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Question #43798

Difficulty: 8/10 | 🟢 Free

The Question

A hydrogen-like ion of mass $M$ and atomic number $Z$ is initially at rest. It absorbs a photon and undergoes a transition from the ground state ( $n=1$ ) to the series limit ( $n=\infty$ ), liberating the electron with zero kinetic energy. Let $\lambda_0$ be the theoretical maximum wavelength of the photon required for this transition, assuming an infinitely massive nucleus. If $\lambda$ is the actual maximum wavelength required when accounting for the recoil of the nucleus, which of the following expressions gives the magnitude of the wavelength shift $|\Delta \lambda| = \lambda_0 - \lambda$ to the first order in the recoil correction? (Assume the internal atomic energy levels are unaffected by the finite mass of the nucleus, $h$ is Planck's constant, and $c$ is the speed of light).

Options

A.
$\frac{h}{2Mc}$
B.
$\frac{h}{Mc}$
C.
$\frac{h Z^2}{2Mc}$
D.
$\frac{h}{4Mc}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Spectral Series Limits|Atomic Recoil|Conservation of Linear Momentum

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume that the photon energy exactly equals the atomic transition energy ( $E_\gamma = \Delta E$ ), ignoring the momentum carried by the photon.

When forced to consider recoil, a common trap is to misapply the Doppler effect sequentially—assuming the atom absorbs the photon, recoils with velocity $v = \Delta E / Mc$ , and experiences a classical Doppler shift $\Delta \lambda = \lambda_0 (v/c) = h/Mc$ .

This sequential reasoning violates simultaneous conservation of energy and momentum.

By rigorously applying both conservation laws, the recoil kinetic energy $K = p^2/2M$ acts as an energy sink, requiring a higher-energy (shorter wavelength) incident photon.

The exact energy partition yields a wavelength shift of exactly half the Compton wavelength of the ion ( $\frac{h}{2Mc}$ ), exposing the flaw in the sequential Doppler approximation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Spectral Series Limits
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free