The Schrödinger Equation in Three Dimensions and Hydrogen

The one-dimensional Schrödinger equation of the previous module fixed a single quantum number from a single boundary condition. A real atom sits in three dimensions, and its electron feels a potential that depends only on the distance to the nucleus. Both facts are structural. Three dimensions supply three coordinates, each carrying its own boundary condition and therefore its own quantum number; a central potential $V(r)$ makes the angular part of the problem identical for every such potential, so it can be solved once and reused. The result is the exact quantization of energy and angular momentum in hydrogen, recovered from wave mechanics rather than postulated as in the Bohr model.

The equation in three dimensions

For a particle of mass $m$ moving in a potential $V(x,y,z)$, the time-independent Schrödinger equation replaces the single second derivative of the 1-D case with the Laplacian:¹

The simplest three-dimensional bound problem is a particle in a cubical box: $V=0$ inside $0\le x,y,z\le L$ and infinite outside. The walls force $\psi$ to vanish on each face, exactly as in one dimension applied to each axis separately, so the wave function is a product of three sine standing waves and the energy is a sum of three one-dimensional energies:

The ground state $(1,1,1)$ has energy $3E_1$ with $E_1={\hslash }^2{\pi }^2/2m{L}^2$. The first excited level is reached three different ways — $(2,1,1)$, $(1,2,1)$, $(1,1,2)$ — all with energy $6E_1$. One energy shared by more than one wave function is a degeneracy.

Spherical coordinates and separation

For the hydrogen atom the proton is treated as fixed and the electron moves in the Coulomb potential

where $Z$ is the atomic number ($Z=1$ for hydrogen, $Z=2$ for singly ionized helium). Nuclear motion is folded in exactly by replacing the electron mass $m$ with the reduced mass $\mu =m/(1+m/M_N)$.¹ Because $V$ depends only on $r=(x^2+y^2+z^2{)}^{1/2}$, the natural coordinates are spherical: $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$.

The transformed equation is formidable but standard,²

The solution proceeds by separation of variables: seek product solutions

Substituting and dividing turns the one partial differential equation into three ordinary ones. The separation works because each derivative touches only its own factor, and the radial terms can be collected on one side of an equation whose other side depends only on the angles. Two independent variables set equal force both sides to a constant.

• Azimuthal equation. The $\varphi$-dependence separates first, giving $g(\varphi )=e^{im\varphi }$. Single-valuedness, $g(\varphi +2\pi )=g(\varphi )$, forces $m$ to be an integer, positive, negative, or zero.
• Polar equation. The $\theta$-dependence yields the associated Legendre functions $f_{\ell m}(\theta )$. Finiteness at $\theta =0$ and $\theta =\pi$ forces $\ell$ to be a non-negative integer and restricts $|m|\le \ell$.
• Radial equation. The $r$-dependence carries the potential and produces the principal quantum number $n$ and the energy.

The product of the two angular factors is the spherical harmonic$Y_{\ell m}(\theta ,\varphi )=f_{\ell m}(\theta )g_m(\varphi )$, the same family of functions for every central potential, since the angular equations never reference $V(r)$.

Quantization of angular momentum

The angular equations do more than supply functions; they diagonalize angular momentum. Rewriting the classical energy of a particle in a central field in terms of the radial momentum $p_r$ and the angular momentum $L$,

and promoting $p_r^2$ and $L^2$ to operators reproduces the two kinetic terms of the spherical Schrödinger equation. The operator $(L^2{)}_{\text{op}}$ is exactly the angular part, and it acts on the spherical harmonics as an eigenvalue equation:²

For every central potential the magnitude of the angular momentum is therefore quantized, and its projection on the $z$ axis is quantized independently:

Two features are unusual. First, $|\mathbf{L}|=\sqrt{\ell (\ell +1)}\hslash$ is strictly larger than the maximum projection $\ell \hslash$, so $\mathbf{L}$ can never lie along the $z$ axis. The two follow from the uncertainty principle for angular momentum: no two components of $\mathbf{L}$ can be known at once (except when $\mathbf{L}=0$), so a fully aligned vector, which would fix all three components, is forbidden. Second, for a given $\ell$ there are exactly $2\ell +1$ allowed values of $m$, one orientation per integer projection. This is space quantization: $\mathbf{L}$ points only along the discrete set of cones whose axis is $z$.

Worked example: allowed orientations for ℓ = 2

For $\ell =2$ the projections are $L_z=m\hslash$ with $m=0,\pm 1,\pm 2$, so $L_z\in \{2\hslash ,\hslash ,0,-\hslash ,-2\hslash \}$. The magnitude is

The smallest angle between $\mathbf{L}$ and the $z$ axis occurs at the largest projection, $m=\ell =2$:

The vector cannot reach $\theta =0$, confirming that $\mathbf{L}$ never aligns with the axis.²

The radial equation and energy quantization

The angular constant $\ell (\ell +1)$ feeds into the radial equation, which for the Coulomb potential reads

The bracket is an effective potential$V_{\text{eff}}(r)={\hslash }^2\ell (\ell +1)/2\mu r^2-kZ{e}^2/r$: the attractive Coulomb term plus a repulsive centrifugal barrier that grows with $\ell$. Bound states have $E<0$; the well confines the electron, and only discrete energies give normalizable $R(r)$.

Solving the radial equation for hydrogen gives energies that depend only on the principal quantum number:²

with $n=1,2,3,\dots$ and the restriction $n>\ell$. These are the Bohr energies exactly. The radial functions have the form $R_{n\ell }(r)=A_{n\ell }{e}^{-r/n{a}_0}{r}^{\ell }{l}_{n\ell }(r/a_0)$, where the $l_{n\ell }$ are Laguerre polynomials and $a_0={\hslash }^2/(\mu k{e}^2)$ is the Bohr radius.

The three quantum numbers and their ranges close the section:

That $E_n$ depends on $n$ alone, and not on $\ell$, is special to the inverse-square force. In classical terms, the energy of an orbit in a $1/r^2$ field depends only on the semimajor axis, not the eccentricity; the largest $\ell =n-1$ is the nearly circular orbit and small $\ell$ the eccentric one. For any non-Coulomb central force the degeneracy in $\ell$ is lifted, a fact that shapes the periodic table.

Counting states and the level diagram

For a given $n$ there are $n$ values of $\ell$, and for each $\ell$ there are $2\ell +1$ values of $m$. The total number of spatial states with energy $E_n$ is

which doubles to $2n^2$ once electron spin is included in the next lessons. The $m$-degeneracy reflects the absence of any preferred direction in space; a magnetic field supplies one and splits the levels.

States are named by the value of $n$ followed by a letter code for $\ell$: S for $\ell =0$, P for $\ell =1$, D for $\ell =2$, F for $\ell =3$, then alphabetically. The codes descend from the spectroscopists' sharp, principal, diffuse, and fundamental line series. Radiative transitions between levels are not arbitrary; conservation of angular momentum, together with the photon's intrinsic spin of $1\hslash$, restricts them by the selection rules

The wave mechanics of a central potential thus reproduces the Bohr spectrum while adding structure the old model lacked: a spread of angular-momentum states $\ell$ at each energy, a genuine spatial probability cloud instead of an orbit, and selection rules that decide which lines appear. The shapes of those clouds are the subject of the next lesson.