link steady; rails on (GEM armed, mirror outward).
You’re mixing units. Fix that and the algebra is trivial.
Pick a convention
Use natural units with c=ℏ=1c=\hbar=1c=ℏ=1. Then energies ↔ angular frequencies:
E [GeV]=ℏ ω⇒1 s−1=6.582 119 57×10−25 GeV.E\,[\text{GeV}] = \hbar\,\omega \quad\Rightarrow\quad 1~\text{s}^{-1} = 6.582\,119\,57\times10^{-25}~\text{GeV}.E[GeV]=ℏω⇒1 s−1=6.58211957×10−25 GeV.
(If you use ordinary frequency ν\nuν in Hz, E=hνE=h\nuE=hν with h=2πℏ=4.135 667 696×10−24 GeV\cdotpsh=2\pi\hbar=4.135\,667\,696\times10^{-24}\,\text{GeV·s}h=2πℏ=4.135667696×10−24GeV\cdotps.)
Convert
Given H=1014H=10^{14}H=1014 GeV:
- as angular frequency: ωH=H/ℏ≈1.519×1038 s−1\omega_H = H/\hbar \approx 1.519\times10^{38}\ \text{s}^{-1}ωH=H/ℏ≈1.519×1038 s−1,
- as frequency: νH=H/h≈2.418×1037 Hz\nu_H = H/h \approx 2.418\times10^{37}\ \text{Hz}νH=H/h≈2.418×1037 Hz,
- natural-units time scale: tH=ℏ/H≈6.58×10−39 st_H=\hbar/H \approx 6.58\times10^{-39}\ \text{s}tH=ℏ/H≈6.58×10−39 s.
Your formula
p=k2H2p=\frac{k^2}{H^2}p=H2k2
is fine
only if kkk and HHH are in the same units. Two clean ways:
A) Keep
If kkk is an ordinary frequency,
Ek=h k=(4.1357×10−24 GeV\cdotps) k (s−1),E_k = h\,k = (4.1357\times10^{-24}\ \text{GeV·s})\;k\ (\text{s}^{-1}),Ek=hk=(4.1357×10−24 GeV\cdotps)k (s−1),
then
p=(EkH)2=(4.1357×10−24 k1014)2.p=\left(\frac{E_k}{H}\right)^2= \left(\frac{4.1357\times10^{-24}\,k}{10^{14}}\right)^2.p=(HEk)2=(10144.1357×10−24k)2.
Example: k=1021k=10^{21}k=1021 Hz ⇒ Ek≃4.14×10−3E_k\simeq 4.14\times10^{-3}Ek≃4.14×10−3 GeV ⇒
p≃(4.14×10−17)2≈1.7×10−33p\simeq (4.14\times10^{-17})^2 \approx 1.7\times10^{-33}p≃(4.14×10−17)2≈1.7×10−33.
B) Convert
Use νH=H/h\nu_H=H/hνH=H/h above, then
p=(kνH)2=(k2.418×1037 Hz)2.p=\left(\frac{k}{\nu_H}\right)^2=\left(\frac{k}{2.418\times10^{37}\ \text{Hz}}\right)^2.p=(νHk)2=(2.418×1037 Hzk)2.
Both give the same number.
What went wrong in your attempt
- You treated 1 GeV∼1024 Hz1~\text{GeV}\sim10^{24}\,\text{Hz}1 GeV∼1024Hz and multiplied by 10−21 s10^{-21}\,\text{s}10−21s to get 10310^3103. That mixes frequency and time and ignores the 2π2\pi2π ( hhh vs ℏ\hbarℏ ) and the actual constants. The right conversion is via hhh or ℏ\hbarℏ as shown; the result is tiny, not 10310^3103.
Can you “convert
Yes—do method
B: convert HHH to νH\nu_HνH using hhh, then use kkk in Hz. Or convert kkk to GeV (method
A) and leave HHH in GeV. What you
can’t do is mix Hz and GeV in the same fraction without converting one side.
If you tell me your exact kkk value (Hz or rad/s), I’ll plug it and give you the numeric ppp in one line.
