Atomic Spectra (Compact)

phy-1109 PHY-1109 Physics

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Compact Note: Atomic Structure and Quantum Principles

Topic 1: Uncertainty Principle

Main Derivation/Equations

1. Heisenberg Uncertainty Principle (Position-Momentum): The product of experimental uncertainties in position ($\Delta x$) and momentum ($\Delta P$) is always larger than or equal to $\frac{1}{2}\hslash$. $\mathbf{\Delta P}\cdot \mathbf{\Delta x}\ge \frac{1}{2}\mathbf{\hslash }$
2. Energy-Time Uncertainty Principle: This principle states that a quantum state existing only for a short time cannot have a definite energy.

Application: Electron Confinement

• Momentum Uncertainty: $\mathbf{\Delta P}\ge \frac{\mathbf{\hslash }}{2\Delta x}$
• Kinetic Energy (Relativistic, for nucleus application): $T=pc$
• Kinetic Energy (Classical, for atom application): $T=\frac{p^2}{2m}$

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Topic 2: Atomic Spectra and Spectral Series

Main Equations (Spectral Series of Hydrogen)

The quantity $\mathbf{R}$ is the Rydberg constant ($\mathbf{R}=1.097\times 10^7{\text{ m}}^{-1}$).

Series	Formula	Transition Level ($n$)	Location
Lyman	$\frac{1}{\lambda }=\mathbf{R}\left(\frac{1}{1^2}-\frac{1}{{\mathbf{n}}^2}\right)$	$n=2,3,4\dots$	Ultraviolet
Balmer	$\frac{1}{\lambda }=\mathbf{R}\left(\frac{1}{2^2}-\frac{1}{{\mathbf{n}}^2}\right)$	$n=3,4,5\dots$	Visible/Near UV
Paschen	$\frac{1}{\lambda }=\mathbf{R}\left(\frac{1}{3^2}-\frac{1}{{\mathbf{n}}^2}\right)$	$n=4,5,6\dots$	Infrared
Brackett	$\frac{1}{\lambda }=\mathbf{R}\left(\frac{1}{4^2}-\frac{1}{{\mathbf{n}}^2}\right)$	$n=5,6,7\dots$	Infrared
Pfund	$\frac{1}{\lambda }=\mathbf{R}\left(\frac{1}{5^2}-\frac{1}{{\mathbf{n}}^2}\right)$	$n=6,7,8\dots$	Infrared

Related PYQs (Topic III.C, III.D)

• (Topic III.C - Q2) State the postulates of Bohr atom model. Obtain the expressions for the radius and electron energy levels in the $n^{th}$ orbit for $H_2$ atom (2024).
• (Implied Answer) Identification of Deuterium.

Answers to PYQs

• Expression for Energy Levels (H atom): $E_n=-\frac{m^{\prime }{e}^4}{8{ϵ}_o^2{h}^2}\left(\frac{1}{n^2}\right)$, where $m^{\prime }$ is the reduced mass.
• Impact of Nuclear Motion (Deuterium): Because the nuclear mass of deuterium is greater than that of ordinary hydrogen, its reduced mass ($m^{\prime }$) is slightly different. This causes the spectral lines of deuterium to be shifted slightly to wavelengths shorter than the corresponding lines of ordinary hydrogen. This spectral shift (e.g., $H_{\alpha }$ at $6561\text{ A˚}$ for deuterium versus $6563\text{ A˚}$ for hydrogen) was the means by which deuterium was identified in 1932.