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Question #43125

Difficulty: 8/10 | 🟢 Free

The Question

A quantum particle of mass $m$ and total energy $E = 100 \text{ eV}$ is projected from the origin $(0,0)$ in the $x-y$ plane. Its initial velocity vector makes an angle $\theta_0$ with the positive $y$ -axis. The particle moves through a region with a conservative potential $V(y)$ given by: $V(y) = \alpha y \quad \text{for } 0 \le y \le d$ $V(y) = V_1 \quad \text{for } y > d$ where $\alpha = 10 \text{ eV/m}$ , $d = 6.4 \text{ m}$ , and $V_1 = 36 \text{ eV}$ . Assuming the particle's trajectory can be described by the Eikonal approximation (where matter waves obey Fermat's principle), what is the maximum value of $\theta_0$ such that the particle successfully enters the region $y > d$ ?

Options

A.
$\sin^{-1}(0.60)$
B.
$\sin^{-1}(0.80)$
C.
$\sin^{-1}(0.64)$
D.
$\sin^{-1}(0.36)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Total Internal Reflection (TIR)|Conservative Force Field|Dot Product Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge geometric optics (Fermat's Principle/Snell's Law) with Hamiltonian mechanics in a non-homogeneous medium.

The primary trap exploits the assumption that the particle travels in a straight line to the interface.

Students commonly apply Snell's Law directly between the initial state and the final state ( $n_0 \sin\theta_0 = n_f \sin\theta_f$ ) to find the critical angle for transmission, yielding $\sin\theta_0 = 0.80$ .

However, they fail to account for the continuous gradient of the conservative potential field, which forces a curved trajectory.

If $\theta_0 > \sin^{-1}(0.60)$ , the particle's $y$ -velocity reaches zero, causing it to turn around (continuous total internal reflection) before it ever reaches the interface at $y = d$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Total Internal Reflection (TIR)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free