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Question #43634

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is trapped in a one-dimensional non-linear potential well given by $U(x) = \frac{1}{4} k x^4$ , where $k$ is a positive constant. Classically, the lowest energy state is at $x=0$ , where the restoring force is zero. However, quantum mechanically, the particle's position spread $a$ and momentum spread $p$ are constrained by the simplified Heisenberg uncertainty relation $a p = \hbar$ . By approximating the particle's total energy as $E \approx \frac{p^2}{2m} + \frac{1}{4} k a^4$ , determine the effective magnitude of the restoring force $F_{\text{eff}} = \left| \frac{dU}{da} \right|$ when the particle is in its state of minimum total energy.

Options

A.
$\hbar \sqrt{\frac{k}{m}}$
B.
$0$
C.
$\hbar \left( \frac{k^2}{m} \right)^{1/3}$
D.
$\frac{\hbar}{2} \sqrt{\frac{k}{m}}$

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

Heisenberg's Limitation|Potential Gradient Direction|Work against Conservative Force|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from classical point-mass equilibrium to quantum variational energy minimization.

The primary trap is the classical assumption: students often incorrectly assume that the minimum energy state must correspond to the minimum of the potential well ( $x=0$ ), leading them to choose a restoring force of $0$ .

However, the Heisenberg Uncertainty Principle dictates that localizing the particle at $x=0$ requires infinite kinetic energy.

Thus, the true ground state is a broadened wave-packet where the 'effective' potential gradient (restoring force) is strictly non-zero.

By forcing the calculation of $F_{\text{eff}}$ , the problem evaluates whether the student correctly applies the work-energy trade-off between quantum kinetic confinement and classical potential energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Heisenberg's Limitation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free