Question

Which of the following has the smallest de Broglie wavelength? a) an electron traveling at \(80 \%\) the speed of light b) a proton traveling at \(20 \%\) the speed of light c) a carbon nucleus traveling at \(70 \%\) the speed of light d) a helium nucleus traveling at \(80 \%\) the speed of light e) a lithium nucleus traveling at \(50 \%\) the speed of light

Short answer

a) an electron at \(80\%\) the speed of light b) a proton at \(20\%\) the speed of light c) a carbon nucleus at \(70\%\) the speed of light d) a helium nucleus at \(80\%\) the speed of light e) a lithium nucleus at \(50\%\) the speed of light Answer: c) a carbon nucleus at \(70\%\) the speed of light

Step-by-step solution

Write down de Broglie wavelength formula

\(\lambda = \frac{h}{mv}\)

Find the mass of each particle

Mass of an electron: \(m_e = 9.11 \times 10^{-31} kg\) (approx.) Mass of a proton: \(m_p = 1.67 \times 10^{-27} kg\) (approx.) Mass of a carbon nucleus: \(m_C = 12 \times m_p\) Mass of a helium nucleus: \(m_{He} = 4 \times m_p\) Mass of a lithium nucleus: \(m_{Li} = 7 \times m_p\)

Calculate the velocity of each particle

Velocity of an electron: \(v_e = 0.8c\) Velocity of a proton: \(v_p = 0.2c\) Velocity of a carbon nucleus: \(v_C = 0.7c\) Velocity of a helium nucleus: \(v_{He} = 0.8c\) Velocity of a lithium nucleus: \(v_{Li} = 0.5c\) Here, \(c\) represents the speed of light.

Calculate the de Broglie wavelength for each particle

Wavelength of an electron: \(\lambda_e = \frac{h}{m_e v_e} = \frac{h}{(9.11 \times 10^{-31} kg)(0.8c)}\) Wavelength of a proton: \(\lambda_p = \frac{h}{m_p v_p} = \frac{h}{(1.67 \times 10^{-27} kg)(0.2c)}\) Wavelength of a carbon nucleus: \(\lambda_C = \frac{h}{m_C v_C} = \frac{h}{(12 \times 1.67 \times 10^{-27} kg)(0.7c)}\) Wavelength of a helium nucleus: \(\lambda_{He} = \frac{h}{m_{He} v_{He}} = \frac{h}{(4 \times 1.67 \times 10^{-27} kg)(0.8c)}\) Wavelength of a lithium nucleus: \(\lambda_{Li} = \frac{h}{m_{Li} v_{Li}} = \frac{h}{(7 \times 1.67 \times 10^{-27} kg)(0.5c)}\)

Compare the de Broglie wavelengths

We can see that the denominators in each of the wavelengths determine their relative sizes. Since all velocities are fractional values of the speed of light, the smallest de Broglie wavelength will correspond to the particle with the highest mass and the highest velocity. Because a carbon nucleus has the highest mass and a velocity of \(70\%\) the speed of light, we expect that the de Broglie wavelength will be smaller than the other particle velocities. Therefore, the smallest de Broglie wavelength is from the carbon nucleus with a velocity of \(70\%\) the speed of light (option c).

Key concepts

Particle Physics

Particle physics is an exciting field of study focusing on the fundamental components of matter and the forces governing their interactions. Understanding these particles, such as electrons, protons, and different atomic nuclei, is crucial in diving deeper into the universe's fabric.
The standard particles we often hear about include quarks, leptons, and bosons. In every atom, particles like electrons zip around the nucleus, which is composed of protons and neutrons, themselves made of quarks.
With particle physics, we aim to uncover mysteries behind the very small portions of matter and how they interact at tiny scales. Here, forces like electromagnetism, quantum mechanics, and strong and weak nuclear forces play pivotal roles.
Mastery over these subjects helps shine light on fundamental processes governing the cosmos, offering answers to profound questions like the origins of mass and energy. Grasping particle interactions forms the backbone for many advanced theories in physics, including quantum mechanics.

Quantum Mechanics

Exploring the world through quantum mechanics can be both thrilling and mind-boggling. It’s the branch of physics that describes nature at the smallest scales of energy levels, typically of atoms and subatomic particles.
This field radically changes how we think about the behavior of particles compared to classical physics. Here, nothing is absolute! Particles like electrons do not have definite positions and pathways but rather probabilistic states. You might have heard of the famous Schrödinger's cat thought experiment – a prime illustration of the bizarre nature of quantum states, where particles can exist in multiple states until measured.
Quantum mechanics introduce us to wave-particle duality, explaining how particles can exhibit characteristics of both waves and particles. Concepts like superposition, entanglement, and uncertainty principle dominate this field, challenging our intuitive understanding of the universe. These quantum principles are not just theoretical; they have practical applications, paving the way for technologies like quantum computing and advanced semiconductors.

Wavelength Calculation

Understanding de Broglie wavelength enables us to peek into the fascinating dual nature of particles. The wavelength here is not about light but rather applied to particles, thanks to Louis de Broglie's groundbreaking work.
According to de Broglie, every moving particle or object has an associated wavelength, given by the formula \[ \lambda = \frac{h}{mv} \]where \(\lambda\) is the wavelength, \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is its velocity. This equation reveals that with decreasing mass or increasing velocity of the particle, the wavelength becomes shorter.
For massive particles like protons or atomic nuclei traveling at high speeds, the de Broglie wavelength is typically very small, making it challenging to observe. Yet, understanding their wavelength is crucial for delving into fields like quantum mechanics.
In everyday experiences with larger objects, this wave property is negligible. However, in atomic and subatomic scales, such calculations become significant, revealing deeper insights into the behavior of particles moving incredibly fast.