Quantum Mechanics · Chapter 04

Position, momentum and wavefunctions

A particle on a line needs a basis with a continuum of states. Position states and the wavefunction they define, probability density and normalisation, the position and momentum operators, the canonical commutation relation, and the Fourier transform that turns a state’s position picture into its momentum picture.

ψ (x) = ⟨ x ∣ ψ ⟩

\int ∣ ψ (x) ∣^{2} d x = 1

p^= - i ℏ \frac{\partial}{\partial x}

[x^, p^] = i ℏ

Lessons

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05The canonical commutation relationMultiply by $x$ and then differentiate, or differentiate and then multiply by $x$: the two orders differ by exactly the function you started with. For position and momentum this is the canonical commutation relation, $\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar$. From it comes Heisenberg’s uncertainty relation, and the proof that no finite matrices can describe a particle on a line.Read lesson →

06Momentum space and the Fourier transformA state can be written in the momentum basis just as well as in the position basis. Its components there form the momentum-space wavefunction, whose modulus squared is the probability density for momentum. The two descriptions are related by the Fourier transform, and the transform has a built-in trade-off: a wavefunction squeezed narrow in position is necessarily spread wide in momentum.Read lesson →

07Gaussian wave packetsA wave packet is a plane wave with an envelope: a phase winding at a steady rate, which gives it a momentum, inside a bump, which gives it a place. With a Gaussian envelope something special happens — the momentum density is a Gaussian too, and the product of the two spreads is exactly $\hbar/2$, the smallest the uncertainty relation allows. The Gaussian is the least uncertain state there is.Read lesson →

08Chapter practiceBring the chapter together: wavefunctions and the delta function, densities and normalisation, the position and momentum operators, their commutator and Heisenberg’s relation, the Fourier transform to momentum space and the Gaussian packet. No worked example sits above the answers.Read lesson →

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Position, momentum and wavefunctions

Build the foundation