Question

There are sparse quantities of atomic hydrogen in interstellar space, the details of whidh are important for studying star formation, galactic dynamics, and cosmology. Use ΔΕιι,-Ca4m,c2 nn to estimate the wavelength of radiation associated with transitions to estimate the wavelength of radiation asociated with transitions am c HFS e1 mp between the singlet and triplet configurations of the 1s state of hydrogen. This transition is the most important signal used in the field of radio astronomy.

Answer 1

The given expression can be written as ,

   $\Delta E_{IFS}=\frac{g_p h^4}{12 \pi^2m_e^2c^2 a_0^2}$

where g_p =5.59 is the gyro-magnetic ratio of proton and a₀ is the Bohr radius. Inserting numerical values of the constants in the above expression, we obtain

$\Delta E_{IFS}=\frac{5.59 \times (6.636 \times10^{-34} )^4}{12 \times (3.14)^2 \times (9.11 \times 10^{-31})^2 \times (3 \times 10^8)^2 \times (0.529 \times 10^{-10})^2}=9.4 \times 10^{-25} J$

Frequency of the emitted photon in this transition is

$\nu=\frac{\Delta E_{IFS}}{h}=\frac{9.4 \times 10^{-25}}{6.626 \times 10^{-34}}=1420 MHz$

Hence wavelength of the emitted radiation is

$\lambda=\frac{c}{\nu}=\frac{3 \times 10^8}{1420 \times 10^6}=21 cm$