Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 38: Quantization

Problem 50

Chapter 38, Problem 50

A muon—a subatomic particle with charge −e and a mass 207 times that of an electron—is confined in a 15-pm-long, one-dimensional box. ( 1pm=1picometer=10⁻¹² m.) What is the wavelength, in nm, of the photon emitted in a quantum jump from n = 2 to n = 1?

Verified step by step guidance

Step 1: Recognize that the problem involves a particle in a one-dimensional box, which is a quantum mechanics problem. The energy levels for a particle in a box are given by the formula:

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

, where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box.

Step 2: Calculate the energy difference between the n=2 and n=1 states. The energy difference is given by:

ΔE = E_{2} - E_{1} = \frac{h}{2} / 8 m L^{2} (2^{2} - 1^{2})

Step 3: Substitute the known values into the formula. Use h = 6.626 × 10^−34 J·s (Planck's constant), m = 207 × 9.109 × 10^−31 kg (mass of the muon), and L = 15 × 10^−12 m (length of the box).

Step 4: Relate the energy difference ΔE to the wavelength of the emitted photon using the formula:

ΔE = \frac{hc}{λ}

, where c is the speed of light (3.00 × 10^8 m/s) and λ is the wavelength. Rearrange the formula to solve for λ:

λ = \frac{hc}{ΔE}

Step 5: Convert the wavelength from meters to nanometers by multiplying by 10^9 (1 nm = 10^−9 m). This will give the final wavelength of the photon emitted during the quantum jump from n=2 to n=1.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It introduces concepts such as quantization, where certain properties, like energy, can only take on discrete values. This framework is essential for understanding phenomena like electron transitions in atoms and the behavior of particles in confined spaces, such as the one-dimensional box in this question.

Energy Levels and Quantum Jumps

In quantum mechanics, particles such as electrons occupy specific energy levels within an atom. A quantum jump refers to the transition of a particle from one energy level to another, which involves the absorption or emission of energy in the form of a photon. The energy difference between these levels determines the wavelength of the emitted photon, which is crucial for solving the problem regarding the muon's transition.

De Broglie Wavelength

The De Broglie wavelength is a fundamental concept that relates the wavelength of a particle to its momentum, given by the formula λ = h/p, where h is Planck's constant and p is the momentum. In the context of a particle in a box, the allowed wavelengths are quantized, leading to specific energy levels. This concept is vital for calculating the wavelength of the photon emitted during the muon's transition from n=2 to n=1.

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Textbook Question

INT The electron interference pattern of Figure 38.12 was made by shooting electrons with 50 keV of kinetic energy through two slits spaced 1.0 μm apart. The fringes were recorded on a detector 1.0 m behind the slits. Figure 38.12 is greatly magnified. What was the actual spacing on the detector between adjacent bright fringes?

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Textbook Question

An electron confined in a one-dimensional box is observed, at different times, to have energies of 12 eV, 27 eV, and 48 eV. What is the length of the box?

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Textbook Question

The cosmic microwave background radiation is light left over from the Big Bang that has been Doppler-shifted to microwave frequencies by the expansion of the universe. It now fills the universe with 450 photons/cm³ at an average frequency of 160 GHz. How much energy from the cosmic microwave background, in MeV, fills a small apartment that has 95 m² of floor space and 2.5-m-high ceilings?

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Textbook Question

The first three energy levels of the fictitious element X are shown in FIGURE P38.54. What wavelengths are observed in the absorption spectrum of element X? Express your answers in nm.

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Textbook Question

The first three energy levels of the fictitious element X are shown in FIGURE P38.54. What is the ionization energy of element X?

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Textbook Question

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?

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A muon—a subatomic particle with charge −e and a mass 207 times that of an electron—is confined in a 15-pm-long, one-dimensional box. ( 1pm=1picometer=10−12 m.) What is the wavelength, in nm, of the photon emitted in a quantum jump from n = 2 to n = 1?

Verified video answer for a similar problem:

Key Concepts

Quantum Mechanics

Energy Levels and Quantum Jumps

De Broglie Wavelength

A muon—a subatomic particle with charge −e and a mass 207 times that of an electron—is confined in a 15-pm-long, one-dimensional box. ( 1pm=1picometer=10⁻¹² m.) What is the wavelength, in nm, of the photon emitted in a quantum jump from n = 2 to n = 1?