Nuclear Motion (Compact)

phy-1109 PHY-1109 Physics

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The Derivation of Energy Levels using Reduced Mass

1. Quantization Condition (Bohr’s Postulate/De Broglie Wavelength)

The condition for a stable orbit requires an integral number ($n$) of de Broglie wavelengths ($\lambda$) to fit the circumference ($2\pi r$): $n\lambda =2\pi r\text{(2)}$ Using the de Broglie wavelength $\lambda =\frac{h}{mv}$, this leads to the angular momentum quantization rule for the electron: $mvr=\frac{nh}{2\pi }=n\hslash \text{(4)}$

2. Incorporating Nuclear Motion (Center of Mass)

If the nucleus ($M$) and electron ($m$) are considered to orbit a common center of mass, the relationship between their distances ($r_N$ and $r_E$) from the center of mass, and the total separation ($r$), is governed by the conservation of momentum: $m{r}_E=M{r}_N\text{(7)}$ $r=r_E+r_N\text{(8)}$ The total angular momentum of the atom must be quantized (Bohr’s first postulate): $\text{Total angular momentum}=m\omega r_E^2+M\omega r_N^2=n\hslash \text{(9)}$ The angular velocity ($\omega$) is the same for both particles.

3. Introduction of Reduced Mass ($m^{\prime }$)

Substituting $r_E=\left(\frac{M}{M+m}\right)r$ and $r_N=\left(\frac{m}{M+m}\right)r$ into the angular momentum equation (9) yields the corrected quantization rule: $\left(\frac{mM}{M+m}\right)\omega r^2=n\hslash \text{(12)}$ The force balance condition must also be generalized: $m{\omega }^2{r}_E=\frac{1}{4\pi {ϵ}_o}\frac{e^2}{r^2}\text{(13)}$ Substituting $r_E$ into equation (13) gives the corrected force balance: $\left(\frac{mM}{M+m}\right){\omega }^{2}r=\frac{1}{4\pi {ϵ}_o}\frac{e^2}{r^2}\text{(14)}$ By defining the reduced mass ($m^{\prime }$) as: ${\mathbf{m}}^{\prime }=\left(\frac{mM}{M+m}\right)\text{(15)}$ The generalized equations become analogous to the standard Bohr model, replacing $m$ with $m^{\prime }$: $m^{\prime }\omega r^2=n\hslash \text{(16)}$ $m^{\prime }{\omega }^{2}r=\frac{1}{4\pi {ϵ}_o}\frac{e^2}{r^2}\text{(17)}$

4. Final Energy Expression

From equations (16) and (17), the expression for the energy levels of the hydrogen atom is obtained: ${\mathbf{E}}_{\mathbf{n}}=-\frac{{\mathbf{m}}^{\prime }{\mathbf{e}}^4}{8{ϵ}_{\mathbf{o}}^2{\mathbf{h}}^2}\left(\frac{1}{{\mathbf{n}}^2}\right)\text{(18)}$

Due to nuclear motion, the energy levels are changed by the fraction $\frac{m^{\prime }}{m}=\left(\frac{M}{M+m}\right)$, which is approximately $\frac{1836}{1837}=0.99945$.

Consequences and Hydrogenic Atoms

The correction for nuclear motion lowers the value of the Rydberg number ($R$).

Because of the greater nuclear mass, the spectral lines of deuterium are shifted slightly to shorter wavelengths compared to ordinary hydrogen. For instance, the $H_{\alpha }$ line ($n=3$ to $n=2$ transition) occurs at $6561\text{ A˚}$ for deuterium versus $6563\text{ A˚}$ for hydrogen.

For Hydrogenic Atoms (atomic number $Z$), the electrostatic force exerted on the orbital electron is $F_e=\frac{1}{4\pi {ϵ}_o}\frac{Z{e}^2}{r^2}$. Consequently, the energy level formula includes $Z^2$: $E_n=-\frac{m^{\prime }{Z}^2{e}^4}{8{ϵ}_o^2{h}^2}\left(\frac{1}{n^2}\right)\text{(20)}$ The general formula for spectral lines is: $\frac{1}{\lambda }=\frac{m^{\prime }{Z}^2{e}^4}{8{ϵ}_o^{2}c{h}^3}\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)\text{(21)}$

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RELATED PYQ TERM QUESTIONS (Atomic Structure and Quantum Mechanics)

Question	Year(s) of Appearance
What are the basic postulates of Bohr Atom Model? Calculate the limiting values of wavelengths of different spectral series of Hydrogen atom.	2018, 2020
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
Find the wavelength of the spectral line corresponding to the transition in the hydrogen atom from $n=4$ to $n=2$ state.	2020
How much energy is required to remove an electron in the $n=2$ state from a Hydrogen atom?	2018
An electron has de-Broglie wavelength of 2 PM. Find its Kinetic energy and the phase & group velocities of its de-Broglie waves.	2021
Calculate the wavelength associated with a thermal neutron of energy 0.025 eV.	2022