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Chapter 39: Q.33 (page 1138)

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

Expert verified

The value of constant a is0.25cm-1

Step by step solution

01

To find the constant value of triangle

The area under the triangle of the graph yx2versusxandequateitequalto∫-∞+∞yx2dx=112heightbase=1124cm--4cm=112a8cm=1a=14cm=0.25cm

The value of constant a is0.25cm-1

02

(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.

03

(c)

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

04

(d)

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

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Most popular questions from this chapter

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?

The wave function of a particle is

ψx=b1+x2-1mm≤x<0mmc1+x20mm≤x≤1mm

and zero elsewhere

FIGURE P39.28 shows a pulse train. The period of the pulse train is T=2Δt, where Δtis the duration of each pulse. What is the maximum pulse-transmission rate (pulses per second) through an electronics system with a 200kHz bandwidth? (This is the bandwidth allotted to each FM radio station.)

What is the smallest one-dimensional box in which you can confine an electron if you want to know for certain that the electron’s speed is no more than 10 m/s?

A particle is described by the wave function c1x2 = b cex/L x … 0 mm ce-x/L x Ú 0 mm where L = 2.0 mm.

a. Sketch graphs of both the wave function and the probability density as functions of x.

b. Determine the normalization constant c.

c. Calculate the probability of finding the particle within 1.0 mm of the origin. d. Interpret your answer to part b by shading the region representing this probability on the appropriate graph in part a
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