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Question #43246

Difficulty: 8/10 | 🟢 Free

The Question

A free, excited atom of mass $m$ is initially at rest at a distance $L$ from a double-slit apparatus with slit separation $d$ . It emits a single photon of wavelength $\lambda$ which passes through the slits and is detected on a screen at a distance $D$ from the slits. An experimenter measures the transverse recoil velocity $v_y$ (parallel to the slit separation) of the atom to an accuracy $\Delta v_y = \frac{h d}{2 m \lambda L}$ in an attempt to determine which slit the photon passed through ( $h$ is Planck's constant). Assuming the atom's transverse position and momentum satisfy the minimum uncertainty condition $\Delta y \Delta p_y = \frac{h}{4\pi}$ , what is the resulting minimum uncertainty in the position of the central interference maximum on the screen?

Options

A.
$\frac{\lambda D}{2\pi d}$
B.
$\frac{\lambda D}{\pi d}$
C.
$\frac{\lambda L}{2\pi d}$
D.
$\frac{\lambda D}{4\pi d}$

Check Your Answer

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

Young's Double Slit Geometry|Wave-Particle Duality Constraint|Recoil of Atom

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of quantum mechanics (Heisenberg's Uncertainty Principle) with classical wave optics (Young's Double Slit Experiment).

Students often treat the photon source as a classical, static point.

By introducing a measurement of the atom's recoil to obtain 'which-path' information, the problem forces the application of HUP to the macroscopic source.

The primary trap is failing to apply the geometric magnification factor ( $D/L$ ) that relates the source's spatial uncertainty to the smearing of the interference pattern on the screen, leading to Option C.

Another common mistake is misapplying the HUP constant as $\hbar$ instead of $\hbar/2$ , leading to Option B.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Young's Double Slit Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free