\subsection{The $1s$-$2s$ transition frequency}


\begin{figure
\centering 
 \includegraphics[width=1.0 \linewidth,angle=0]
   {omega-Eides-2002-2018.eps}
\caption{\footnotesize 
  The $1s$-$2s$ hydrogen transition frequency.  Red point:
  $f_{1s,2s}^{\mathrm{(exp)}} = 2466061413187.035 (10)$~kHz \cite{Parthey},
  Eq.~(\ref{eq:omega_exp}).
  Purple point,
  $f_{1s,2s}^{\mathrm{(Eides,Sick)}} = ( 2466061413141 \pm 47 )$~kHz,
  Eq.~(\ref{eq:f_1s2s-Eides-Sick}).
  Blue point,
  $f_{1s,2s}^{(\mathrm{Eides)}} = ( 2466061412396 \pm 21 )$~kHz,
  see Eq.~(\ref{eq:f_1s2s-Eides}).
  Azure point,
  $\tilde{f}_{1s,2s}^{(2002)} = ( 2466061413996 \pm 23 )$~kHz,
  see Eq.~(\ref{eq:f_1s2s-2002-34}). 
  Green point,
  $\tilde{f}_{1s,2s}^{(2018)} = ( 2466061413163.087 \pm 6.4 \big)$~kHz,
  see Eq.~(\ref{eq:f_1s2s-2018-34}).
  Magenta point,
  $f_{1s,2s}^{(2018,\mathrm{fit)}} = f_{1s,2s}^{\mathrm{(exp)}}$ with
  an uncertainty of 6.4 kHz.
  The discrepancies of the calculated frequencies from the measured
  frequency are as follows:
  $\Delta f_{1s,2s}^{\mathrm{(Eides,Sick)}} = -46$~kHz
  in Eq.~(\ref{eq:Delta-f_1s2s-Eides-Sick}),
  $\Delta f_{1s,2s}^{\mathrm{(Eides)}} = 2$~kHz
  in Eq.~(\ref{eq:Delta-f_1s2s-Eides}),
  $\Delta \tilde{f}_{1s,2s}^{(2002)} = 19.522$~kHz
  in Eq.~(\ref{eq:Delta-f_1s2s-2002-34}),
  $\Delta \tilde{f}_{1s,2s}^{(2018)} = -23.948$~kHz
  in Eq.~(\ref{eq:Delta-f_1s2s-2018-34}), and 
  $\Delta f_{1s,2s}^{(2018,\mathrm{fit)}} = 0$.
  }
\label{Fig-Delta-omega}
\end{figure}


\textit{Transition frequency using 2002 CODATA~\cite{CODATA-2002}}:
The values of the fundamental constants used in Ref.~\cite{CODATA-2002}
are given in Eq.~(\ref{subeqs:2002-CODATA}), and the Dirac frequency 
$f_{1s, 2s}^{\mathrm{(Dirac, 2002)}}$ is given in Eq.~(\ref{eq:f-Dirac-2002}).
Using the technique in Ref.~\cite{CODATA-2002} (see SM~\cite{supplementary}), 
we find the following QED radiative and FNS correction $f_{1s, 2s}^{\mathrm{(QED, FNS, 2002)}}$
to the $1s$-$2s$ transition frequency:
\begin{equation}
  \label{eq:f-QED-FNS-2002}
  f_{1s, 2s}^{\mathrm{(QED, FNS, 2002)}} =
  ( -7127027.610 \pm 16 ) ~ \text{kHz}.
\end{equation}
Using Eq.~(\ref{eq:energy-ns-Dirac-recoil}) in Eq.~(\ref{eq:f=f^Dirac+f^QED-FNS}), and the proton 
RMS charge radius in Eq.~(\ref{eq:proton-radius-num-2002}), we get the transition frequency 
$f_{1s, 2s}^{\mathrm{(2002)}}$,
\begin{equation}
  \label{eq:f_1s2s-2002}
  f_{1s,2s}^{(2002)} =
  ( 2466061413995.950 \pm 23 ) ~ \text{kHz},
\end{equation}
and the discrepancy $\Delta f_{1s, 2s}^{(2002)} = %
f_{1s, 2s}^{(2002)} - f_{1s, 2s}^{\mathrm{(exp)}}$,
\begin{equation}
  \label{eq:Delta-f_1s2s-2002}
  \Delta f_{1s,2s}^{(2002)} = 808.915 ~ \text{kHz} .
\end{equation}
$| \Delta f_{1s, 2s}^{(2002)} |$ is much larger than the uncertainty of $f_{1s, 2s}^{(2002)}$,
$\delta f_{1s, 2s}^{(2002)} = 23$~kHz, and much larger than the uncertainty of 
$f_{1s, 2s}^{\mathrm{(exp)}}$, $\delta f_{1s, 2s}^{\mathrm{(exp)}} = 0.010$~kHz.

Using Eq.~(\ref{eq:energy-ns-Dirac-recoil-34}) in Eq.~(\ref{eq:f=f^Dirac+f^QED-FNS}),
we get
\begin{equation}
  \label{eq:f_1s2s-2002-34}
  \tilde{f}_{1s,2s}^{(2002)} =
  ( 2466061413206.557 \pm 23 ) ~ \text{kHz},
\end{equation}
and the discrepancy $\Delta \tilde{f}_{1s, 2s}^{(2002)} = %
\tilde{f}_{1s, 2s}^{(2002)} - f_{1s, 2s}^{\mathrm{(exp)}}$ is
\begin{equation}
  \label{eq:Delta-f_1s2s-2002-34}
  \Delta \tilde{f}_{1s,2s}^{(2002)} = 19.522 ~ \text{kHz} .
\end{equation}
$| \Delta \tilde{f}_{1s, 2s}^{(2002)} |$ is smaller than the uncertainty of $f_{1s, 2s}^{(2002)}$,
$\delta f_{1s, 2s}^{(2002)} = 23$~kHz, but much larger than the uncertainty of 
$f_{1s, 2s}^{\mathrm{(exp)}}$, $\delta f_{1s, 2s}^{\mathrm{(exp)}} = 0.010$~kHz.

\textit{Transition frequency using 2018 CODATA~\cite{proton_radius}}:
The values of the Rydberg frequency $c R_{\infty}$, the electron to proton mass ratio $\mu_{ep}$,
the electron to muon mass ratio $\mu_{e \mu}$, the fine structure constant $\alpha$, 
the RMS proton radius $r_p$, and the reduced electron Compton wavelength $\lambdabar_C$, from Ref.~\cite{proton_radius} are:
\begin{subequations}   \label{subeqs:2018-CODATA}
\begin{eqnarray}
  c R_{\infty}^{(2018)} &=& 3.289 841 960 2508 (64) \times 10^{15} ~ \text{Hz},
  \label{eq:Ridberg-num-2018}
  \\
  \mu_{ep}^{(2018)} &=& 5.446 170 214 87 (33) \times 10^{-4},
  \label{eq:mu_e-num-2018}
  \\
  \mu_{e \mu}^{(2018)} &=& 4.836 331 69 (11) \times 10^{-3},
  \label{eq:m-_emu-num-2018}
  \\
  \alpha^{(2018)} &=& 7.297 352 5693 (11) \times 10^{-3},
  \label{eq:alpha-num-2018}
  \\
  r_{p}^{(2018)} &=& 0.8414 (19) \times 10^{-15} ~ \text{m},
  \label{eq:proton-radius-num-2018}
  \\
  \lambdabar_{C}^{(2018)} &=& 3.861 592 6796 (12) \times 10^{-13} \, \text{m}.
  \label{eq:Compton-num-2018}
\end{eqnarray}
\end{subequations}
Using Eq.~(\ref{eq:energy-ns-Dirac-recoil}) in Eq.~(\ref{eq:f=f^Dirac+f^QED-FNS}),
we get the Dirac frequency $f_{1s, 2s}^{\mathrm{(Dirac, 2018)}}$,
\begin{equation}    \label{eq:f-Dirac-2018}
  f^{\mathrm{(Dirac, 2018)}} = ( 2466068540913.7 \pm 4.8) ~ \text{kHz} .
\end{equation}
Using Eq.~(\ref{eq:energy-ns-Dirac-recoil-34}) in Eq.~(\ref{eq:f=f^Dirac+f^QED-FNS}), we get
\begin{equation}
  \label{eq:f-Dirac-2018-34}
  \tilde{f}^{\mathrm{(Dirac, 2018)}} =
  ( 2466068540152.921 \pm 4.8) ~ \text{kHz} .
\end{equation}
Using the technique in Ref.~\cite{proton_radius} (see the SM~\cite{supplementary}), 
we find the following QED radiative and FNS correction $f_{1s, 2s}^{\mathrm{(QED, FNS, 2018)}}$
to the $1s$-$2s$ transition frequency:
\begin{equation}
  \label{eq:f-QED-FNS-2018}
  f_{1s, 2s}^{\mathrm{(QED, FNS, 2018)}} =
  ( -7126679.6 \pm 4.4 ) ~ \text{kHz}.
\end{equation}
The hydrogen $1s$-$2s$ transition frequency $f_{1s, 2s}^{(2018)}$
for the proton RMS charge radius in Eq.~(\ref{eq:proton-radius-num-2018})
and the Dirac frequency in Eq.~(\ref{eq:f-Dirac-2018}) is
\begin{equation}
  \label{eq:f_1s2s-2018}
  f_{1s,2s}^{(2018)} =
  ( 2466061413923.866 \pm 6.4 ) ~ \text{kHz},
\end{equation}
and the discrepancy $\Delta f_{1s, 2s}^{(2002)} = 
f_{1s, 2s}^{(2018)} - f_{1s, 2s}^{\mathrm{(exp)}}$ is
\begin{equation}
  \label{eq:Delta-f_1s2s-2018}
  \Delta f_{1s,2s}^{(2018)} = 736.831 ~ \text{kHz} .
\end{equation}
$| \Delta f_{1s, 2s}^{(2018)} |$ is much larger than the uncertainty of $f_{1s, 2s}^{(2018)}$,
$\delta f_{1s, 2s}^{(2018)} = 6.4$~kHz, and much larger than the experimental uncertainty 
$\delta f_{1s, 2s}^{\mathrm{(exp)}} = 0.010$~kHz.

The hydrogen $1s$-$2s$ transition frequency $f_{1s, 2s}^{(2018)}$
using the proton RMS charge radius in Eq.~(\ref{eq:proton-radius-num-2018})
and the Dirac frequency in Eq.~(\ref{eq:f-Dirac-2018-34}) is
\begin{equation}
  \label{eq:f_1s2s-2018-34}
  \tilde{f}_{1s,2s}^{(2018)} =
  ( 2466061413187.035 \pm 6.4 ) ~ \text{kHz},
\end{equation}
and the discrepancy $\Delta \tilde{f}_{1s, 2s}^{(2002)} = 
f_{1s, 2s}^{(2018)} - f_{1s, 2s}^{\mathrm{(exp)}}$ is
\begin{equation}
  \label{eq:Delta-f_1s2s-2018-34}
  \Delta \tilde{f}_{1s,2s}^{(2018)} = -23.948 ~ \text{kHz} .
\end{equation}
$| \Delta \tilde{f}_{1s, 2s}^{(2018)} |$ is larger than the uncertainty of $\tilde{f}_{1s, 2s}^{(2018)}$,
$\delta \tilde{f}_{1s, 2s}^{(2018)} = 6.4$~kHz, and much larger than the experimental uncertainty 
$\delta f_{1s, 2s}^{\mathrm{(exp)}} = 0.010$~kHz.

\textit{Fitting $r_{p}$ to obtain optimal transition frequency}:
Next, since $r_{p}$ is the least well-known of the fundamental constants that enter
into the expression for $\tilde{f}_{1s, 2s}$, we fit $r_{p}$ so as to minimize 
$\Delta {\tilde f}_{1s,2s} \equiv {\tilde f}_{1s, 2s} - f_{1s, 2s}^{\mathrm{(exp)}}$.
We expand $\Delta \tilde{f}_{1s, 2s}$ in powers of
$\varrho_p \equiv (r_{p} - r_{p}^{(2018)})/\delta r_{p}^{(2018)}$, and get
\begin{equation}
  \Delta \tilde{f}_{1s, 2s} =
  \Delta \tilde{f}_{1s, 2s}^{(2018)} -
  \mathcal{B}_{1} \, \varrho_p -
  \mathcal{B}_{2} \, \varrho_{p}^{2} ,
\end{equation}
where $\mathcal{B}_{1} = 429.337 \, \delta f_{1s, 2s}^{\mathrm{(exp)}}$
and $\mathcal{B}_{2} = 0.484751 \, \delta f_{1s, 2s}^{\mathrm{(exp)}}$.
We can solve for the value of $r_p$ which gives $\Delta \tilde{f}_{1s, 2s} = 0$ and
we find that $r_{p}^{\mathrm{(fit)}} = 0.830734$~fm, so that $\Delta r_{p}^{\mathrm{(fit)}} \equiv
r_{p}^{\mathrm{(fit)}} - r_{p}^{(2018)} = -5.613 \, \delta r_{p}^{(2018)} = -0.01066$~fm,
where $\delta r_{p}^{(2018)}$ is the uncertainty of 
$r_{p}^{(2018)}$ in Eq.~(\ref{eq:proton-radius-num-2018}).

Solving the equations $\Delta \tilde{f}_{1s, 2s} = \delta f_{1s, 2s}^{\mathrm{(exp)}}$
and $\Delta \tilde{f}_{1s, 2s} = -\delta f_{1s, 2s}^{\mathrm{(exp)}}$ for $r_p$, we estimate
the uncertainty $\delta r_{p}^{\mathrm{(fit)}}$ due to the uncertainty
$\delta f_{1s, 2s}^{\mathrm{(exp)}}$ of $f_{1s, 2s}^{\mathrm{(exp)}}$, and we find
$\delta r_{p}^{\mathrm{(fit)}} = 0.002 \, \delta r_{p}^{(2018)}$.  Note that
$\delta r_{p}^{\mathrm{(fit)}}$ is much smaller than $\delta r_{p}^{(2018)}$.
Hence, we take $\delta r_{p}^{(2018)}$ as the uncertainty for $r_p$.

{\textit{Summary and Conclusion}}:  
We have calculated the $1s$-$2s$ quantum transition frequency of the hydrogen atom using QED 
radiative and finite nuclear size and mass corrections, and evaluated the uncertainty of the 
calculated result due to uncertainties in the fundamental constants using four methods. 
(1) The procedure outlined in Ref.~\cite{Eides-hydrogen-atom} yields $f_{1s,2s}^{\mathrm{(Eides)}} = ( 2466061412396 \pm 21 )$~kHz and its uncertainty 
$\delta f_{1s,2s}^{\mathrm{(Eides)}} = 2$~kHz. (2) The 2002 CODATA of
Ref.~\cite{CODATA-2002} yields $f_{1s,2s}^{\mathrm{(2002)}} = ( 2466061413996 \pm 23 )$~kHz and
its uncertainty $\delta f_{1s,2s}^{\mathrm{(2002)}}  = 19.522$~kHz. (3) The 2018 CODATA of 
Ref.~\cite{proton_radius} yields $f_{1s,2s}^{\mathrm{(2018)}}  = ( 2466061413163.087 \pm 6.4 \big)$~kHz and
its uncertainty $\delta f_{1s,2s}^{\mathrm{(2018)}} = -23.948$~kHz.  (4) using method (3) but
fitting the proton RMS charge radius to obtain 
$f_{1s,2s}^{(2018,\mathrm{fit)}} = f_{1s,2s}^{(\mathrm{exp)}}$,  
which has an uncertainty due to the uncertainty of the fundamental constants of $\delta f_{1s, 2s}^{(2018,\mathrm{fit})} = 6.4$~kHz.

To conclude, atomic hydrogen $1s$-$2s$ spectroscopy has been crucial for the development of quantum
mechanics. Due to hydrogen's simple structure, combined with the long lifetime of the 
$2s$ level and the Doppler suppression of the two-photon transition, it is crucial for
testing the accuracy of quantum mechanics.  Moreover, it will likely be crucial
for studying CPT symmetry using anti-hydrogen \cite{Ahmadi, Kostelecky, Bluhm, Rasmussen}.


A useful interaction with V.A. Yerokhin and K. Pachucki is gratefully acknowledged.



