Question

Let \(\kappa\) be the magnitude of the wave number of a particle moving in one dimension with velocity \(v\). If the velocity of the particle is doubled, to \(2 v,\) then the wave number is: a) \(\kappa\) b) \(2 \kappa\) c) \(\kappa / 2\) d) none of these

Short answer

Question: When the velocity of a particle moving in one dimension is doubled, its wave number: a) stays the same b) is doubled c) is halved d) increases by a factor of 4 Answer: b) is doubled

Step-by-step solution

Recall the de Broglie wavelength equation

The de Broglie wavelength can be determined using the equation: \( \lambda = \dfrac{h}{m \cdot v} \) where \(\lambda\) is the wavelength, \(h\) is the Planck's constant, \(m\) is the mass of the particle, and \(v\) is its velocity.

Calculate the initial wave number

The wave number \(\kappa\) is the reciprocal of the wavelength. We can find the initial wave number using the following equation: \( \kappa = \dfrac{1}{\lambda} \) By substituting the de Broglie wavelength equation, we get: \( \kappa = \dfrac{m \cdot v}{h} \)

Determine the new wavelength after doubling the velocity

Now we need to find the new wavelength when the velocity is doubled to \(2v\). Using the de Broglie wavelength equation, we get: \( \lambda' = \dfrac{h}{m \cdot (2v)} = \dfrac{h}{2m \cdot v} \)

Calculate the new wave number

Now we need to find the new wave number. We can use the following equation: \( \kappa' = \dfrac{1}{\lambda'} \) Substituting the new wavelength, we get: \( \kappa' = \dfrac{1}{\dfrac{h}{2m \cdot v}} = \dfrac{2m \cdot v}{h} = 2 \kappa \) The new wave number when the velocity is doubled is \(2 \kappa\). Hence, the correct answer is option (b).

Key concepts

wave number

The concept of wave number, denoted as \( \kappa \), is essential in understanding wave phenomena. In simple terms, it tells us how many wavelengths fit into a certain distance in space. The wave number is inversely related to the wavelength \( \lambda \). Mathematically, it's expressed as:

• \( \kappa = \dfrac{1}{\lambda} \)

When dealing with particle motion, the wave number indicates the quantum mechanical properties of the particle as it behaves like a wave.
This relationship comes from the idea that matter exhibits both particle-like and wave-like characteristics, known as wave-particle duality. Understanding this helps when calculating changes in the wave number in response to varying particle velocities.

Planck's constant

Planck's constant is a fundamental constant in physics, symbolized by \( h \). It plays a crucial role when discussing the quantum mechanics of particles. Its value, approximately \( 6.626 \times 10^{-34} \text{Js} \), was pivotal in the development of quantum theory.
Planck's constant establishes a relationship between energy and frequency for electromagnetic waves through:

• \( E = h \cdot f \)

In the context of de Broglie wavelengths, Planck's constant helps link a particle's momentum with its wavelength:

• \( \lambda = \dfrac{h}{m \cdot v} \)

Thus, understanding \( h \) is instrumental for calculating the behavior of particles as waves.

particle velocity

Particle velocity, represented as \( v \), plays a significant role in wave mechanics. It is the speed at which a particle travels in a particular direction.
According to the de Broglie hypothesis, the velocity of a particle affects its wavelength and wave number unfailingly. When the velocity is doubled, both the wavelength and wave number change:

• Initial velocity \( = v \)
• Doubled velocity \( = 2v \)

The effect of changing velocity helps in understanding how dynamic systems behave at quantum level and demonstrate the wave-particle duality.

wavelength calculation

Wavelength calculation is vital in determining how waves behave in different scenarios. For a particle, the de Broglie wavelength \( \lambda \) can be calculated using:

• \( \lambda = \dfrac{h}{m \cdot v} \)

This formula connects the physical properties of the particle such as mass \( m \) and velocity \( v \) to its wave characteristics.
Upon increasing the velocity of the particle to \( 2v \), the formula adapts to:

• \( \lambda' = \dfrac{h}{m \cdot 2v} = \dfrac{h}{2m \cdot v} \)

This inverse relationship between velocity and wavelength means as velocity increases, the wavelength decreases, and vice versa. Accurate wavelength calculation is essential for deeper insights into the behavior of microscopic and macroscopic systems alike.