Why Classical Mechanics Fails
Classical mechanics, founded on Newton's laws of motion, describes macroscopic objects — a falling stone, an orbiting planet — with complete success because such bodies behave essentially like particles. The same framework collapses when turned on microscopic objects such as electrons, atoms and molecules. The reason is twofold: classical mechanics ignores the dual wave–particle behaviour of matter that becomes pronounced for sub-atomic particles, and it takes no account of the Heisenberg uncertainty principle.
The branch of science that explicitly incorporates this dual behaviour is quantum mechanics — a theoretical science dealing with the motion of microscopic objects that display both observable wave-like and particle-like properties. It was developed independently in 1926 by Werner Heisenberg and Erwin Schrödinger. Reassuringly, when quantum mechanics is applied to macroscopic objects, for which wave-like properties are insignificant, its results reduce exactly to those of classical mechanics.
The version studied here is the wave-mechanical formulation built on the idea of wave motion. Its fundamental equation was given by Schrödinger, work for which he shared the 1933 Nobel Prize in Physics with P.A.M. Dirac.
The dual nature of matter and the uncertainty principle that motivate this model are built up in Towards the Quantum Mechanical Model.
The Schrödinger Wave Equation
For a system whose energy does not change with time — for instance an atom or a molecule — the (time-independent) Schrödinger equation is written compactly as
$\hat{H}\,\psi = E\,\psi$
Here $\hat{H}$ is a mathematical operator called the Hamiltonian. Schrödinger gave a recipe for constructing this operator from the expression for the total energy of the system. That total energy accounts for the kinetic energies of all the sub-atomic particles (electrons and nuclei), the attractive potential between electrons and nuclei, and the repulsive potential among the electrons and among the nuclei individually. Solving the equation yields the allowed energies $E$ and the corresponding wavefunctions $\psi$.
When the equation is solved for the hydrogen atom, the solution delivers the possible energy levels the electron can occupy together with the wavefunction $\psi$ associated with each level. These quantised energy states and their wavefunctions emerge characterised by a set of three quantum numbers — the principal quantum number $n$, the azimuthal quantum number $l$ and the magnetic quantum number $m_l$ — which arise as a natural consequence of the solution rather than being imposed by hand, as they had to be in the Bohr model.
"Schrödinger derived the equation, so he could solve every atom."
The equation can be solved exactly only for one-electron (hydrogen-like) species such as $\ce{H}$, $\ce{He+}$ and $\ce{Li^2+}$. For multi-electron atoms it is solved only approximately. Examiners often phrase this as a statement to be judged true or false.
Exact analytical solution ⇒ one-electron systems only. Multi-electron atoms ⇒ approximate methods.
Orbital as a One-Electron Wavefunction
When an electron is in a particular energy state, the wavefunction $\psi$ corresponding to that state contains all the information about the electron. The wavefunction is a mathematical function whose value depends only upon the coordinates of the electron in the atom; by itself it carries no physical meaning. For hydrogen or hydrogen-like species with a single electron, such wavefunctions are called atomic orbitals, and the species are called one-electron systems.
Because many such wavefunctions are possible for an electron, there are many atomic orbitals in an atom. These "one-electron orbital wavefunctions", or orbitals, form the basis of the electronic structure of atoms. In each orbital the electron has a definite energy, and a single orbital cannot hold more than two electrons. In a multi-electron atom, electrons fill the various orbitals in order of increasing energy, so each electron is described by an orbital wavefunction characteristic of the orbital it occupies.
Orbit versus orbital — not synonyms
A Bohr orbit is a circular path of fixed radius; the uncertainty principle forbids defining such a path, so Bohr orbits have no real meaning and can never be shown experimentally. An orbital is a quantum-mechanical one-electron wavefunction $\psi$ characterised by three quantum numbers $(n, l, m_l)$, describing only where the electron is most probably found.
Orbit ⇒ definite path (rejected). Orbital ⇒ probability region defined by $\psi$.
Probability Density and |ψ|²
Although $\psi$ has no physical meaning, its square does. According to Max Born, the square of the wavefunction at a point gives the probability density of the electron at that point. The probability of finding an electron at a point within an atom is proportional to $|\psi|^2$ there, and crucially $|\psi|^2$ is always positive.
Probability density $|\psi|^2$ is the probability per unit volume. Multiplying it by a small volume element gives the probability of finding the electron within that volume — a small element is specified because $|\psi|^2$ varies from region to region but may be treated as constant over a sufficiently small piece of space. Summing all such products of $|\psi|^2$ and their volume elements gives the total probability and, with it, the probable distribution of the electron in an orbital.
Key Features of the Model
The quantum mechanical model is, in NCERT's words, the picture of atomic structure that emerges from applying the Schrödinger equation to atoms. Its essential features are best held side by side.
| # | Feature | What it means |
|---|---|---|
| 1 | Quantised energy | The energy of electrons bound to the nucleus can take only certain specific values. |
| 2 | Origin of quantisation | These quantised levels are a direct result of the wave-like properties of electrons and are the allowed solutions of the Schrödinger wave equation. |
| 3 | Uncertainty | Exact position and exact velocity cannot be known simultaneously; the electron's path can never be determined, so one speaks only of probability. |
| 4 | Orbital = ψ | An atomic orbital is the wavefunction $\psi$ for an electron; all information about the electron is stored in $\psi$, and an orbital holds at most two electrons. |
| 5 | Probability via |ψ|² | The probability of finding an electron at a point is proportional to $|\psi|^2$, the probability density, which is always positive. |
Feature 3 is the conceptual hinge of the whole model. Because the uncertainty principle forbids a known path, the orbital can never be a trajectory; it can only be a map of probability. Features 4 and 5 then turn that abstract statement into something computable — every observable property of the electron is locked inside $\psi$, and quantum mechanics is precisely the tool that extracts it.
Radial Probability and Nodes
Qualitatively, orbitals are distinguished by their size, shape and orientation. A smaller orbital means a greater chance of finding the electron near the nucleus, while shape and orientation describe the directions along which the electron is more likely to lie. To make this quantitative, one plots how $\psi$, and more usefully $|\psi|^2$, varies with the distance $r$ from the nucleus.
For a 1s orbital the probability density is maximum at the nucleus and then decreases sharply as $r$ increases. For a 2s orbital the behaviour is richer: $|\psi|^2$ first falls sharply to zero, then rises again to a small maximum, before decreasing once more and approaching zero at large $r$. The radial position at which the probability density function drops to zero is a nodal surface, or simply a node.
The pattern generalises: an $ns$ orbital has $(n-1)$ nodes. Thus 1s has none, 2s has one, 3s has two, and so on — the number of nodes increases with the principal quantum number. The same kind of zero-crossing occurs for $p$ and $d$ orbitals too, where the probability density passes through zero (besides at the nucleus and at infinity) as $r$ grows.
Counting Radial, Angular and Total Nodes
NCERT separates nodes into two kinds. Radial (spherical) nodes are the spherical shells where $|\psi|^2 = 0$ as $r$ varies. Angular nodes are the planes passing through the nucleus on which the probability density is zero — for example, the $xy$-plane is a nodal plane for the $p_z$ orbital, and the $d_{xy}$ orbital has two nodal planes through the origin. The counting rules are exact and worth memorising.
| Node type | Formula | Depends on |
|---|---|---|
| Angular (nodal planes) | $l$ | Azimuthal quantum number only |
| Radial (spherical) | $n - l - 1$ | Both $n$ and $l$ |
| Total nodes | $n - 1$ | Principal quantum number only |
The arithmetic is internally consistent: angular nodes plus radial nodes give $l + (n - l - 1) = n - 1$, the total. So one angular node belongs to every $p$ orbital, two to every $d$ orbital, and the radial count for a $p$ orbital is $n-2$ (one for 3p, two for 4p, and so on).
Find the number of radial, angular and total nodes in the 4d orbital.
For 4d, $n = 4$ and $l = 2$.
Angular nodes $= l = 2$.
Radial nodes $= n - l - 1 = 4 - 2 - 1 = 1$.
Total nodes $= n - 1 = 4 - 1 = 3$, and indeed $2 + 1 = 3$. ✔
| Orbital | $n,\ l$ | Angular = $l$ | Radial = $n-l-1$ | Total = $n-1$ |
|---|---|---|---|---|
| 1s | 1, 0 | 0 | 0 | 0 |
| 2s | 2, 0 | 0 | 1 | 1 |
| 2p | 2, 1 | 1 | 0 | 1 |
| 3p | 3, 1 | 1 | 1 | 2 |
| 3d | 3, 2 | 2 | 0 | 2 |
| 4s | 4, 0 | 0 | 3 | 3 |
Mixing up the three node formulae
A common error is to call $n-1$ the radial-node count. It is the total. Radial nodes are $n-l-1$ and angular nodes are $l$. The 3d and 3p orbitals both have 2 total nodes, but their split differs: 3d is 2 angular + 0 radial, while 3p is 1 angular + 1 radial.
Angular $= l$ · Radial $= n-l-1$ · Total $= n-1$.
Multi-Electron Atoms
Applying the Schrödinger equation to multi-electron atoms presents a real difficulty: the equation cannot be solved exactly because of electron–electron interactions. The difficulty is overcome with approximate methods, and calculations on modern computers show that orbitals in atoms other than hydrogen do not differ in any radical way from the hydrogen orbitals. The principal difference is a consequence of increased nuclear charge, which contracts all the orbitals somewhat.
One distinction is decisive for chemistry. In hydrogen or hydrogen-like species the orbital energy depends only on the principal quantum number $n$; in multi-electron atoms the energies of the orbitals depend on both $n$ and $l$. This single shift is what gives rise to the familiar filling order of subshells and underpins the entire scheme of electronic configurations.
Quantum Mechanical Model in One Glance
- Classical mechanics fails for electrons because it ignores wave–particle duality and the uncertainty principle; quantum mechanics restores both.
- The Schrödinger equation $\hat{H}\psi = E\psi$ yields allowed energies $E$ and wavefunctions $\psi$; $\hat{H}$ is the Hamiltonian operator.
- An orbital is a one-electron wavefunction $\psi$ characterised by three quantum numbers $(n, l, m_l)$; it holds at most two electrons and has no physical meaning by itself.
- $|\psi|^2$ is the probability density — always positive — and gives the probability of finding the electron per unit volume.
- Nodes: angular $= l$, radial $= n-l-1$, total $= n-1$. An $ns$ orbital has $n-1$ radial nodes.
- Exact solution holds only for one-electron species; in multi-electron atoms orbital energy depends on both $n$ and $l$.