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Question #43828

Difficulty: 8/10 | 🟢 Free

The Question

A stationary nucleus undergoes $\beta^{-}$ -decay, releasing a total energy $Q$ . Assume the daughter nucleus is infinitely massive (zero recoil energy) and the antineutrino is massless. In the ultra-relativistic limit where the electron's rest mass energy is negligible ( $Q \gg m_e c^2$ ), the probability density function for the electron's momentum $p$ is given by Fermi's theory as $P(p) = k p^2 (Q - pc)^2$ for $0 \le p \le Q/c$ , where $k$ is a normalization constant and $c$ is the speed of light. Which of the following represents the expectation value of the de Broglie wavelength $\langle \lambda \rangle$ of the emitted electron?

Options

A.
$\frac{5hc}{2Q}$
B.
$\frac{2hc}{Q}$
C.
$\frac{hc}{Q}$
D.
$\frac{3hc}{2Q}$

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

Energy Distribution in Beta Decay|De Broglie Wavelength Scaling|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from discrete decay kinematics (like alpha or gamma decay) to continuous spectrum kinematics (beta decay).

A common trap is assuming the electron carries away the full energy $Q$ as a fixed value, leading to $\lambda = hc/Q$ (Option C).

A more subtle, secondary trap is calculating the average momentum $\langle p \rangle = Q/2c$ and incorrectly assuming that the average wavelength is $\langle \lambda \rangle = h / \langle p \rangle = 2hc/Q$ (Option B).

The problem forces students to correctly apply the mathematical definition of an expectation value for an inversely proportional variable, integrating $\int (h/p) P(p) dp$ over the momentum space.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Energy Distribution in Beta Decay
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free