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0. Math Review31m
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1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
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 40.20

Textbook Question

An electron is in a box of width 3.0*10^-10 m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the n = 1 level; (b) the n = 2 level; (c) the n = 3 level? In each case how does the wavelength compare to the width of the box?

Verified step by step guidance

Understand the concept of quantization in a box: An electron in a box is a model where the electron is confined to a one-dimensional region with fixed boundaries. The energy levels are quantized, meaning the electron can only occupy certain discrete energy states.

Use the formula for the de Broglie wavelength: The de Broglie wavelength $ \lambda $ of a particle is given by $ \lambda = \frac{h}{p} $, where $ h $ is Planck's constant and $ p $ is the momentum of the particle.

Determine the momentum of the electron: The momentum $ p $ of an electron in a box is quantized and can be expressed as $ p = \frac{n h}{2L} $, where $ n $ is the quantum number, $ h $ is Planck's constant, and $ L $ is the width of the box.

Calculate the de Broglie wavelength for each energy level: Substitute the expression for momentum into the de Broglie wavelength formula to find $ \lambda = \frac{2L}{n} $. Calculate $ \lambda $ for $ n = 1 $, $ n = 2 $, and $ n = 3 $.

Compare the wavelength to the width of the box: For each energy level, compare the calculated de Broglie wavelength $ \lambda $ to the width of the box $ L = 3.0 \times 10^{-10} $ m to understand how the wavelength changes with different quantum levels.

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Key Concepts

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

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that describes the wave-like nature of particles. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. This concept is crucial for understanding how particles like electrons exhibit both particle and wave characteristics, especially in confined systems like a box.

Particle in a Box Model

The particle in a box model is a quantum mechanical system where a particle is confined to move within a perfectly rigid and impenetrable box. The energy levels of the particle are quantized, meaning the particle can only occupy certain discrete energy states. The energy levels are determined by the quantum number n, and the wave function solutions are sinusoidal, reflecting the standing wave nature of the particle within the box.

Quantum Numbers and Energy Levels

Quantum numbers are integers that describe the quantized energy levels of a system. In the context of a particle in a box, the principal quantum number n determines the energy level and the corresponding wave function of the particle. Higher quantum numbers correspond to higher energy levels and shorter wavelengths, as the particle's wave function must fit more nodes within the box's fixed width.

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Related Practice

Textbook Question

A proton is in a box of width $L$ . What must the width of the box be for the ground-level energy to be $5.0$ MeV, a typical value for the energy with which the particles in a nucleus are bound? Compare your result to the size of a nucleus — that is, on the order of $10^{- 14}$ m.

Textbook Question

An electron in a one-dimensional box has ground state energy $2.00$ eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

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Textbook Question

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width $0.360$ nm.

(b) The electron makes a transition from the $n equals 1$ to $n equals 4$ level by absorbing a photon. Calculate the wavelength of this photon.

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Textbook Question

An electron is in a box of width $3.0 \times 10^{- 10}$ m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the $n equals 1$ level; (b) the $n equals 2$ level; (c) the $n equals 3$ level? In each case how does the wavelength compare to the width of the box?

Textbook Question

(a) An electron with initial kinetic energy $32$ eV encounters a square barrier with height $41$ eV and width $0.25$ nm. What is the probability that the electron will tunnel through the barrier?

(b) A proton with the same kinetic energy encounters the same barrier. What is the probability that the proton will tunnel through the barrier?

Textbook Question

An electron with initial kinetic energy $6.0$ eV encounters a barrier with height $11.0$ eV. What is the probability of tunneling if the width of the barrier is (a) $0.80$ nm and (b) $0.40$ nm?

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Textbook Question

An electron with initial kinetic energy $5.0$ eV encounters a barrier with height $U sub 0$ and width $0.60$ nm. What is the transmission coefficient if (a) $U sub 0 equals 7.0$ eV; (b) $U sub 0 equals 9.0$ eV; (c) $U sub 0 equals 13.0$ eV?

Textbook Question

While undergoing a transition from the $n equals 1$ to the $n equals 2$ energy level, a harmonic oscillator absorbs a photon of wavelength $6.50$ $μ$ m. What is the wavelength of the absorbed photon when this oscillator undergoes a transition (a) from the $n equals 2$ to the $n equals 3$ energy level and (b) from the $n equals 1$ to the $n equals 3$ energy level?

(c) What is the value of $\sqrt{(k^{'} / m)}$ , the angular oscillation frequency of the corresponding Newtonian oscillator?

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