When an electron makes a transition between energy levels of...

When an electron makes a transition between energy levels of a hydrogen atom, there are no restrictions on the initial and final values of the principal quantum number $n$. However, there is a quantum mechanical rule that restricts the initial and final values of the orbital angular momentum $\ell$. This is the selection rule, which states that $\Delta \ell=\pm 1$; that is, in a transition, the value of $\ell$ can increase or decrease by only 1 . According to this rule, which of the following transitions are allowed: (a) $1 s \longrightarrow 2 s$ (b) $2 p \longrightarrow 1 s$ (c) $1 s \longrightarrow 3 d$ (d) $3 d \longrightarrow 4 f(\mathrm{e}) 4 d \longrightarrow 3 s ?$

When an electron makes a transition between energy levels of a hydrogen atom, there are no restrictions on the initial and final values of the principal quantum number $n$. However, there is a quantum mechanical rule that restricts the initial and final values of the orbital angular momentum $\ell$. This is the selection rule, which states that $\Delta \ell=\pm 1$; that is, in a transition, the value of $\ell$ can increase or decrease by only 1 . According to this rule, which of the following transitions are allowed:
(a) $1 s \longrightarrow 2 s$ (b) $2 p \longrightarrow 1 s$ (c) $1 s \longrightarrow 3 d$ (d) $3 d \longrightarrow 4 f(\mathrm{e}) 4 d \longrightarrow 3 s ?$

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as n, describes the overall energy level or shell of an electron in an atom. It is a fundamental quantum number that determines the electron's energy and its average distance from the nucleus, but unlike the orbital angular momentum quantum number, there are no strict rules governing the allowed changes in n during atomic transitions. This concept is crucial in understanding the structure of atomic energy levels.

Orbital Angular Momentum Quantum Number

The orbital angular momentum quantum number, represented by ?, specifies the shape of the electron's orbital and is directly related to the angular momentum of the electron within the atom. It has integer values ranging from 0 to n-1 for each principal quantum number n. Changes in ? are more restricted during transitions, forming the basis for quantum mechanical selection rules.

Selection Rules for Electric Dipole Transitions

Selection rules are criteria derived from conservation laws and symmetries that dictate the allowed transitions between quantum states in atoms. For electric dipole transitions, a key selection rule requires that the orbital angular momentum quantum number changes by ?? = ±1. This rule governs which transitions between energy levels are permissible, ensuring that only transitions conforming to this criterion occur with significant probability.

Electron Transitions in Atoms

Electron transitions in atoms refer to the movement of electrons between discrete energy levels, often observed as the absorption or emission of photons. These transitions are governed by quantum numbers and selection rules, with the allowed changes in the orbital angular momentum quantum number (?? = ±1 for electric dipole transitions) being a critical factor. Understanding these transitions is essential for analyzing phenomena such as spectral lines and atomic emission or absorption.

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