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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
7. Friction, Inclines, Systems2h 44m
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9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
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 1a

Textbook Question

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. What kind of photons are they—infrared, visible, or ultraviolet?

Verified step by step guidance

Determine the energy difference between the two energy levels (n=3 and n=2) using the formula for energy levels in a rigid box:

E

_n = n²h² / 8mL², where n is the quantum number, h is Planck's constant, m is the mass of the electron, and L is the length of the box. The energy difference is given by

ΔE = E

₃ - E₂.

Relate the energy difference to the wavelength of the emitted photon using the equation

ΔE = hc / λ

, where h is Planck's constant, c is the speed of light, and λ is the wavelength of the photon. Rearrange the equation to find the energy difference:

ΔE = (6.626 × 10

^-34 J·s)(3.00 × 10⁸ m/s) / (1484 × 10^-9 m).

Convert the energy difference from joules to electronvolts (eV) using the conversion factor

1 eV = 1.602 × 10

^-19 J. This will give the energy of the photon in eV.

Compare the energy of the photon to the typical energy ranges for infrared, visible, and ultraviolet light. Infrared photons typically have energies less than 1.65 eV, visible photons range from about 1.65 eV to 3.1 eV, and ultraviolet photons have energies greater than 3.1 eV.

Based on the calculated energy, classify the photon as infrared, visible, or ultraviolet. This classification depends on the energy range it falls into.

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Key Concepts

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

Photon Wavelength and Electromagnetic Spectrum

Photons are particles of light that can be characterized by their wavelength, which determines their position in the electromagnetic spectrum. The spectrum ranges from radio waves with long wavelengths to gamma rays with very short wavelengths. Wavelengths in the range of 700 nm to 400 nm correspond to visible light, while wavelengths longer than 700 nm are classified as infrared.

Energy Levels in Atoms

Electrons in an atom occupy specific energy levels, and transitions between these levels result in the emission or absorption of photons. When an electron moves from a higher energy level to a lower one, it emits a photon with energy equal to the difference between the two levels. The wavelength of the emitted photon can be calculated using the energy-wavelength relationship, which is inversely proportional.

Identifying Photon Types

To classify photons based on their wavelength, one must refer to the electromagnetic spectrum. Photons with wavelengths longer than 700 nm are considered infrared, while those between 400 nm and 700 nm are visible light, and wavelengths shorter than 400 nm fall into the ultraviolet range. The specific wavelength of 1484 nm indicates that the emitted photons from the electron transition are in the infrared range.

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Related Practice

Textbook Question

A particle is described by the wave function $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

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Textbook Question

A particle is described by the wave function $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ mm where L = 2.0 mm. Determine the normalization constant c.

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Textbook Question

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. A ²³⁸U nucleus, which decays by alpha emission, is 15 fm in diameter. Model an alpha particle within a ²³⁸U nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have?

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Textbook Question

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. The probability that a nucleus will undergo alpha decay is proportional to the frequency with which the alpha particle reflects from the walls of the nucleus. What is that frequency (reflections/s) for a maximum-speed alpha particle within a ²³⁸U nucleus?

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Textbook Question

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. How long is the box in which the electrons are confined?

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Textbook Question

A 16-nm-long box has a thin partition that divides the box into a 4-nm-long section and a 12-nm-long section. An electron confined in the shorter section is in the n = 2 state. The partition is briefly withdrawn, then reinserted, leaving the electron in the longer section of the box. What is the electron’s quantum state after the partition is back in place?

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Textbook Question

A finite potential well has depth U₀ = 2.00 eV. What is the penetration distance for an electron with energy (a) 0.50 eV, (b) 1.00 eV, and (c) 1.50 eV?

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Textbook Question

The graph in FIGURE EX40.15 shows the potential-energy function U(x) of a particle. Solution of the Schrödinger equation finds that the n = 3 level has E₃ = 0.5 eV and that the n = 6 level has E₆ = 2.0 eV. Redraw this figure and add to it the energy lines for the n = 3 and n = 6 states.

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