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Chapter 38: Q. 53 (page 1116)

An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n=2 to n =1. What is the length of the box?

Short Answer

Expert verified

Tthe length of the box is $L = 426.4 pm$

Step by step solution

01

Given information

An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n=2 to n =1.

02

Explanation

Energy levels of the particle in the box:

$E_{n} = \frac{h^{2} n^{2}}{8 m L^{2}}$

Energy of the photon:

$Δ E_{12} = \frac{h c}{λ}$

Substitute energy levels:

role="math" localid="1648514586435" $E_{2} - E_{1} = \frac{h^{2} \cdot 2^{2}}{8 m L^{2}} - \frac{h^{2} \cdot 1^{2}}{8 m L^{2}} E_{2} - E_{1} = \frac{3 h^{2}}{8 m L^{2}}$

Substitute energy of the photon:

$\frac{h c}{λ} = \frac{3 h^{2}}{8 m L^{2}}$

03

Calculations:

Express the above equation in terms of L:

$L = \sqrt{\frac{3 h^{2} λ}{8 m h c}}$

Substitute the values:

$L = \sqrt{\frac{3 {(6.63 \cdot 10^{- 34})}^{2} \cdot 200 \cdot 10^{- 9}}{8 \cdot 9.11 \cdot 10^{- 31} \cdot 6.63 \cdot 10^{- 34} \cdot 3 \cdot 10^{8}}} L = 426.4 pm$

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Most popular questions from this chapter

Very large, hot stars—much hotter than our sun—can be identified by the way in which ${He}^{+}$ ions in their atmosphere absorb light. What are the three longest wavelengths, in nm, in the Balmer series of ${He}^{+}$ ?

Very large, hot stars—much hotter than our sun—can be identified by the way in which He+ ions in their atmosphere absorb light. What are the three longest wavelengths, in nm, in the Balmer series of He+?

A $100 W$ incandescent lightbulb emits about $5 W$ of visible light. (The other $95 W$ are emitted as infrared radiation or lost as heat to the surroundings.) The average wavelength of the visible light is about $600 n m$ , so make the simplifying assumption that all the light has this wavelength. How many visible-light photons does the bulb emit per second?

What about the winding of the frauliens $3 \to 2.4 \to 2$ , and $5 \to 2$ in the hydroponics $O^{- 7}$ In what spectral range do these lie?

The electron interference pattern of Figure 38.12 was made by shooting electrons with $50 keV$ of kinetic energy through two slits spaced role="math" localid="1650737433408" $1 .0 μm$ apart. The fringes were recorded on a detector $1.0 m$ behind the slits.

a. What was the speed of the electrons? (The speed is large enough to justify using relativity, but for simplicity do this as a nonrelativistic calculation.)

b. Figure 38.12 is greatly magnified. What was the actual spacing on the detector between adjacent bright fringes?

See all solutions

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