Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo

Knight Calc 5th Edition

Ch 38: Quantization

Problema 20

Capitolo 38, Problema 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?

Guida verificata passo dopo passo

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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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.

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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.

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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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