Discussion
For a simple harmonic oscillator the probability of finding the particle, if the measurement is made at random time, is inversely proportional to the speed. If the amplitude of oscillation is A, the probability of finding the particle is :
Answer: A
The probability is maximum close to ± A.
The time spent in a small interval \( dx \) is \( dx/|v| \), and for SHM the speed is \( v = \omega\sqrt{A^{2} - x^{2}} \). So
\[ P(x)\,dx = \frac{dx}{\pi\sqrt{A^{2} - x^{2}}} \]
which is smallest at \( x = 0 \), where the particle moves fastest, and rises steeply near the turning points \( \pm A \), where it momentarily stops.
So the probability is a minimum, not a maximum, at zero; it is not constant over [−A, A]; and nothing special happens at ±A/2.
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