Question

Show that the energy eigenvalues for the free particle, \(E=\hbar^{2} k^{2} / 2 m,\) are consistent with the classical result \(E=(1 / 2) m \mathrm{v}^{2}\)

Short answer

The quantum mechanical energy eigenvalue formula for a free particle is given by \(E = \frac{\hbar^2 k^2}{2m}\). Using the De Broglie relation, we can relate the wave number (k) to the momentum (p) as \(p = \hbar k\), giving \(E = \frac{p^2}{2m}\). The classical momentum-velocity relation is \(p = mv\), and substituting this into the energy expression gives \(E = \frac{(mv)^2}{2m}\). Simplifying this expression, we obtain \(E =\frac{1}{2}mv^2\), which is the classical kinetic energy formula, showing the consistency between the quantum mechanical energy eigenvalues and the classical result.

Step-by-step solution

Relate the wave number k to the momentum p

The relationship between the wave number (k) and the momentum (p) of a particle in quantum mechanics is given by the De Broglie relation: \(p = \hbar k\).

Substitute the p expression into the energy eigenvalue formula

Now, we'll use the momentum expression to rewrite the quantum mechanical energy formula in terms of momentum: \[E = \frac{\hbar^2 k^2}{2m} = \frac{(\hbar k)^2}{2m} = \frac{p^2}{2m}\]

Relate the momentum p to the velocity v

The classic momentum-velocity relation is given by: \(p = mv\).

Substitute the v expression into the rewritten energy eigenvalue formula

Now, we'll replace the momentum (p) with the velocity (v) in terms of mass (m) in the quantum mechanical energy formula: \[E = \frac{p^2}{2m} = \frac{(mv)^2}{2m}\]

Simplify the expression

Simplify the energy formula expression to show that the energy eigenvalues in quantum mechanics are consistent with the classical result. \[E = \frac{(mv)^2}{2m} = \frac{m^2v^2}{2m} = \frac{1}{2}mv^2\] And there we have it! The quantum mechanical energy eigenvalue formula for a free particle, \(E = \frac{\hbar^2 k^2}{2m}\), has been shown to be consistent with classical expression \(E = \frac{1}{2}mv^2\) through a series of substitutions and simplifications.

Key concepts

De Broglie Wavelength

The De Broglie wavelength is a fundamental concept in quantum mechanics that links the wave-like behavior of particles with their particle-like properties. According to physicist Louis de Broglie, every particle with momentum also behaves like a wave, a groundbreaking idea that fused the worlds of classical physics and quantum mechanics.

The De Broglie wavelength, denoted as \(\lambda\), of a particle is inversely proportional to its momentum (p), and is given by the formula \(\lambda = \frac{h}{p}\), where \(h\) is Planck’s constant. For example, as a moving particle increases in speed, its momentum goes up and accordingly, its De Broglie wavelength decreases. This dual behavior is at the heart of wave-particle duality, one of the most intriguing aspects of quantum mechanics.

Momentum

In physics, momentum, denoted by \(p\), is a measure of the quantity of motion of a moving body and is a fundamental concept not only in classical mechanics but also in quantum mechanics. For a particle of mass \(m\) moving with velocity \(v\), the momentum is given by \(p = mv\).

Quantum mechanically, momentum is related to the wave aspect of particles through the De Broglie wavelength, as shown in the relationship \(p = \frac{h}{\lambda}\). This means that we can discuss the properties of particles in terms of their wave-like characteristics, like wavelength, and these properties will then have direct implications on the particle's momentum.

Free Particle Model

The free particle model in quantum mechanics describes a particle that is not subject to any external forces or potential energies; it is free to move through space. This idealized concept is important for understanding fundamental quantum behavior, as it simplifies the analysis by minimizing complications from interactions with the environment.

For a free particle, the energy is purely kinetic, and the energy eigenvalues explain the possible energy levels that a particle can have. These eigenvalues arise from the solution of the Schrödinger equation for a free particle and are represented as \(E = \frac{\hbar^2 k^2}{2m}\), where \(\hbar\) is the reduced Planck constant, \(k\) is the wave number, and \(m\) is the mass of the particle. Understanding this model is critical for deciphering more complex quantum systems where potential energy also plays a role.

Energy-Momentum Relationship

The energy-momentum relationship is at the core of uniting classical physics with quantum theory. It defines the total energy of a particle in terms of its momentum and other quantities like rest mass. In a quantum context, for a free particle, energy is related to momentum by the equation \(E = \frac{p^2}{2m}\), with \(E\) being the energy of the particle, \(p\) its momentum, and \(m\) its mass.

This relationship shows that if we know the momentum of a quantum particle, we can determine its energy. This is a central concept in quantum mechanics and is particularly striking when we compare it to the classical formula for kinetic energy, \(E = \frac{1}{2}mv^2\), which is the energy of an object in motion in classical mechanics. Despite the apparent differences, these two formulas are consistent with each other when we take into account the wave-particle duality expressed by the De Broglie hypothesis.