A hydrogen atom of mass m , initially at rest in an excited state with principal quantum number n , transitions to the ground state. It emits a photon and recoils with a speed v . If R is the Rydberg constant and h is Planck's constant, the value of n is :

Step-by-step solution

By the principle of conservation of linear momentum, the recoil momentum of the hydrogen atom must be equal in magnitude to the momentum of the emitted photon. p_ atom = p_ photon mv = h 1 = mv h Using the Rydberg formula for the hydrogen atom ( Z=1 ) transitioning from state n to the ground state ( n_f = 1 ): 1 = R ( 1 1^2 - 1 n^2 ) Equating the two expressions for 1 : mv h = R (1 - 1 n^2 ) mv hR = 1 - 1 n^2 1 n^2 = 1 - mv hR = hR - mv hR Taking the reciprocal and the square root: n = hR hR - mv Answer: hR hR - mv