Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 39: Particles Behaving as Waves

Problem 3

Chapter 38, Problem 3

An electron has a de Broglie wavelength of $2.80 \times 10^{- 10}$ m. Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).

Verified step by step guidance

Step 1: Recall the de Broglie wavelength formula, which relates the wavelength (λ) of a particle to its momentum (p): λ = h / p, where h is Planck's constant (6.626 × 10^-34 J·s). Rearrange the formula to solve for momentum: p = h / λ.

Step 2: Substitute the given de Broglie wavelength (λ = 2.80 × 10^-10 m) and Planck's constant (h = 6.626 × 10^-34 J·s) into the formula p = h / λ to calculate the magnitude of the electron's momentum.

Step 3: Use the relationship between momentum and kinetic energy for a non-relativistic particle: KE = p² / (2m), where m is the mass of the electron (9.109 × 10^-31 kg). Substitute the calculated momentum (p) and the mass of the electron into this formula to find the kinetic energy in joules.

Step 4: Convert the kinetic energy from joules to electron volts (eV) using the conversion factor: 1 eV = 1.602 × 10^-19 J. Divide the kinetic energy in joules by this factor to express the result in electron volts.

Step 5: Summarize the process: First, calculate the momentum using the de Broglie wavelength formula. Then, use the momentum to find the kinetic energy in joules. Finally, convert the kinetic energy to electron volts for the second part of the problem.

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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 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. This concept illustrates the wave-particle duality of matter, indicating that particles like electrons exhibit both wave-like and particle-like properties.

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity (p = mv). In quantum mechanics, momentum can also be expressed in terms of the de Broglie wavelength, allowing for the calculation of a particle's momentum using its wavelength. Understanding momentum is crucial for analyzing the motion and behavior of particles at the quantum level.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 1/2 mv² for classical mechanics. In the context of quantum mechanics, the kinetic energy of a particle can also be derived from its momentum using the relation KE = p²/2m. This concept is essential for understanding how energy is related to the motion of particles, particularly in quantum systems.

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Textbook Question

Calculate the de Broglie wavelength of a $5.00$ -g bullet that is moving at $340$ m/s. Will the bullet exhibit wavelike properties?

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Textbook Question

For crystal diffraction experiments (discussed in Section $39.1$ ), wavelengths on the order of $0.20$ nm are often appropriate. Find the energy in electron volts for a particle with this wavelength if the particle is a photon.

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Textbook Question

An alpha particle ( $m = 6.64 \times 10^{- 27}$ kg) emitted in the radioactive decay of uranium- $238$ has an energy of $4.20$ MeV. What is its de Broglie wavelength?

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Textbook Question

(a) An electron moves with a speed of $4.70 \times 10^{6}$ m/s. What is its de Broglie wavelength?

(b) A proton moves with the same speed. Determine its de Broglie wavelength.

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Textbook Question

An electron is moving with a speed of $8.00 \times 10^{6}$ m/s. What is the speed of a proton that has the same de Broglie wavelength as this electron?

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An electron has a de Broglie wavelength of 2.80×10−102.80\(\times\)10^{-10} m. Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).

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Key Concepts

de Broglie Wavelength

Momentum

Kinetic Energy

An electron has a de Broglie wavelength of $2.80 \times 10^{- 10}$ m. Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).