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Chapter 40: Q. 20 (page 1175)

An electron in a harmonic potential well absorbs a photon with a wavelength of $400 n m$ as it undergoes a $1 鈫� 2$ quantum jump. What wavelength is absorbed in a $1 鈫� 3$ quantum jump?

Short Answer

Expert verified

Wavelength absorbed in $1 鈫� 2$ quantum jump = $200 n m$

Wavelength absorbed in $1 鈫� 3$ quantum jump= $200 n m$

Step by step solution

01

Step 1. Given information

Energy level of harmonic quantum oscillator,

$E_{n} = (n - \frac{1}{2}) \frac{h 蝇}{2 蟺}$

Here,

n denotes the state,

$h =$ Planck's constant and

$蝇 =$ classical angular frequency.

02

Step 2. Quantum jump between 1→2

$E_{1} = (1 - \frac{1}{2}) \frac{h 蝇}{2 蟺}$

$= \frac{1}{2} (\frac{h 蝇}{2 蟺})$

$E_{2} = (2 - \frac{1}{2}) \frac{h 蝇}{2 蟺}$

$= \frac{3}{2} (\frac{h 蝇}{2 蟺})$

Now,

$位_{photon, 1 鈫� 2} = \frac{h c}{\frac{3}{2} (\frac{h 蝇}{2 蟺}) - \frac{1}{2} (\frac{h 蝇}{2 蟺})}$

$= \frac{h c}{(\frac{h 蝇}{2 蟺})}$

$= \frac{2 蟺 c}{蝇}$

$\frac{2 蟺 c}{蝇} = 400 nm$

$\frac{蟺 c}{蝇} = 200 nm$

03

Step 3. Quantum jump between 1→3

$E_{1} = \frac{1}{2} (\frac{h 蝇}{2 蟺})$

$E_{3} = (3 - \frac{1}{2}) \frac{h 蝇}{2 蟺}$

$= \frac{5}{2} (\frac{h 蝇}{2 蟺})$

$位_{photon, 1 鈫� 3} = \frac{h c}{\frac{5}{2} (\frac{h 蝇}{2 蟺}) - \frac{1}{2} (\frac{h 蝇}{2 蟺})}$

$= \frac{h c}{(\frac{4}{2}) (\frac{h 蝇}{2 蟺})}$

$= \frac{蟺 c}{蝇}$

As found earlier.

$\frac{蟺 c}{蝇} = 200 nm$

位_{photon, 1 鈫� 3} = 200 nm

Thus, the wavelength absorbed in $1 鈫� 3$ quantum jump = $200 n m$

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Most popular questions from this chapter

A neutron is confined in a $10 fm$ -diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than $2.0 fm$ from the edge of the nucleus?

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a. What happens to the spacing between the nodes of the wave function as |x| increases? Why?

b. What happens to the heights of the antinodes of the wave function as |x| increases? Why?

c. Sketch a reasonably accurate graph of the n=8 wave function of a quantum harmonic oscillator.

A particle confined in a rigid one-dimensional box of length $10 fm$ has an energy level $E_{n} = 32.9 MeV$ and an adjacent energy level $E_{n + 1} = 51.4 MeV$ .

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b. Draw an energy-level diagram showing all energy levels from 1 through n + 1. Label each level and write the energy beside it.

c. Sketch the n + 1 wave function on the n + 1 energy level.

d. What is the wavelength of a photon emitted in the $n + 1 鈫� n$ transition? Compare this to a typical visible-light wavelength.

e. What is the mass of the particle? Can you identify it?

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