1 < E_2 < E_3 < \cdots$ gives a decaying, square-integrable $\psi_n$. Bound-state energies are quantised. Two universal features:

• There is a lowest energy $E_1 > V_{\min}$ — the ground state. Confinement plus $\Delta x\,\Delta p \gtrsim \hbar/2$ forbids a state of zero kinetic energy (zero-point energy).
• The $n$th state has $n-1$ nodes (interior zeros), and the $\psi_n$ can be chosen real and are mutually orthogonal, $\int\psi_m^*\psi_n\,dx = \delta_{mn}$.

For energies above the top of the potential the particle is unbound: the spectrum is continuous and the eigenfunctions are non-normalisable scattering states (Chapter 6).

6. The Free Particle and Wave Packets

With $V = 0$ the TISE is $-\dfrac{\hbar^2}{2m}\psi'' = E\psi$, solved by $\psi_k(x) = e^{ikx}$ with $E = \dfrac{\hbar^2 k^2}{2m}$, so the stationary solutions are

These are not normalisable ($|\Psi_k|^2 = 1$ everywhere), so no free particle is in a single plane-wave state. Physical states are wave packets — continuous superpositions,

with $\phi(k)$ the Fourier transform of $\Psi(x,0)$. A packet peaked around $k_0$ with spread $\Delta k$ describes a particle localised to $\Delta x \sim 1/\Delta k$ around a moving centre, with $\Delta x\,\Delta p \gtrsim \hbar/2$.

7. Phase Velocity, Group Velocity, and Spreading

For the free particle the dispersion relation is $\omega(k) = \hbar k^2/2m$, which is nonlinear in $k$. Two velocities follow:

The group velocity is the speed of the packet's envelope, and it equals the classical particle speed $p_0/m$ — as it must, matching $d\langle x\rangle/dt = \langle p\rangle/m$ from Chapter 2. The phase velocity is half of it and has no direct physical meaning for a particle.

Because $\omega''(k) = \hbar/m \ne 0$, different Fourier components travel at different group speeds, so the packet spreads as it moves: an initial width $\Delta x_0$ grows, for large $t$, as $\Delta x(t) \approx \dfrac{\hbar t}{2m\,\Delta x_0}$. A tightly localised particle disperses fastest. This spreading is a real, observed effect and is absent only for special potentials (the harmonic oscillator, whose linear dispersion in $n$ keeps packets coherent).

8. Hermiticity, Conservation, and the Role of the Energy Basis

Because $\hat H$ is Hermitian:

• its eigenvalues $E_n$ are real (energies are measurable);
• its eigenfunctions form a complete orthonormal set, so the expansion $\Psi = \sum_n c_n\psi_n e^{-iE_nt/\hbar}$ is always possible;
• time evolution is unitary: $\sum_n |c_n|^2 = 1$ is preserved, since each $|c_n|^2$ is separately constant (the phases have unit modulus). Probabilities of energy outcomes never change in time — energy is conserved in the statistical sense.

More generally, for any operator $\hat Q$ with no explicit time dependence,

so $\langle Q\rangle$ is a constant of the motion precisely when $[\hat H,\hat Q] = 0$. Energy itself ($\hat Q = \hat H$) is always conserved for a time-independent Hamiltonian.

9. Common Pitfalls

• Calling $\psi_E(x)e^{-iEt/\hbar}$ "time-dependent physics". The phase is unobservable; every measurable quantity in a stationary state is constant. It is a state of definite energy, nothing about it evolves.
• Using the TISE when $V$ depends on time. Separation of variables fails if $V = V(x,t)$; you must solve the full TDSE (or use time-dependent perturbation theory, Chapter 12).
• Forgetting the phase factors when evolving a superposition. $\Psi(x,t) = \sum c_n\psi_n e^{-iE_n t/\hbar}$ — each term carries its own $E_n$. Dropping them (or using a common phase) loses all the dynamics.
• Normalising a plane wave. $e^{ikx}$ has infinite norm; build a wave packet. Quantities like $\langle x\rangle$ are undefined for a bare plane wave.
• Confusing phase and group velocity. The particle moves at $v_g = \hbar k_0/m = p_0/m$; $v_p = \hbar k/2m$ is not the particle speed.
• Assuming the energy is sharp in a general state. Only stationary states have $\Delta E = 0$. A superposition of eigenstates has a spread of energies with probabilities $|c_n|^2$.
• Expecting a free wave packet to hold its shape. Free packets spread, because $\omega(k)$ is nonlinear. Only the harmonic oscillator (and other special systems) avoid dispersion.
• Thinking quantisation is imposed by hand. It emerges: the normalisability/boundary conditions on the TISE admit solutions only for discrete $E$.

Summary

• TDSE: $i\hbar\,\partial_t\Psi = \hat H\Psi$ with $\hat H = -\dfrac{\hbar^2}{2m}\partial_x^2 + V$. First-order in time (deterministic), linear (superposition), complex/unitary (norm-preserving).
• Separation (for $V = V(x)$): $\Psi = \psi(x)e^{-iEt/\hbar}$ with the TISE $-\dfrac{\hbar^2}{2m}\psi'' + V\psi = E\psi$, an eigenvalue problem $\hat H\psi = E\psi$.
• Stationary states: $|\Psi|^2$, expectation values and $j$ are time-independent; energy is sharp, $\Delta E = 0$.
• General solution: $\Psi(x,t) = \sum_n c_n\psi_n(x)e^{-iE_n t/\hbar}$, $c_n = \int\psi_n^*\Psi(x,0)\,dx$. Evolve by expanding, phasing, resumming.
• Bound states: normalisability quantises the energy, $E_1 < E_2 < \cdots$; ground-state energy $> V_{\min}$ (zero-point energy); $\psi_n$ has $n-1$ nodes and $\int\psi_m^*\psi_n = \delta_{mn}$.
• Free particle: $\Psi_k = e^{i(kx-\omega t)}$, $\omega = \hbar k^2/2m$, non-normalisable; physical states are wave packets $\Psi = \tfrac{1}{\sqrt{2\pi}}\int\phi(k)e^{i(kx-\omega t)}\,dk$.
• Velocities: $v_p = \hbar k/2m$, $v_g = \hbar k_0/m = p_0/m$ (the particle speed). Nonlinear $\omega(k)$ makes packets spread as $\Delta x(t) \sim \hbar t/(2m\Delta x_0)$.
• Conservation: $\hat H$ Hermitian $\Rightarrow$ real $E_n$, complete basis, $|c_n|^2$ constant in time; $\dfrac{d\langle Q\rangle}{dt} = \dfrac{i}{\hbar}\langle[\hat H,\hat Q]\rangle$, so $[\hat H,\hat Q] = 0$ means $\langle Q\rangle$ is conserved.