The shape of an orbital is defined by the quantum numbers associated with an electron in an atom.

More specifically, the shape of an orbital is fundamentally derived from the solutions to the Schrödinger equation for the hydrogen atom. This equation describes the behavior of quantum systems, and its solutions provide the possible energy levels and spatial distributions of an electron within an atom. We interpret these spatial distributions as the shape of the orbital.

The solutions to the Schrödinger equation are characterized by three quantum numbers: the principal quantum number ($n$), the azimuthal quantum number ($l$), and the magnetic quantum number ($m$). Each of these quantum numbers conveys distinct information about the electron’s state.

1. Principal Quantum Number ($n$): This quantum number specifies the energy level of the electron and the size of the orbital. It can take any positive integer value. As the value of $n$ increases, the energy level also increases, resulting in a larger orbital.

2. Azimuthal Quantum Number ($l$): This quantum number defines the shape of the orbital. It can take integer values ranging from $0$ to $n-1$. For a specific value of $n$, the value of $l$ indicates the number of nodes in the orbital—regions where there is zero probability of locating the electron. The distinct shapes associated with varying values of $l$ are commonly referred to as $s$, $p$, $d$, and $f$ orbitals.

3. Magnetic Quantum Number ($m$): This quantum number determines the orientation of the orbital in space. It can take integer values from $-l$ to $+l$. For a given value of $l$, different values of $m$ correspond to different spatial orientations of the same orbital shape.

In conclusion, the shape of an orbital is determined by the quantum numbers associated with an electron in an atom, which arise from the solutions of the Schrödinger equation. The principal quantum number ($n$) indicates the size of the orbital, the azimuthal quantum number ($l$) defines its shape, and the magnetic quantum number ($m$) determines its orientation in space.