Question

An accelerator boosts a proton's kinetic energy so that the de Broglie wavelength of the proton is \(3.5 \cdot 10^{-15} \mathrm{~m}\) What are the momentum and energy of the proton?

Short answer

Answer: The kinetic energy of the proton is approximately \(1.071 \times 10^{-13} \mathrm{J}\).

Step-by-step solution

Find the de Broglie wavelength formula

The de Broglie wavelength formula relates the wavelength (λ) of a particle to its momentum (p) by h, the Planck's constant. The formula is given by: $$ \lambda = \frac{h}{p} $$ Where: - λ (lambda) is the de Broglie wavelength - h is the Planck's constant, approximately \(6.626 \times 10^{-34} \mathrm{Js}\) - p is the momentum of the particle We are given the de Broglie wavelength as \(3.5 \cdot 10^{-15} \mathrm{~m}\). We can use this value to find the momentum of the proton.

Calculate the momentum of the proton

Now we will use the de Broglie wavelength formula to calculate the momentum of the proton. Rearrange the formula for momentum: $$ p = \frac{h}{\lambda} $$ Plug in the values of the Planck constant and the de Broglie wavelength: $$ p = \frac{6.626 \times 10^{-34} \mathrm{Js}}{3.5 \cdot 10^{-15} \mathrm{m}} $$ Now calculate the momentum: $$ p \approx 1.894 \times 10^{-19} \mathrm{kg \cdot m/s} $$ The momentum of the proton is approximately \(1.894 \times 10^{-19} \mathrm{kg \cdot m/s}\).

Calculate the kinetic energy of the proton

The kinetic energy of the proton can be found using its momentum and mass. The formula for the kinetic energy is given by: $$ KE = \frac{p^2}{2m} $$ Where: - KE is the kinetic energy - p is the momentum of the proton (calculated in the previous step) - m is the mass of the proton, approximately \(1.673 \times 10^{-27} \mathrm{kg}\) Now, we will plug in the momentum and mass of the proton into the formula: $$ KE = \frac{(1.894 \times 10^{-19} \mathrm{kg \cdot m/s})^2}{2(1.673 \times 10^{-27} \mathrm{kg})} $$ Finally, calculate the kinetic energy of the proton: $$ KE \approx 1.071 \times 10^{-13} \mathrm{J} $$ The kinetic energy of the proton is approximately \(1.071 \times 10^{-13} \mathrm{J}\).

Key concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. Unlike classical physics, where objects follow deterministic paths, quantum mechanics introduces a probabilistic approach to understanding the states of particles.
A key feature of quantum mechanics is the wave-particle duality, which means that particles like electrons and protons exhibit both wave-like and particle-like properties. The de Broglie wavelength is a direct result of this duality, stating that any moving particle has an associated wavelength, which is inversely proportional to its momentum. This has profound implications for our understanding of the microscopic world, as it links the concepts of momentum and wave behavior, allowing for calculations of particle positions and momenta with varying degrees of certainty due to the Heisenberg uncertainty principle.

Particle Momentum

In the realm of both classical and quantum mechanics, momentum is a fundamental property of moving objects. It's defined as the product of an object's mass and its velocity. Momentum has both magnitude and direction, making it a vector quantity.
The calculation of a particle's momentum in quantum mechanics becomes particularly interesting due to the de Broglie hypothesis, which allows us to relate momentum to wavelength. Based on the formula \( p = \frac{h}{\lambda} \), where \( p \) is the momentum, \( h \) is Planck's constant, and \( \lambda \) is the de Broglie wavelength, we can determine the momentum of particles that exhibit wave-like properties such as electrons or protons when accelerated.

Kinetic Energy Calculation

Kinetic energy represents the energy an object possesses due to its motion. It's a scalar quantity, which means it only has magnitude and no direction. For classical objects, kinetic energy is calculated using the expression \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object.
In the context of a subatomic particle such as a proton in quantum mechanics, the calculation of kinetic energy can be related to particle momentum. Using the relation \( KE = \frac{p^2}{2m} \), where \( p \) is the momentum and \( m \) is the mass, we are able to compute the kinetic energy of particles. This formula is especially useful when we recall that the particle's momentum is not always readily observable, but its de Broglie wavelength can be.

Planck's Constant

Planck's constant (\( h \)) is a fundamental constant in quantum mechanics that has the dimensions of action, which is energy multiplied by time. Its value is approximately \( 6.626 \times 10^{-34} \) joule-seconds.
This constant is pivotal because it quantifies the relationship between the energy of a photon (the quantum of light) and the frequency of its associated electromagnetic wave. In essence, Planck's constant bridges the gap between the quantized world of subatomic particles and the more familiar realm of classical physics. It is also central to the calculation of the de Broglie wavelength, which showcases its crucial role in understanding quantum behavior and how energy and momentum at the quantum level relate to broader physical phenomena.