Quantum Number Selection Rules Calculator
Check whether a hydrogen-like electric dipole transition is allowed using the exact Δl = ±1, Δm = 0, ±1 selection rules.
🚦 What is the Quantum Number Selection Rules Calculator?
This selection rules calculator checks whether a radiative transition between two hydrogen-like quantum states is allowed by electric dipole (E1) radiation, the fastest and dominant transition mechanism in atomic physics. Enter the initial and final n, l, and m quantum numbers, and it returns a clear allowed-or-forbidden verdict along with the exact reason.
The rule comes from symmetry: an electric dipole transition matrix element is only nonzero when the orbital quantum number changes by exactly Δl = ±1 and the magnetic quantum number changes by Δm_l = 0 or ±1. When either condition fails, the transition is exactly zero to this order and is called "forbidden."
This is not just a bookkeeping exercise. Selection rules explain real, measurable atomic behavior, most famously why hydrogen's 2s state is metastable (living about a hundred million times longer than 2p) simply because its decay to 1s violates the Δl rule.
This calculator is useful for physics and chemistry students learning atomic spectroscopy, and anyone curious why some atomic transitions are bright, fast lines while others are dim or effectively forbidden.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - 2p → 1s (Lyman-alpha, allowed)
Example 2 - 2s → 1s (forbidden, hence metastable)
Example 3 - 3d → 2s (forbidden, Δl too large)
❓ Frequently Asked Questions
🔗 Related Calculators
What are selection rules in atomic physics?
Selection rules are conditions on the quantum numbers of two atomic states that determine whether a radiative transition between them can occur through a particular mechanism, such as electric dipole (E1) radiation. They arise from evaluating the transition matrix element, symmetry forces it to vanish exactly unless the quantum numbers change in specific allowed ways.
What is the electric dipole (E1) selection rule?
For a single-electron transition, the electric dipole selection rules are Δl = ±1 and Δm_l = 0, ±1, where l is the orbital angular momentum quantum number and m_l is the magnetic quantum number. Both conditions must hold simultaneously for the transition to be E1-allowed.
Why does hydrogen's 2s state live so much longer than 2p?
The 2p state decays to 1s almost instantly (about 1.6 nanoseconds) via an allowed E1 transition (Δl = -1). The 2s state cannot decay to 1s by E1 radiation at all, because Δl = 0 fails the selection rule, so it is 'metastable' and instead decays via a much slower two-photon process, with a lifetime around 0.15 seconds, roughly a hundred million times longer.
Why is Δl = ±1 required, but Δl = 0 or ±2 is not?
The electric dipole operator is proportional to position x, which has odd parity, it flips sign under spatial inversion. A transition matrix element is only nonzero if the initial and final states have opposite parity, and since a state's parity is (-1)^l, that means l must change by an odd number. Combined with angular momentum conservation (the emitted photon carries away one unit of angular momentum), this narrows the rule to exactly Δl = ±1.
What does Δm_l = 0, ±1 mean physically?
The photon emitted in an E1 transition carries one unit of angular momentum, and its component along the quantization axis can be -1, 0, or +1 (corresponding to left-circular, linear, or right-circular polarization). Conservation of the z-component of angular momentum then requires the atomic magnetic quantum number to change by exactly that amount, Δm_l ∈ {-1, 0, +1}.
Does a 'forbidden' transition mean it never happens?
It means the transition cannot happen via the dominant electric dipole (E1) mechanism. It can still occur through weaker processes, magnetic dipole (M1), electric quadrupole (E2), or two-photon emission, which are typically thousands to millions of times slower, but not literally impossible.
Why does this calculator check n1 = n2 separately?
In the simple non-relativistic hydrogen model, energy depends only on the principal quantum number n, so if n1 equals n2 there is no energy difference and therefore no photon to emit, regardless of whether Δl and Δm happen to satisfy the selection rule. This calculator flags that case as a reminder, even though real atoms have small fine-structure energy differences within the same n.
What are the valid ranges for l and m?
For a given principal quantum number n, the orbital quantum number l can be any whole number from 0 to n-1, and the magnetic quantum number m_l can be any whole number from -l to +l. This calculator validates both the initial and final states against these ranges before checking the transition.
Is the 2p→1s transition allowed?
Yes. Going from n=2, l=1 to n=1, l=0 gives Δl = -1, satisfying the rule, and with m=0 for both states, Δm_l = 0 also satisfies the rule. This is the hydrogen Lyman-alpha transition, one of the strongest, fastest lines in the hydrogen spectrum.
Is the 3d→2s transition allowed?
No. Going from n=3, l=2 to n=2, l=0 gives Δl = -2, which fails the Δl = ±1 rule, so this transition is E1-forbidden. A 3d electron can decay to 2p (Δl = -1, allowed) but not directly to 2s.
Does this calculator include spin-orbit coupling or total angular momentum J?
No, it is deliberately scoped to the simpler single-electron orbital selection rules (Δl, Δm_l) taught first in every quantum mechanics course. Multi-electron atoms with fine structure use the related but more involved rules ΔJ = 0, ±1 (excluding J=0 to J=0) and a required parity change, which follow the same underlying physics.