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Chapter 39: Q5. (page 1136)

What is the value of the constant a in FIGURE Q39.5?

Short Answer

Expert verified

a = 2mm-1

Step by step solution

01

Given parameters

A graph of probability density of a particle.

the graph forms a triangle with base length of 1mm (from 1mm to 2mm)

and height of a.

02

Finding a

Now we know that probability of finding the particle from -∞to+∞is 1.

And here all the probability density is distributed from x= 1mm to x = 2mm.

So probability of finding the particle between 1mm to 2mm is 1.

this means integral of this wave function from 1mm to 2mm will give 1.

And integral of a function also tells about area under the graph.

⇒areaofthisgraph=1

⇒areaofthetriangle=1⇒12×1mm×a=1⇒a=2mm-1

Hence, the value of a is 2mm-1

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Most popular questions from this chapter

Physicists use laser beams to create an atom trap in which atoms are confined within a spherical region of space with a diameter of about 1mm. The scientists have been able to cool the atoms in an atom trap to a temperature of approximately 1nK, which is extremely close to absolute zero, but it would be interesting to know if this temperature is close to any limit set by quantum physics. We can explore this issue with a onedimensional model of a sodium atom in a 1.0-mm-long box.
a. Estimate the smallest range of speeds you might find for a sodium atom in this box.
b. Even if we do our best to bring a group of sodium atoms to rest, individual atoms will have speeds within the range you found in part a. Because there's a distribution of speeds, suppose we estimate that the root-mean-square speed vmsof the atoms in the trap is half the value you found in part a. Use this vrms to estimate the temperature of the atoms when they've been cooled to the limit set by the uncertainty principle.


FIGURE P39.32 shows |ψ(x)|2for the electrons in an experiment.

a. Is the electron wave function normalized? Explain.

b. Draw a graph of ψ(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=0.00cm? At x=0.50cm? At x=0.999cm?

d. If 104electrons are detected, how many are expected to land in the interval -0.30cm≤x≤0.30cm?

A small speck of dust with mass 1.0×10-13ghas fallen into the hole shown in FIGURE P39.46 and appears to be at rest. According to the uncertainty principle, could this particle have enough energy to get out of the hole? If not, what is the deepest hole of this width from which it would have a good chance to escape?

When 5 X 1012 photons pass through an experimental apparatus, 2.0 X 109 land in a 0.10-mm-wide strip. What is the probability density at this point?

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?
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