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Chapter 39: Q. 1 (page 1136)

FIGURE Q 39.1 shows the probability density for photons to be detected on the x-axis.

a. Is a photon more likely to be detected at x = 0 m or at x=1 m? Explain.

b. One million photons are detected. What is the expected number of photons in a 1 mm-wide interval at x = 0.50 m?

Short Answer

Expert verified

The expected number of photons is: 1,000

Step by step solution

01

Given information

FIGURE Q 39.1 shows the probability density for photons to be detected on the x-axis.

02

Explanation

a) The photon is detected most likely at x≈1, because the probability density is higher there.

03

Explanation

b)P(inδxatx)=P(x)δx.

If 1,000,000 photons are detected then the number expected in al-mm-wide interval is 1,000,000P(x)δxwhere whereδx=1.0mm.Forx=0.50m,P(x)=1m-1and the expected number is(1,000,000)1m-1(0.001m)=1,000.

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Most popular questions from this chapter

a. Starting with the expressionΔfΔt≈1for a wave packet, find an expression for the product

ΔEΔtfor a photon.

b. Interpret your expression. What does it tell you?

c. The Bohr model of atomic quantization says that an atom in an excited state can jump to a lower-energy state by emitting a photon. The Bohr model says nothing about how long this process takes. You'll learn in Chapter 41 that the time any particular atom spends in the excited state before emitting a photon is unpredictable, but the average lifetime Δtof many atoms can be determined. You can think of Δtas being the uncertainty in your knowledge of how long the atom spends in the excited state. A typical value is Δt≈10ns. Consider an atom that emits a photon with a 500nmwavelength as it jumps down from an excited state. What is the uncertainty in the energy of the photon? Give your answer in eV.

d. What is the fractional uncertainty ΔE/Ein the photon's energy?

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?

FIGURE P39.46

Consider the electron wave function

ψX=0x<0nm1.414nm-12e-x1.0nmx≥0nm

where x is in cm.

a. Determine the normalization constant c.

b. Draw a graph of c1x2 over the interval -2 cm ≤x ≤2 cm. Provide numerical scales on both axes.

c. Draw a graph of 0 c1x2 0 2 over the interval -2 cm ≤x ≤2 cm. Provide numerical scales.

d. If 104 electrons are detected, how many will be in the interval 0.00 cm ≤x ≤0.50 cm?

The wave function of a particle is

ψx=bπx2+b2

where b is a positive constant. Find the probability that the particle is located in the interval -b≤x ≤b

A proton is confined within an atomic nucleus of diameter 4.0m. Use a one-dimensional model to estimate the smallest range of speeds you might find for a proton in the nucleus.
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