Recoil Momentum of Atoms emitting Photons: Applying conservation of linear momentum to calculate the recoil velocity of an excited hydrogen atom when it transitions to a lower state and emits a photon.

Recoil Momentum of Atoms Emitting Photons

📘 Concept & Complete Theory

When an excited atom changes from a higher energy state to a lower energy state, it emits a photon. Since a photon possesses momentum, the atom receives an equal and opposite recoil momentum.

Excited Hydrogen Atom

H

Photon → p = E/c

Atom Recoil ←

Core Concept:
Initial momentum of atom + system = 0.
After emission:
Atom recoil momentum + photon momentum = 0.

Photon Momentum

p_γ = E_γ/c = hν/c = h/λ

Conservation of Momentum

p_atom + p_photon = 0

Therefore:

|p_atom| = |p_photon|

Recoil Velocity

Mv = h/λ

Therefore:

v = h/(Mλ)

Using Photon Energy

v = E_γ/(Mc)

Recoil Kinetic Energy

K_R = p²/(2M) = E_γ²/(2Mc²)

Hydrogen Atom

E_γ = 13.6 [1/n_f² − 1/n_i²] eV

JEE/NEET Shortcut:

Step 1 → Find transition energy.
Step 2 → Find photon momentum using p = E/c.
Step 3 → Atomic recoil momentum = photon momentum.
Step 4 → Use v = p/M.

Quantity	Formula
Photon Energy	E = hν = hc/λ
Photon Momentum	p = E/c = h/λ
Recoil Momentum	pR = −pγ
Recoil Velocity	v = h/(Mλ)
Recoil Energy	K = p²/(2M)

🧮 Fully Solved Example

Question:
A hydrogen atom initially at rest undergoes transition from n = 2 to n = 1. Calculate the recoil velocity.

Hydrogen transition energy:

E = 13.6(1 − 1/4)
E = 10.2 eV

E = 10.2 × 1.602×10⁻¹⁹ J
E ≈ 1.634×10⁻¹⁸ J

Photon momentum:
p = E/c
p ≈ 5.45×10⁻²⁷ kg m/s

For atom:
Mv = p
v = p/M
v ≈ 3.26 m/s

Final Answer ≈ 3.26 m/s opposite to photon.

🎯 30 JEE / NEET / Competitive MCQs

⚡ 30 Assertion–Reason Questions

🧮 30 Numerical Practice Questions

⭐ 1 Mark Questions — 30

⭐ 2 Marks Questions — 30

⭐ 3 Marks Questions — 30

⭐ 4 Marks Questions — 30

⭐ 5 Marks Questions — 30

⭐ 6 Marks Questions — 30

🚀 Final Revision Formula Sheet

p_γ = E/c = hν/c = h/λ

p_atom = −p_γ

v = h/(Mλ)

v = E/(Mc)

K_R = E²/(2Mc²)

E_γ = 13.6[1/n_f² − 1/n_i²] eV

Remember:

Photon → momentum forward
Atom → recoil momentum backward
Total momentum → conserved
Shorter wavelength → larger recoil
Higher photon energy → larger recoil
Heavier atom → smaller recoil