Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 38: Quantization

Problem 20

Chapter 38, Problem 20

The diameter of the nucleus is about 10 fm. What is the kinetic energy, in MeV, of a proton with a de Broglie wavelength of 10 fm?

Verified step by step guidance

Step 1: Understand the relationship between the de Broglie wavelength and the momentum of a particle. The de Broglie wavelength (λ) is given by the formula:

λ = \frac{h}{p}

, where

h

is Planck's constant and

p

is the momentum of the particle.

Step 2: Rearrange the formula to solve for the momentum

p

p = \frac{h}{λ}

. Substitute the values for Planck's constant (

h = 4.1357 \times 10 ⁻ 15 eV \cdot s

) and the de Broglie wavelength (

λ = 10 fm

, where

1 fm = 10 ⁻ 15 m

Step 3: Convert the momentum

p

into kinetic energy using the relativistic energy-momentum relation. For non-relativistic speeds, the kinetic energy

K

can be approximated as:

K = \frac{p^{2}}{2 m}

, where

m

is the mass of the proton (

m = 938.27 MeV / c^{2}

Step 4: Substitute the calculated momentum

p

and the mass of the proton into the kinetic energy formula. Ensure all units are consistent (e.g., converting

p

to MeV/c if necessary).

Step 5: Simplify the expression to find the kinetic energy

K

in MeV. This will give the final result for the proton's kinetic energy based on its de Broglie wavelength.

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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 of the particle. This concept illustrates the wave-particle duality of matter, indicating that particles like protons exhibit wave-like properties at small scales.

Kinetic Energy of a Particle

Kinetic energy is the energy that an object possesses due to its motion, calculated using the formula KE = (1/2)mv², where m is the mass and v is the velocity of the object. In quantum mechanics, the kinetic energy of a particle can also be expressed in terms of its momentum, using the relation KE = p²/(2m). Understanding this relationship is crucial for calculating the kinetic energy of particles at the atomic scale.

Nuclear Scale and Units

The nuclear scale refers to dimensions on the order of femtometers (fm), which are used to measure atomic nuclei. One femtometer is 10^-15 meters, a scale where quantum effects dominate. In nuclear physics, energy is often expressed in mega-electronvolts (MeV), a unit that reflects the energy of particles at this scale. Recognizing these units is essential for converting and interpreting results in nuclear physics problems.

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