When a photon is emitted from an atom, the atom recoils. The kinetic energy of recoil and the energy of the photon come from the difference in energies between the states involved in the transition. Suppose, a hydrogen atom changes its state from *n* = 3 to *n* = 2. Calculate the fractional change in the wavelength of light emitted, due to the recoil.

#### Solution

Difference in energy in the transition from *n* = 3 to *n* = 2 is 1.89 eV ( = *E*).

If all this energy is used up in emitting a photon (i.e. recoil energy is zero).

Then,

`E = (hc)/lamda`

`rArr lamda = (hc)/E` ..........(i)

If difference of energy is used up in emitting a photon and recoil of atom, then let *E _{R}* be the recoil energy of atom.

`E = (hc)/lamda + E_R`

`rArr lamda ' = (hc)/(E - E_R)` ............(ii)

Fractional change in the wavelength is given as,

`(Delta lamda)/lamda = (lamda'-lamda)/lamda`

`rArr (Deltalamda)/lamda = 1/lamda((hc)/(E-E_R ) - (hc)/E)`

`rArr (Delta lamda)/(lamda)=E/(hc) (hcE_R)/(E(E-E_R)) (therefore lamda = (hc)/E)`

`rArr (Deltalamda)/lamda = ((E_R)/(E- E_R))`