Chapter 39: Q.34 (page 1138)

The probability density for finding a particle at position x is P1x2 = • a 11 - x2 -1 mm … x 6 0 mm b11 - x2 0 mm … x … 1 mm and zero elsewhere. a. You will learn in Chapter 40 that the wave function must be a continuous function. Assuming that to be the case, what can you conclude about the relationship between a and b? b. Determine values for a and b. c. Draw a graph of the probability density over the interval -2 mm … x … 2 mm. d. What is the probability that the particle will be found to the left of the origin?

Short Answer

Expert verified

The wave function of a particle confined between $x = - 4 m m a n d x = 4 m m$

Step by step solution

(a)The wave function of a particle confirmed between x=-4mm and x=4mm is shown in the figure below

Using the concept of similar triangles, we have

$\frac{c}{ψ (x)} = \frac{4}{x} ψ (x) = \frac{c}{4} x \int_{- \infty}^{+ \infty} {|ψ (x)|}^{2} d x = 1 \int_{- 4}^{+ 4} {|ψ (x)|}^{2} d x = 1 \int_{- 4}^{0} {|ψ (x)|}^{2} d x + \int_{0}^{4} {|ψ (x)|}^{2} d x = 1$

The statement particle has to land somewhere on the x-axis is expressed mathematically as

Now in the case

$\int_{- 4}^{0} {(\frac{C}{4} x)}^{2} d x + \int_{0}^{4} {(\frac{C}{4} x)}^{2} d x = 1 \frac{C^{2}}{16} \int_{- 4}^{0} x^{2} d x + \int_{0}^{4} \frac{C^{2}}{16} x^{2} d x = 1 \frac{C^{2}}{16} {[\frac{x^{3}}{3}]}_{- 4}^{0} + \frac{C^{2}}{16} {[\frac{x^{3}}{3}]}_{0}^{4} = 1 \frac{C^{2}}{16} (\frac{64}{3}) + \frac{C^{2}}{16} (\frac{64}{4 = 1}) = 1 \frac{4 C^{2}}{3} + \frac{4 C^{2}}{3} = 1 8 C^{2}$

(b)The probability density curve ψx2 of the particle is shown in figure below

(c)

(d)The probability of finding the particle in the interval -2.0mm ≤x≤2.0mm is

$P (- 2.0 m m \leq x \leq 2.0 m m) = \int_{- 2}^{2} {|ψ (x)|}^{2} d x = \int_{- 2}^{2} {(\frac{C}{4} x)}^{2} d x = \frac{C^{2}}{16} \int_{- 2}^{2} x^{2} d x = \frac{3}{128} {[\frac{x^{3}}{3}]}_{- 2}^{2} = (\frac{3}{128}) (\frac{16}{3}) = \frac{1}{8} = 0.125$

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Short Answer

Step by step solution

(a)The wave function of a particle confirmed between x=-4mm and x=4mm is shown in the figure below

Using the concept of similar triangles, we have

Now in the case

(b)The probability density curve ψx2 of the particle is shown in figure below

(c)

(d)The probability of finding the particle in the interval -2.0mm ≤x≤2.0mm is

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