An electron is in a box of width 3.0×10−103.0\$$times10^{-10} m. $$What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the n=1n = 1 level; (b) the n=2n = 2 level; (c) the n=3n = 3 level? In each case how does the wavelength compare to the width of the box?

Ch 40: Quantum Mechanics I: Wave Functions

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 40: Quantum Mechanics I: Wave Functions

문제 19

40장, 문제 19

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?

검증된 단계별 안내

Step 1: Understand the problem. The electron is confined in a one-dimensional box of width $ L = 3.0 \times 10^{-10} \; \text{m} $. The energy levels of the electron are quantized, and its de Broglie wavelength and momentum are related to the quantum number $ n $. We need to calculate the de Broglie wavelength $ \lambda $, the momentum $ p $, and compare $ \lambda $ to the width of the box for $ n = 1, 2, 3 $.

Step 2: Recall the relationship between the de Broglie wavelength and the width of the box. For a particle in a box, the de Broglie wavelength is given by $ \lambda = \frac{2L}{n} $, where $ n $ is the quantum number. Substitute $ L = 3.0 \times 10^{-10} \; \text{m} $ into this formula to calculate $ \lambda $ for each value of $ n $.

Step 3: Use the relationship between momentum and wavelength. The de Broglie wavelength is related to the momentum by $ \lambda = \frac{h}{p} $, where $ h $ is Planck's constant $ h = 6.626 \times 10^{-34} \; \text{J·s} $. Rearrange this formula to find the momentum: $ p = \frac{h}{\lambda} $. Use the values of $ \lambda $ from Step 2 to calculate $ p $ for each $ n $.

Step 4: Compare the de Broglie wavelength to the width of the box. For each quantum number $ n $, calculate the ratio $ \frac{\lambda}{L} $ to determine how the wavelength compares to the box's width. This will help us understand the relationship between the quantum state and the spatial confinement of the electron.

Step 5: Summarize the results. For each $ n $, report the calculated de Broglie wavelength $ \lambda $, the momentum $ p $, and the ratio $ \frac{\lambda}{L} $. Discuss how the wavelength decreases as $ n $ increases and how this reflects the increasing energy and momentum of the electron in higher quantum states.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that relates the wavelength of a particle to its momentum. 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 illustrates the wave-particle duality of matter, indicating that particles like electrons exhibit wave-like properties.

Quantum Levels

Quantum levels, or energy levels, refer to the discrete states of energy that an electron can occupy in a quantum system, such as an atom or a box. Each level is associated with a specific quantum number (n), which determines the energy and spatial distribution of the electron. The allowed wavelengths of the electron in a box are quantized, meaning only certain wavelengths corresponding to these levels are permitted.

Momentum of a Particle

Momentum is a vector quantity defined as the product of an object's mass and its velocity (p = mv). In quantum mechanics, the momentum of a particle like an electron can also be expressed in terms of its wavelength using the de Broglie relation. Understanding momentum is crucial for analyzing the behavior of particles in confined spaces, such as a box, where their movement is restricted.