Part I — From Spins to Particles: A New Physical Setting
Throughout Lectures 1–7, we worked exclusively with finite-dimensional Hilbert spaces: the qubit ($\mathbb{C}^2$), multi-spin systems ($\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \cdots$). We now make a fundamental leap to an infinite-dimensional Hilbert space: a particle moving on a line.
The new setting introduces two key continuous observables:
| Observable | Operator | Spectrum | Eigenstates |
|---|---|---|---|
| Position | $\hat{x}$ | $x \in \mathbb{R}$ (continuous) | $|x\rangle$: delta-localized |
| Momentum | $\hat{p}$ | $p \in \mathbb{R}$ (continuous) | $|p\rangle$: plane waves |
The state is represented by a wave function $\psi(x) = \langle x|\psi\rangle$ — a complex-valued function on $\mathbb{R}$. The physical content: $|\psi(x)|^2$ is a probability density, not a probability. The probability of finding the particle in the interval $[a,b]$ is:
Part II — The Fourier Transform
The Fourier transform is the mathematical bridge between the position and momentum representations. It converts a function of $x$ into a function of $k$ (wavenumber, with $p = \hbar k$) and vice versa. Using the symmetric convention with $\hbar = 1$:
Parseval's theorem states that the Fourier transform preserves norms:
This is essential: it means $|\tilde{\psi}(k)|^2$ is a valid probability density in momentum space. The physical meaning of $\tilde{\psi}(k)$: it is the amplitude to have momentum $p = \hbar k$. The key duality: if $\psi(x)$ is sharply peaked (definite position), then $\tilde{\psi}(k)$ is spread out (indefinite momentum) — the Heisenberg uncertainty principle expressed as Fourier duality.
The canonical Fourier pair: Gaussian ↔ Gaussian. If $\psi(x) \sim e^{-x^2/(4\sigma^2)}$ has width $\sigma$ in position, then $\tilde{\psi}(k) \sim e^{-k^2\sigma^2}$ has width $1/(2\sigma)$ in momentum, with the product $\sigma \cdot \frac{1}{2\sigma} = \frac{1}{2}$ — saturating the Heisenberg bound.
Part III — The Momentum Operator
In position space, the action of the momentum operator on the wave function is derived using completeness:
Since $p\,e^{ipx} = -i\,\dfrac{d}{dx}\,e^{ipx}$, we can pull the derivative outside the integral:
Part IV — Free Particle Dynamics
The Hamiltonian for a free particle (no potential) is purely kinetic:
The time-dependent Schrödinger equation $i\hbar\,\partial\psi/\partial t = H\psi$ becomes:
The plane wave $\psi_k(x,t) = e^{i(kx - \omega t)}$ is a solution, provided the dispersion relation is satisfied:
This is a quadratic dispersion — different wavenumber components travel at different speeds. This is what causes wave packets to spread over time. A plane wave is not normalizable (it extends to $\pm\infty$), so physical states must be superpositions of plane waves.
Part V — Wave Packets
A physical particle is described by a wave packet — a superposition of plane waves localized in both position and momentum:
At $t = 0$, a Gaussian wave packet centered at $x = 0$ with width $\sigma$ and mean momentum $p_0 = \hbar k_0$:
The dynamics are governed by two effects:
- Translation: the center moves at the group velocity $v_g = \hbar k_0/m = p_0/m$
- Spreading: the width grows as $\sigma(t)^2 = \sigma^2 + \left(\dfrac{\hbar t}{2m\sigma}\right)^2$
Heavy particles spread slowly; light particles spread rapidly. This spreading is purely quantum — it has no classical analogue.
Part VI — Group Velocity vs Phase Velocity
A plane wave $e^{i(kx-\omega t)}$ has two velocities:
| Velocity | Definition | Formula ($\omega = \hbar k^2/2m$) | Physical meaning |
|---|---|---|---|
| Phase velocity | $v_\mathrm{ph} = \omega/k$ | $\hbar k/(2m) = p/(2m)$ | Speed of phase fronts — not physical |
| Group velocity | $v_g = d\omega/dk$ | $\hbar k/m = p/m$ | Speed of wave packet envelope — physical |
For the free non-relativistic particle: $v_g = 2v_\mathrm{ph}$. The particle moves at the group velocity — this is what corresponds to the classical particle velocity $p/m$. The phase velocity of $p/(2m)$ has no direct physical interpretation.
The expectation value of position moves at the group velocity: $\langle x\rangle(t) = \langle x\rangle(0) + v_g\,t$, recovering Newton's first law for the quantum average.
Part VII — The Hamiltonian $H = (\omega/2)\sigma_z$ as Linear Dispersion
Our familiar spin Hamiltonian $H = (\omega/2)\sigma_z$ has exactly two energy levels $\pm\omega/2$ — analogous to a "two-site dispersion relation." This connects to a broader principle about dispersion. For a linear dispersion $\omega = vk$ (like a relativistic photon or massless Dirac fermion):
- Group velocity = phase velocity = $v$ for all $k$
- The Schrödinger equation becomes $i\hbar\partial\psi/\partial t = -i\hbar v\,\partial\psi/\partial x$
- Exact solution: $\psi(x,t) = f(x - vt)$ — any initial packet translates rigidly at speed $v$, no spreading
The spreading of the non-relativistic wave packet comes from the quadratic dispersion $\omega = \hbar k^2/2m$: different wavenumber components travel at different group velocities $\hbar k/m$, so the packet stretches over time. Linear dispersion is dispersionless — all components travel at the same speed.
Part VIII — Connection to Classical Mechanics (Ehrenfest)
Quantum mechanics must recover classical mechanics for macroscopic systems. Ehrenfest's theorem makes this precise. For a free particle:
Together: $\langle x\rangle(t) = \langle x\rangle(0) + \dfrac{\langle p\rangle(0)}{m}\,t$ — Newton's first law for expectation values.
Adding a potential $V(x)$:
In the narrow-packet limit, the quantum averages satisfy exactly the classical equations of motion. Classical mechanics is the limit of quantum mechanics when wave packets are so narrow that $V'(x)$ is approximately constant across the packet width.