Question

shows the probability density for finding a particle at position x. a. Determine the value of the constant a, as defined in the figure. b. At what value of x are you most likely to find the particle? Explain. c. Within what range of positions centered on your answer to part b are you 75% certain of finding the particle? d. Interpret your answer to part c by drawing the probability density graph and shading the appropriate region.

Short answer

The value of constant a is$0.25c{m}^{-1}$

Step-by-step solution

To find the constant value of triangle

The area under the triangle of the graph $$\begin{gathered} {\left|y\left(x\right)\right|}^{2}versusxandequateitequalto \\ {\int }_{-\infty }^{+\infty }{\left|y\left(x\right)\right|}^{2}dx=1 \\ \left(\frac{1}{2}\right)\left(height\right)\left(base\right)=1 \\ \left(\frac{1}{2}\right)\left(4cm-\right)\left(-4cm\right)=1 \\ \left(\frac{1}{2}\right)\left(a\right)\left(8cm\right)=1 \\ a=\frac{1}{4cm} \\ =0.25cm \end{gathered}$$

The value of constant a is$0.25c{m}^{-1}$

(b)

The probability function determines the position of the particle at that particular instant. The probability of finding the particle is more at the origin. Therefore, the maximum probability of finding the particle is observed at x=0 cm.

(c)

The probability of finding the particle is 75 % in the region from x=-2cm to x=2 cm centre about the origin.

(d)

The probability of finding the particle for the region mentioned in part (c) is shown by the shaded area below.