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0. Math Review31m
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1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
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6. Intro to Forces (Dynamics)3h 22m
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8. Centripetal Forces & Gravitation7h 26m
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10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
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16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
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20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
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31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

35. Special Relativity

Inertial Reference Frames

Physics

35. Special Relativity

Inertial Reference Frames

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Problem 50

Textbook Question

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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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?

Key Concepts

Quantum Mechanics

Energy Levels and Quantum Jumps

De Broglie Wavelength

Watch next

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?