Question

Let \(\kappa\) be 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 magnitude of the wave number is a) \(\kappa\). b) \(2 \kappa\). c) \(\kappa / 2\) d) none of these.

Short answer

Answer: The magnitude of the new wave number is \(2 \kappa\), when the velocity of the particle is doubled.

Step-by-step solution

Write the given information and the equation relating wave number and momentum

We are given: - Initial wave number, \(\kappa\) - Initial velocity, \(v\) - Final velocity, \(2v\) We have to find the magnitude of the new wave number, \(\kappa'\). We know that the wave number is related to the momentum as - \(\kappa = \frac{2 \pi p}{h}\).

Write down the initial and final momentum

We know that momentum, \(p=mv\). Therefore, the initial momentum \(p_1=mv\) and the final momentum \(p_2 = m(2v) = 2mv\).

Calculate the initial and final wave number using the momentum

Using the initial momentum (\(p_1 = mv\)) and the equation relating momentum and wave number, we can calculate the initial wave number \(\kappa\): $$\kappa = \frac{2 \pi p_1}{h} = \frac{2 \pi mv}{h}$$ Similarly, using the final momentum (\(p_2 = 2mv\)): $$\kappa' = \frac{2 \pi p_2}{h} = \frac{2 \pi (2mv)}{h} = \frac{4 \pi mv}{h}$$

Compare the initial and final wave numbers

Now, we observe that the final wave number \(\kappa'\) is obtained by multiplying the initial wave number \(\kappa\) by a factor of 2: $$\kappa' = 2 \kappa$$ Thus, when the velocity of the particle doubles, the magnitude of the wave number also doubles.

Conclusion

The correct option is (b) \(2 \kappa\). When the velocity of the particle is doubled, the magnitude of the wave number is also doubled.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. In stark contrast to classical mechanics, quantum mechanics introduces the ideas that energy, momentum, and other quantities of a particle are restricted to discrete values (quantitization), objects have characteristics of both particles and waves (wave-particle duality), and there are limits to how precisely quantities can be known (uncertainty principle).

One of the quantum mechanical properties of a particle is described by its wave function, which provides information about the probability of finding a particle in a certain position or state. The wave number, represented by \(\kappa\), is related to the wavelength of the associated wave function and is an important concept when studying the dynamics of particles in quantum mechanics.

Momentum-Wavenumber Relationship

The momentum-wavenumber relationship is central to understanding wave mechanics and quantum mechanics. In quantum mechanics, the wave number \(\kappa\) of a particle, which is the number of wavelengths per unit distance, is directly proportional to the particle's momentum \(p\). The relationship is given by the equation:\[\kappa = \frac{2 \pi p}{h}\]

The Planck constant \(h\) is a fundamental physical constant in quantum mechanics that relates energy carried by a photon to the frequency of its associated electromagnetic wave. This equation signifies a deep connection between wave-like and particle-like properties since momentum is traditionally a classical particle property, while the wave number comes from wave description.

Particle Velocity

Particle velocity in classical mechanics simply refers to the rate at which a particle changes its position. However, in the context of quantum mechanics, particle velocity is also connected to the momentum and hence the energy of the particle. Due to the De Broglie hypothesis, we can associate wave characteristics with particles in motion, which establishes that every moving particle has an associated wavelength and therefore a wave number.

Relation to Wave Number

As demonstrated in the textbook example, when the velocity of a particle doubles, its momentum also doubles given by \(p=mv\), where \(m\) is the mass of the particle. Following from the momentum-wavenumber relationship, the wave number \(\kappa\) will also double, underscoring the interdependence between particle velocity, momentum, and wave number in quantum mechanical descriptions.

• Initial velocity \(v\) results in momentum \(mv\) and wave number \(\kappa\).
• Doubled velocity \(2v\) leads to doubled momentum \(2mv\) and hence wave number \(2\kappa\).