Quantum Mechanics · Chapter 11

Three dimensions and central potentials

Where the one-dimensional techniques meet angular momentum. The Schrödinger equation in space, separation of variables, the central potentials whose rotational symmetry splits every problem into spherical harmonics and a radial equation, the effective potential with its centrifugal barrier, and the reduction of two bodies to one.

\nabla^{2} ψ + \frac{2 m}{ℏ ^{2}} (E - V) ψ = 0

ψ (r, θ, ϕ) = R (r) Y_{ℓ m}

\int ∣ ψ (r) ∣^{2} d^{3} r = 1

L^^{2} Y_{ℓ m} = ℏ^{2} ℓ (ℓ + 1) Y_{ℓ m}

Lessons

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05The effective potentialThe radial equation is a one-dimensional problem in an effective potential: the real one plus the centrifugal barrier $\hbar^{2}\ell(\ell+1)/2mr^{2}$, which grows without limit at the centre. For an attractive Coulomb potential the sum has a minimum, at the radius where a classical particle with that angular momentum would circle. The higher $\ell$, the further out the minimum and the shallower it is — which is why high angular momentum states sit far from the nucleus and are loosely bound.Read lesson →

06Quantum numbers in a central potentialEach value of $\ell$ gives its own radial equation, and each radial equation has its own ladder of solutions, counted by the number of nodes of $u$. So a state of a central potential is labelled by three numbers: the radial number, $\ell$ and $m$. The energy depends on the first two. And the half-line brings a surprise: in three dimensions a shallow well can have no bound state at all, because the $s$-wave equation is the odd half of a one-dimensional well.Read lesson →

07The two-body problemA hydrogen atom is two particles, not one electron round a fixed nucleus. When two particles interact through a force depending only on their separation, the problem splits: the centre of mass moves freely, as a particle of the total mass, and the separation moves as a single particle of the reduced mass in the potential. Since the proton is 1836 times heavier than the electron, the reduced mass is almost the electron’s, and the correction it makes is small but measured — it distinguishes hydrogen from deuterium in a spectrum.Read lesson →

08Chapter practiceBring the chapter together: the Schrödinger equation in space, separable potentials and the cubic box, central potentials and the conservation of angular momentum, the radial equation, the effective potential, the labels of states and the two-body problem. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Three dimensions and central potentials

Build the foundation