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For a simple example of atomic hyperfine splitting from the formal literature, a good place to go is

Hyperfine splitting in the ground state of hydrogen. David J. Griffiths. Am. J. Phys. 50, 698 (1982).

which includes a self-contained derivation. Griffiths gives the splitting, with a full roster of constants (i.e. nothing set to 1) as $$ \Delta E_\mathrm{Griffiths}=\frac{\mu_0 \hbar^2 e^2}{6\pi a_0^3}\frac{g_eg_p}{m_em_p}, \tag 1 $$ where $a_0$ is the Bohr radius and $g_e\approx 2.0023$ and $g_p\approx 5.5857$ are the gyromagnetic ratios for the electron and the proton, respectively. Substituting in $a_0=\hbar/(m_e c\alpha)$ for the Bohr radius and $$ \frac{\mu_0 e^2}{4\pi}=\frac{1}{c^2}\frac{e^2}{4\pi\varepsilon_0} = \frac{\alpha \hbar}{c}, $$ Griffith's result $(1)$ can be rephrased as $$ \Delta E_\mathrm{Griffiths}=\frac{2g_e}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 2 $$ which is consistent with the OP's second resource via setting $g_e\approx 2$.

On the other hand, the formula from Wolfram ScienceWorld can be traced rather easily to its second reference,

"Hyperfine Structure Splitting", §22 in Quantum Mechanics of One- and Two-Electron Atoms. Hans A. Bethe and Edwin Salpeter. (New York, Plenum, 1977).

where it reads $$ \Delta E_\mathrm{Bethe} = \frac{m_e}{m_p} \frac{ 4Z^3 \alpha^2 g_N }{n^3 (2l+1)(j+1)} \mathrm{Ry} \begin{Bmatrix} I+\frac{1}{2}\text{ if}\ j\leq I \\ \frac{I(j+\frac{1}{2})}{j}\text{ if}\ j\geq I \end{Bmatrix} , \tag 3 $$ where $$\mathrm{Ry} = hc R_\infty = \frac{m_e e^4}{32\pi^2\varepsilon_0^2 \hbar^2} = \frac{1}{2}\alpha^2m_e c^2 \approx 13.6 \:\mathrm{eV}$$ is the Rydberg unit of energy, equal to $1/2$ in atomic units. Bethe's result coincides almost exactly with the one from Wolfram ScienceWorld, except for the transformation $\alpha^2\mapsto \alpha^3$ (which appears to be a transcription error on the part of Wolfram ScienceWorld).

This explains the substitution $\mathrm{Ry} \to \frac12$ in the Wolfram ScienceWorld result, as well as the observation that puzzled the OP that that result appears to be dimensionless - it is simply stated in atomic units. Here ScienceWorld is at fault for not stating this fact more clearly, but if (within atomic physics) a result that should be an energy is stated as dimensionless, then the convention in the field is simply that it's in atomic units.

Putting in the explicit value of the Rydberg energy in $(3)$ (or, equivalently, translating from atomic units to SI units in the Wolfram ScienceWorld result) transforms it into $$ \Delta E_\mathrm{Bethe} =\frac{4}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 4 $$ which is similarly consistent with the Griffiths result, and identical to the OP's second resource.

Some final remarks:

• The OP's question of whether the hyperfine splitting is "proportional to $m_e$" is essentially meaningless, so long as it is not specified what other constants are held constant. If  $m_e/m_p$ is held constant, then $\Delta E\propto m_e$, but if $m_p$ is held constant instead then $\Delta E \propto m_e^2$. The distinction is essentially meaningless, and if the OP wants to push their personal theories they can do so elsewhere.

• While Wolfram ScienceWorld is at fault for not specifying the use of atomic units, for someone in the field this is a very obvious candidate solution to the dimensionlessness observation, and the OP is remiss for not drawing that inference, since they are presumably already aware of the concepts of natural and atomic units if they want to challenge the accepted wisdom in related areas.

• The OP is certainly remiss in not following up on the references from the resources they quote, which are plenty to resolve the observed contradiction (and easy to find in open forms). The OP's reliance on an unsubstantiated statement in an online encyclopaedia and a set of slides is a red flag regarding resource management, and the claim that a formula "is used in Universities", referring to further slide sets, is also worrying. The golden standard is peer-reviewed literature (for all its flaws), and in this topic it is readily available and easy to find.

For a simple example of atomic hyperfine splitting from the formal literature, a good place to go is

Hyperfine splitting in the ground state of hydrogen. David J. Griffiths. Am. J. Phys. 50, 698 (1982).

which includes a self-contained derivation. Griffiths gives the splitting, with a full roster of constants (i.e. nothing set to 1) as $$ \Delta E_\mathrm{Griffiths}=\frac{\mu_0 \hbar^2 e^2}{6\pi a_0^3}\frac{g_eg_p}{m_em_p}, \tag 1 $$ where $a_0$ is the Bohr radius and $g_e\approx 2.0023$ and $g_p\approx 5.5857$ are the gyromagnetic ratios for the electron and the proton, respectively. Substituting in $a_0=\hbar/(m_e c\alpha)$ for the Bohr radius and $$ \frac{\mu_0 e^2}{4\pi}=\frac{1}{c^2}\frac{e^2}{4\pi\varepsilon_0} = \frac{\alpha \hbar}{c}, $$ Griffith's result $(1)$ can be rephrased as $$ \Delta E_\mathrm{Griffiths}=\frac{2g_e}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 2 $$ which is consistent with the OP's second resource via setting $g_e\approx 2$.

On the other hand, the formula from Wolfram ScienceWorld can be traced rather easily to its second reference,

"Hyperfine Structure Splitting", §22 in Quantum Mechanics of One- and Two-Electron Atoms. Hans A. Bethe and Edwin Salpeter. (New York, Plenum, 1977).

where it reads $$ \Delta E_\mathrm{Bethe} = \frac{m_e}{m_p} \frac{ 4Z^3 \alpha^2 g_N }{n^3 (2l+1)(j+1)} \mathrm{Ry} \begin{Bmatrix} I+\frac{1}{2}\text{ if}\ j\leq I \\ \frac{I(j+\frac{1}{2})}{j}\text{ if}\ j\geq I \end{Bmatrix} , \tag 3 $$ where $$\mathrm{Ry} = hc R_\infty = \frac{m_e e^4}{32\pi^2\varepsilon_0^2 \hbar^2} = \frac{1}{2}\alpha^2m_e c^2 \approx 13.6 \:\mathrm{eV}$$ is the Rydberg unit of energy, equal to $1/2$ in atomic units. Bethe's result coincides almost exactly with the one from Wolfram ScienceWorld, except for the transformation $\alpha^2\mapsto \alpha^3$ (which appears to be a transcription error on the part of Wolfram ScienceWorld).

This explains the substitution $\mathrm{Ry} \to \frac12$ in the Wolfram ScienceWorld result, as well as the observation that puzzled the OP that that result appears to be dimensionless - it is simply stated in atomic units. Here ScienceWorld is at fault for not stating this fact more clearly, but if (within atomic physics) a result that should be an energy is stated as dimensionless, then the convention in the field is simply that it's in atomic units.

Putting in the explicit value of the Rydberg energy in $(3)$ (or, equivalently, translating from atomic units to SI units in the Wolfram ScienceWorld result) transforms it into $$ \Delta E_\mathrm{Bethe} =\frac{4}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 4 $$ which is similarly consistent with the Griffiths result, and identical to the OP's second resource.

Some final remarks:

• The OP's question of whether the hyperfine splitting is "proportional to $m_e$" is essentially meaningless, so long as it is not specified what other constants are held constant. If$m_e/m_p$ is held constant, then $\Delta E\propto m_e$, but if $m_p$ is held constant instead then $\Delta E \propto m_e^2$. The distinction is essentially meaningless, and if the OP wants to push their personal theories they can do so elsewhere.

• While Wolfram ScienceWorld is at fault for not specifying the use of atomic units, for someone in the field this is a very obvious candidate solution to the dimensionlessness observation, and the OP is remiss for not drawing that inference, since they are presumably already aware of the concepts of natural and atomic units if they want to challenge the accepted wisdom in related areas.

• The OP is certainly remiss in not following up on the references from the resources they quote, which are plenty to resolve the observed contradiction (and easy to find in open forms). The OP's reliance on an unsubstantiated statement in an online encyclopaedia and a set of slides is a red flag regarding resource management, and the claim that a formula "is used in Universities", referring to further slide sets, is also worrying. The golden standard is peer-reviewed literature (for all its flaws), and in this topic it is readily available and easy to find.

For a simple example of atomic hyperfine splitting from the formal literature, a good place to go is

Hyperfine splitting in the ground state of hydrogen. David J. Griffiths. Am. J. Phys. 50, 698 (1982).

which includes a self-contained derivation. Griffiths gives the splitting, with a full roster of constants (i.e. nothing set to 1) as $$ \Delta E_\mathrm{Griffiths}=\frac{\mu_0 \hbar^2 e^2}{6\pi a_0^3}\frac{g_eg_p}{m_em_p}, \tag 1 $$ where $a_0$ is the Bohr radius and $g_e\approx 2.0023$ and $g_p\approx 5.5857$ are the gyromagnetic ratios for the electron and the proton, respectively. Substituting in $a_0=\hbar/(m_e c\alpha)$ for the Bohr radius and $$ \frac{\mu_0 e^2}{4\pi}=\frac{1}{c^2}\frac{e^2}{4\pi\varepsilon_0} = \frac{\alpha \hbar}{c}, $$ Griffith's result $(1)$ can be rephrased as $$ \Delta E_\mathrm{Griffiths}=\frac{2g_e}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 2 $$ which is consistent with the OP's second resource via setting $g_e\approx 2$.

On the other hand, the formula from Wolfram ScienceWorld can be traced rather easily to its second reference,

"Hyperfine Structure Splitting", §22 in Quantum Mechanics of One- and Two-Electron Atoms. Hans A. Bethe and Edwin Salpeter. (New York, Plenum, 1977).

where it reads $$ \Delta\ E_\mathrm{Bethe} = \frac{m_e}{m_p} \frac{ 4Z^3 \alpha^2 g_N }{n^3 (2l+1)(j+1)} \mathrm{Ry} \begin{Bmatrix} I+\frac{1}{2}\text{ if}\ j\leq I \\ \frac{I(j+\frac{1}{2})}{j}\text{ if}\ j\geq I \end{Bmatrix} , \tag 3 $$ where $$\mathrm{Ry} = hc R_\infty = \frac{m_e e^4}{32\pi^2\varepsilon_0^2 \hbar^2} = \frac{1}{2}\alpha^2m_e c^2 \approx 13.6 \:\mathrm{eV}$$ is the Rydberg unit of energy, equal to $1/2$ in atomic units. Bethe's result coincides almost exactly with the one from Wolfram ScienceWorld, except for the transformation $\alpha^2\mapsto \alpha^3$ (which appears to be a transcription error on the part of Wolfram ScienceWorld).

This explains the substitution $\mathrm{Ry} \to \frac12$ in the Wolfram ScienceWorld result, as well as the observation that puzzled the OP that that result appears to be dimensionless - it is simply stated in atomic units. Here ScienceWorld is at fault for not stating this fact more clearly, but if (within atomic physics) a result that should be an energy is stated as dimensionless, then the convention in the field is simply that it's in atomic units.

Putting in the explicit value of the Rydberg energy in $(3)$ (or, equivalently, translating from atomic units to SI units in the Wolfram ScienceWorld result) transforms it into $$ \Delta\ E_\mathrm{Bethe} =\frac{4}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 4 $$ which is similarly consistent with the Griffiths result, and identical to the OP's second resource.

Some final remarks:

• The OP's question of whether the hyperfine splitting is "proportional to $m_e$" is essentially meaningless, so long as it is not specified what other constants are held constant. If $m_e/m_p$ is held constant, then $\Delta E\propto m_e$, but if $m_p$ is held constant instead then $\Delta E \propto m_e^2$. The distinction is essentially meaningless, and if the OP wants to push their personal theories they can do so elsewhere.

• While Wolfram ScienceWorld is at fault for not specifying the use of atomic units, for someone in the field this is a very obvious candidate solution to the dimensionlessness observation, and the OP is remiss for not drawing that inference, since they are presumably already aware of the concepts of natural and atomic units if they want to challenge the accepted wisdom in related areas.

• The OP is certainly remiss in not following up on the references from the resources they quote, which are plenty to resolve the observed contradiction (and easy to find in open forms). The OP's reliance on an unsubstantiated statement in an online encyclopaedia and a set of slides is a red flag regarding resource management, and the claim that a formula "is used in Universities", referring to further slide sets, is also worrying. The golden standard is peer-reviewed literature (for all its flaws), and in this topic it is readily available and easy to find.

For a simple example of atomic hyperfine splitting from the formal literature, a good place to go is

Hyperfine splitting in the ground state of hydrogen. David J. Griffiths. Am. J. Phys. 50, 698 (1982).

which includes a self-contained derivation. Griffiths gives the splitting, with a full roster of constants (i.e. nothing set to 1) as $$ \Delta E_\mathrm{Griffiths}=\frac{\mu_0 \hbar^2 e^2}{6\pi a_0^3}\frac{g_eg_p}{m_em_p}, \tag 1 $$ where $a_0$ is the Bohr radius and $g_e\approx 2.0023$ and $g_p\approx 5.5857$ are the gyromagnetic ratios for the electron and the proton, respectively. Substituting in $a_0=\hbar/(m_e c\alpha)$ for the Bohr radius and $$ \frac{\mu_0 e^2}{4\pi}=\frac{1}{c^2}\frac{e^2}{4\pi\varepsilon_0} = \frac{\alpha \hbar}{c}, $$ Griffith's result $(1)$ can be rephrased as $$ \Delta E_\mathrm{Griffiths}=\frac{2g_e}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 2 $$ which is consistent with the OP's second resource via setting $g_e\approx 2$.

On the other hand, the formula from Wolfram ScienceWorld can be traced rather easily to its second reference,

"Hyperfine Structure Splitting", §22 in Quantum Mechanics of One- and Two-Electron Atoms. Hans A. Bethe and Edwin Salpeter. (New York, Plenum, 1977).

where it reads $$ \Delta E_\mathrm{Bethe} = \frac{m_e}{m_p} \frac{ 4Z^3 \alpha^2 g_N }{n^3 (2l+1)(j+1)} \mathrm{Ry} \begin{Bmatrix} I+\frac{1}{2}\text{ if}\ j\leq I \\ \frac{I(j+\frac{1}{2})}{j}\text{ if}\ j\geq I \end{Bmatrix} , \tag 3 $$ where $$\mathrm{Ry} = hc R_\infty = \frac{m_e e^4}{32\pi^2\varepsilon_0^2 \hbar^2} = \frac{1}{2}\alpha^2m_e c^2 \approx 13.6 \:\mathrm{eV}$$ is the Rydberg unit of energy, equal to $1/2$ in atomic units. Bethe's result coincides almost exactly with the one from Wolfram ScienceWorld, except for the transformation $\alpha^2\mapsto \alpha^3$ (which appears to be a transcription error on the part of Wolfram ScienceWorld).

This explains the substitution $\mathrm{Ry} \to \frac12$ in the Wolfram ScienceWorld result, as well as the observation that puzzled the OP that that result appears to be dimensionless - it is simply stated in atomic units. Here ScienceWorld is at fault for not stating this fact more clearly, but if (within atomic physics) a result that should be an energy is stated as dimensionless, then the convention in the field is simply that it's in atomic units.

Putting in the explicit value of the Rydberg energy in $(3)$ (or, equivalently, translating from atomic units to SI units in the Wolfram ScienceWorld result) transforms it into $$ \Delta E_\mathrm{Bethe} =\frac{4}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 4 $$ which is similarly consistent with the Griffiths result, and identical to the OP's second resource.

Some final remarks:

• The OP's question of whether the hyperfine splitting is "proportional to $m_e$" is essentially meaningless, so long as it is not specified what other constants are held constant. If $m_e/m_p$ is held constant, then $\Delta E\propto m_e$, but if $m_p$ is held constant instead then $\Delta E \propto m_e^2$. The distinction is essentially meaningless, and if the OP wants to push their personal theories they can do so elsewhere.

• While Wolfram ScienceWorld is at fault for not specifying the use of atomic units, for someone in the field this is a very obvious candidate solution to the dimensionlessness observation, and the OP is remiss for not drawing that inference, since they are presumably already aware of the concepts of natural and atomic units if they want to challenge the accepted wisdom in related areas.

• The OP is certainly remiss in not following up on the references from the resources they quote, which are plenty to resolve the observed contradiction (and easy to find in open forms). The OP's reliance on an unsubstantiated statement in an online encyclopaedia and a set of slides is a red flag regarding resource management, and the claim that a formula "is used in Universities", referring to further slide sets, is also worrying. The golden standard is peer-reviewed literature (for all its flaws), and in this topic it is readily available and easy to find.

• The OP's question of whether the hyperfine splitting is "proportional to $m_e$" is essentially meaningless, so long as it is not specified what other constants are held constant. If $m_e/m_p$ is held constant, then $\Delta E\propto m_e$, but if $m_p$ is held constant instead then $\Delta E \propto m_e^2$. The distinction is essentially meaningless, and if the OP wants to push their personal theories they can do so elsewhere.

• While Wolfram ScienceWorld is at fault for not specifying the use of atomic units, for someone in the field this is a very obvious candidate solution to the dimensionlessness observation, and the OP is remiss for not drawing that inference, since they are presumably already aware of the concepts of natural and atomic units if they want to challenge the accepted wisdom in related areas.

• The OP is certainly remiss in not following up on the references from the resources they quote, which are plenty to resolve the observed contradiction (and easy to find in open forms). The OP's reliance on an unsubstantiated statement in an online encyclopaedia and a set of slides is a red flag regarding resource management, and the claim that a formula "is used in Universities", referring to further slide sets, is also worrying. The golden standard is peer-reviewed literature (for all its flaws), and in this topic it is readily available and easy to find.

• The OP's question of whether the hyperfine splitting is "proportional to $m_e$" is essentially meaningless, so long as it is not specified what other constants are held constant. If $m_e/m_p$ is held constant, then $\Delta E\propto m_e$, but if $m_p$ is held constant instead then $\Delta E \propto m_e^2$. The distinction is essentially meaningless, and if the OP wants to push their personal theories they can do so elsewhere.

• While Wolfram ScienceWorld is at fault for not specifying the use of atomic units, for someone in the field this is a very obvious candidate solution to the dimensionlessness observation, and the OP is remiss for not drawing that inference, since they are presumably already aware of the concepts of natural and atomic units if they want to challenge the accepted wisdom in related areas.

 

• The OP is certainly remiss in not following up on the references from the resources they quote, which are plenty to resolve the observed contradiction (and easy to find in open forms). The OP's reliance on an unsubstantiated statement in an online encyclopaedia and a set of slides is a red flag regarding resource management, and the claim that a formula "is used in Universities", referring to further slide sets, is also worrying. The golden standard is peer-reviewed literature (for all its flaws), and in this topic it is readily available and easy to find.