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Chapter 28: Problem 51

Many lasers, including the helium-neon, can produce beams at more than one wavelength. Photons can stimulate emission and cause transitions between the \(20.66-\mathrm{eV}\) metastable state and several different states of lower energy. One such state is 18.38 eV above the ground state. What is the wavelength for this transition? If only these photons leave the laser to form the beam, what color is the beam?

Short Answer

Expert verified
Answer: The wavelength of the emitted photon is approximately 542.8 nm, and the color of the beam is green.

Step by step solution

01

Calculate the energy difference between metastable and lower energy states

The energy difference between the metastable state (E1) and the lower energy state (E2) can be determined by the subtraction E1 - E2. Given E1 = 20.66 eV and E2 = 18.38 eV, the energy difference is: ΔE = E1 - E2 = 20.66 eV - 18.38 eV = 2.28 eV
02

Convert energy difference to Joules

In order to use Planck's equation to determine the wavelength of the emitted photon, we need to convert the energy difference from electron volts (eV) to Joules (J). We can use the following conversion factor: 1 eV = 1.602 x 10^{-19} J So, ΔE in Joules is: ΔE (J) = 2.28 eV × (1.602 × 10^{-19} J/eV) ≈ 3.6536 × 10^{-19} J
03

Calculate the wavelength using Planck's equation

Planck's equation relates the energy of a photon (E) to its wavelength (λ), through the equation: E = hc/λ Here, h is Planck's constant (6.626 x 10^{-34} Js), and c is the speed of light (3.00 x 10^8 m/s). To find the wavelength, we can rearrange the equation and plug in the energy value: λ = hc/ΔE λ ≈ (6.626 × 10^{-34} Js × 3.00 × 10^8 m/s) / (3.6536 × 10^{-19} J) ≈ 5.428 × 10^{-7} m
04

Convert wavelength to nanometers

It is more common to express wavelengths in nanometers (nm) rather than meters. To convert from meters to nanometers, multiply by 1 × 10^9: λ (nm) = 5.428 × 10^{-7} m × (1 × 10^9 nm/m) ≈ 542.8 nm
05

Determine the color of the beam

Now that we have the wavelength of the emitted photon, we can use the visible light spectrum to determine the color of the beam. Wavelengths of visible light range from approximately 400 nm (violet) to 700 nm (red). The given wavelength of 542.8 nm falls within the green portion of the visible spectrum. Hence, the color of the beam emitted by the helium-neon laser is green.

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Most popular questions from this chapter

Before the discovery of the neutron, one theory of the nucleus proposed that the nucleus contains protons and electrons. For example, the helium-4 nucleus would contain 4 protons and 2 electrons instead of - as we now know to be true- 2 protons and 2 neutrons. (a) Assuming that the electron moves at nonrelativistic speeds, find the ground-state energy in mega-electron- volts of an electron confined to a one-dimensional box of length $5.0 \mathrm{fm}\( (the approximate diameter of the \)^{4} \mathrm{He}$ nucleus). (The electron actually does move at relativistic speeds. See Problem \(80 .)\) (b) What can you conclude about the electron-proton model of the nucleus? The binding energy of the \(^{4} \mathrm{He}\) nucleus - the energy that would have to be supplied to break the nucleus into its constituent particles-is about \(28 \mathrm{MeV} .\) (c) Repeat (a) for a neutron confined to the nucleus (instead of an electron). Compare your result with (a) and comment on the viability of the proton-neutron theory relative to the electron-proton theory.
At a baseball game, a radar gun measures the speed of a 144-g baseball to be \(137.32 \pm 0.10 \mathrm{km} / \mathrm{h} .\) (a) What is the minimum uncertainty of the position of the baseball? (b) If the speed of a proton is measured to the same precision, what is the minimum uncertainty in its position?
An electron is confined in a one-dimensional box of length \(L\). Another electron is confined in a box of length 2 \(L\). Both are in the ground state. What is the ratio of their energies $E_{2 l} / E_{L} ?$
The particle in a box model is often used to make rough estimates of energy level spacings. For a metal wire \(10 \mathrm{cm}\) long, treat a conduction electron as a particle confined to a one-dimensional box of length \(10 \mathrm{cm} .\) (a) Sketch the wave function \(\psi\) as a function of position for the electron in this box for the ground state and each of the first three excited states. (b) Estimate the spacing between energy levels of the conduction electrons by finding the energy spacing between the ground state and the first excited state.
(a) Show that the number of electron states in a subshell is \(4 \ell+2 .\) (b) By summing the number of states in each of the subshells, show that the number of states in a shell is \(2 n^{2} .\) [Hint: The sum of the first \(n\) odd integers, from 1 to \(2 n-1,\) is \(n^{2} .\) That comes from regrouping the sum in pairs, starting by adding the largest to the smallest: \(1+3+5+\dots+(2 n-5)+(2 n-3)+(2 n-1)\) \(=[1+(2 n-1)]+[3+(2 n-3)]+[5+(2 n-5)]+\cdots\) \(=2 n+2 n+2 n+\cdots=2 n \times \frac{n}{2}=n^{2}\)
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