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Chapter 39: Q.38 (page 1139)

A particle is described by the wave function c1x2 = b cex/L x 鈥� 0 mm ce-x/L x 脷 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

Short Answer

Expert verified

the normalization constant c is 0.707 mm^-1/2

prob= $= 2 c^{2} (- \frac{L}{2}) {[e^{- \frac{2 x}{L}}]}_{0}^{1.0 m m}$

Thus the probability of finding the particle within 1.0mm for the origin is 63.2%

Step by step solution

01

To find the probability of a particle

The expression for the probability of a particle at position x is given from probability density is as follows:

$p (x) = {|蠄 (x)|}^{2}$

Here ${|蠄 (x)|}^{2}$ is the probability density

A particle is described by the wave function as follows: $蠄 (x) = \{\overset{C e^{\frac{x}{L}} X 鈮� 0 n m}{C e^{\frac{- x}{L}} X 鈮� 0 n m}\}$

Here L is 2.0nm

The probability density of the particle at $x 鈮� 0 n m$ is as follows:

${|蠄 (x)|}^{2} = c^{2} e^{\frac{2 蟺}{L}}$

The probability density of the particle at $x 鈮� 0 n m$ is as follows:

${|蠄 (x)|}^{2} = c^{2} e^{\frac{2 蟺}{L}}$

02

Table for the values of ψ and ψ2 is as follows:

03

(b)Determine the normalization constant c∫-∞+∞ψx2dx=12∫-∞+∞ψx2dx=12c3-L2e-2xl=1 Lc21-0=1 c=1Lsubstitute 2.0mm for Lc=12.0mm =0.707 mm-12Thus the normalization constant c is 0.707 mm-1/2

These are the constant

04

(c) Calculate the probability of finding the particle within 1.0mmof the origin.

$P r o b (- 1.0 m m 鈮� x 鈮� + 1.0 m m) = {鈭�}_{- 1.0 m m}^{+ 1.0 m m} {|蠄 (x)|}^{2} d x = 2 {鈭�}_{- 1.0 m m}^{+ 1.0 m m} {|蠄 (x)|}^{2} d x = 2 c^{2} (- \frac{L}{2}) {[e^{- \frac{2 x}{L}}]}_{0}^{1.0 m m}$

05

Simplify the above equation furtherProb-1.0mm≤x≤+1.0mm=Lc2e0-e2.0mm/Lsubstitute 2.0mm for L and 0.707 mm-1/2 for cprob-1.0mm≤x≤+1.0mm=2.0mm0.707mm-1/22e0-e-2.0mm 2.0mm =63.2%

Thus the probability of finding the particle within 1.0mm for the origin is 63.2%

06

(d)The following is the graph showing the shaded region representing the finding of the particle within 1.0 mm of the origin.

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Most popular questions from this chapter

Andrea, whose mass is $50 k g$ , thinks she鈥檚 sitting at rest in her $5.0 m$ long dorm room as she does her physics homework. Can Andrea be sure she鈥檚 at rest? If not, within what range is her velocity likely to be?

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 $10 m / s$ ?

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

FIGURE P39.32 shows $| 蠄 (x) |^{2}$ for the electrons in an experiment.

a. Is the electron wave function normalized? Explain.

b. Draw a graph of $蠄 (x)$ over this same interval. Provide a numerical scale on both axes. (There may be more than one acceptable answer.)

c. What is the probability that an electron will be detected in a $0.0010 - cm$ -wide region at $x = 0.00 cm$ ? At $x = 0.50 cm$ ? At $x = 0.999 c m$ ?

d. If $10^{4}$ electrons are detected, how many are expected to land in the interval $- 0.30 cm 鈮� x 鈮� 0.30 cm$ ?

Consider a single-slit diffraction experiment using electrons. Using Figure 39.5 as a model, draw

a. A dot picture showing the arrival positions of the first $40$ or $50$ electrons.

b. A graph of ${|蠄 (x)|}^{2}$ for the electrons on the detection screen.

c. A graph of $蠄 (x)$ for the electrons. Keep in mind that $蠄$ , as a wave-like function, oscillates between positive and negative.

See all solutions

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