Chapter 39: Q.49 (page 1140)

The wave function of a particle is
$ψ (x) = \sqrt{\frac{b}{π (x^{2} + b^{2})}}$
where b is a positive constant. Find the probability that the particle is located in the interval -b $\leq$ x $\leq$ b

Short Answer

Expert verified

$\int_{- \infty}^{+ \infty} |ψ {(x)}^{2}| d x = 1$

the probability of finding the particle within the given interval $- b \leq x \leq b$ is 50%

Step by step solution

First check the normalization condition for the given wave function of the given particle within the limits -∞ to +∞

$\int_{- \infty}^{+ \infty} |ψ {(x)}^{2}| d x = \int_{- \infty}^{+ \infty} {(\sqrt{\frac{b}{π (x^{2} + b^{2})}})}^{2} d x = \int_{- \infty}^{+ \infty} \frac{b}{π (x^{2} + b^{2})} = \frac{b}{π} {(\frac{1}{b} \tan^{- 1} (\frac{x}{b}))}_{- \infty}^{\infty} = \frac{1}{π} ((\frac{π}{2}) - (\frac{- π}{2})) = 1$

The probability of finding the particle within the given interval -b≤x≤b is,

$\int_{- \infty}^{+ \infty} |ψ {(x)}^{2}| d x = \int_{- b}^{+ b} {(\sqrt{\frac{b}{π (x^{2} + b^{2})}})}^{2} d x = \int_{- b}^{+ b} \frac{b}{π (x^{2} + b^{2})} = \frac{b}{π} {(\frac{1}{b} \tan^{- 1} (\frac{x}{b}))}_{- b}^{b} = \frac{1}{π} ((\frac{π}{4}) - (\frac{- π}{4})) = \frac{1}{2} = \frac{1}{2} (100 %) = 50 %$

Therefore, the probability of finding the particle within the given intervalrole="math" localid="1649761340195" $- b \leq x \leq b$ is 50%

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The wave function of a particle is
$ψ (x) = \sqrt{\frac{b}{π (x^{2} + b^{2})}}$
where b is a positive constant. Find the probability that the particle is located in the interval -b $\leq$ x $\leq$ b

Short Answer

Step by step solution

First check the normalization condition for the given wave function of the given particle within the limits -∞ to +∞

The probability of finding the particle within the given interval -b≤x≤b is,

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The wave function of a particle isψx=bπx2+b2where b is a positive constant. Find the probability that the particle is located in the interval -b≤x ≤b

Short Answer

Step by step solution

First check the normalization condition for the given wave function of the given particle within the limits -∞ to +∞

The probability of finding the particle within the given interval -b≤x≤b is,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Torque and Rotational Motion

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Study anywhere. Anytime. Across all devices.

The wave function of a particle is
$ψ (x) = \sqrt{\frac{b}{π (x^{2} + b^{2})}}$
where b is a positive constant. Find the probability that the particle is located in the interval -b $\leq$ x $\leq$ b