The language of the atom is governed by the principles of quantum mechanics. Within this framework, electrons are assigned a unique “address” using four distinct quantum numbers. These numbers define the electron’s state and location probability within the atom. These quantum numbers must adhere to strict mathematical constraints derived from the fundamental equations of quantum physics. When a combination of these numbers violates these constraints, it describes an electron state that is physically impossible, or “not allowed.”
The Role and Definition of the Four Quantum Numbers
The first quantum number, the principal quantum number (\(n\)), primarily dictates the electron’s energy level and its probable distance from the nucleus. Higher values of \(n\) correspond to shells farther from the nucleus and greater energy. The azimuthal quantum number (\(l\)) defines the shape of the electron’s orbital, often referred to as a subshell. Orbitals with \(l=0\) are spherical, while \(l=1\) orbitals are dumbbell-shaped.
The magnetic quantum number (\(m_l\)) specifies the orientation of the orbital in three-dimensional space. This number accounts for the possible spatial arrangements for a given orbital shape. The final number is the spin quantum number (\(m_s\)), which is independent of the first three and describes the intrinsic angular momentum of the electron.
Rules Governing Individual Quantum Number Values
The principal quantum number, \(n\), must be a positive whole number, starting at \(1\). A value of \(n=0\) is forbidden because it would imply an electron with zero energy residing within the nucleus, which is not a bound state.
The azimuthal quantum number, \(l\), must be zero or a positive integer. A negative value for \(l\) is an impossible state, as it would contradict the physical definition of orbital angular momentum. The spin quantum number, \(m_s\), has the simplest constraint: it can only take one of two possible values: \(+1/2\) or \(-1/2\). Any proposed value outside this pair describes a spin state that is physically disallowed for an electron.
Forbidden Relationships Between Quantum Numbers
Beyond the individual limitations, the quantum numbers are hierarchically linked. The azimuthal quantum number, \(l\), is strictly limited by the value of \(n\). The allowed values for \(l\) range from \(0\) up to \(n-1\), meaning that \(l\) can never be equal to or greater than \(n\). For instance, a set starting with \(n=2\) cannot have an \(l\) value of \(2\) or higher, because \(l\) must be \(0\) or \(1\).
This constraint means that a combination like \(\{n=3, l=3\}\) is not allowed, as the orbital shape cannot be “larger” than the energy level it occupies. The magnetic quantum number, \(m_l\), is then constrained by \(l\), taking any integer value from \(-l\) through zero up to \(+l\). If an orbital has a shape defined by \(l=1\), the only allowed orientations \(m_l\) can take are \(-1, 0,\) and \(+1\).
A set such as \(\{n=1, l=0, m_l=1\}\) is forbidden, even though the first two numbers are individually valid. Since \(n=1\) only allows \(l=0\), the magnetic quantum number \(m_l\) must be restricted to \(m_l=0\). The proposed \(m_l=1\) violates the maximum constraint that \(|m_l|\) cannot exceed \(l\), making the entire set impossible.
The Pauli Exclusion Principle and Identical Sets
A different type of “not allowed” state is defined by the Pauli Exclusion Principle, which governs the occupancy of orbitals by electrons. This principle states that no two electrons within the same atom can possess an identical set of all four quantum numbers. The first three quantum numbers—\(n\), \(l\), and \(m_l\)—together define a specific atomic orbital. For example, the set \(\{n=2, l=1, m_l=0\}\) defines one of the \(2p\) orbitals.
According to the principle, two electrons can occupy this single, allowed orbital, but only if their fourth quantum number, \(m_s\), is different. This means if one electron has \(m_s=+1/2\), the second electron in the same orbital must have \(m_s=-1/2\). A proposed scenario where a third electron attempts to enter this orbital, or where two electrons in the same orbital both have \(m_s=+1/2\), is a forbidden electron configuration. The Pauli Exclusion Principle is what limits the occupancy of any single orbital to a maximum of two electrons, ensuring that every electron in a multi-electron atom maintains a unique quantum identity.