Animated Solution for Physics - Dual Nature of Matter and Radiation: An electron of mass $m$ and magnitude of charge $|e|$ initially at rest gets accelerated by a constant electric field $E$. The rate of change of de Broglie wavelength of this electron at time $t$ ignoring relativistic effects is

Imagine an electron, a tiny fundamental particle, placed gently into a uniform electric field. At first, it is perfectly still. But because it carries a negative charge, it cannot remain at rest. The electric field exerts an invisible, relentless pull on it, forcing it to move in the direction exactly opposite to the field lines. This simple setup is the gateway to understanding one of the most profound concepts in modern physics: the dual nature of matter.

In this problem, we are tasked with finding how the wave-like nature of this accelerating electron changes over time. Let's break down the physics and the math step-by-step.

Force and Acceleration

We begin with classical mechanics. When a charge $q$ is placed in an electric field $E$, it experiences an electrostatic force given by $F=qE$. For our electron, we are interested in the magnitude of this force to determine its acceleration. Using the magnitude of the electron's charge $|e|$, the force is:

According to Newton's second law of motion, a net force causes an object to accelerate. The magnitude of this acceleration $a$ is simply the force divided by the mass $m$ of the electron:

Because the electric field is uniform (constant), the force is constant, and therefore, the acceleration is also constant. This is a crucial realization because it allows us to use the standard equations of kinematics.

Velocity and Momentum

The electron starts from rest, meaning its initial velocity $u=0$. We want to find its velocity $v$ at any given time $t$. Using the first equation of motion, $v=u+at$, we can substitute our known values:

Now that we have the velocity, we can easily find the electron's linear momentum $p$. Momentum is the product of mass and velocity ($p=mv$). Substituting our expression for $v$:

Notice how elegantly the mass $m$ cancels out! The momentum of the electron in this specific setup depends only on its charge, the electric field strength, and the time elapsed.

Here is where classical mechanics meets quantum physics. In 1924, Louis de Broglie proposed that all moving particles have an associated wave nature. The wavelength $\lambda$ of this "matter wave" is inversely proportional to the particle's momentum $p$, connected by Planck's constant $h$:

By substituting the momentum we just calculated into de Broglie's equation, we get the wavelength of our accelerating electron as a function of time:

The core of the question asks for the rate of change of this de Broglie wavelength with respect to time. In the language of calculus, we need to find the derivative $\frac{d\lambda }{dt}$.

Let's set up the differentiation:

Since Planck's constant $h$, the charge magnitude $|e|$, and the electric field $E$ are all constants, we can pull them out of the derivative operator:

Using the power rule for differentiation, we know that the derivative of $t^{-1}$ is $-1\cdot t^{-2}$, or $-\frac{1}{t^2}$. Substituting this back in gives us our final result:

What does this final mathematical expression actually tell us? The negative sign is not just a mathematical artifact; it carries deep physical meaning. It signifies that the rate of change is negative, meaning the de Broglie wavelength is continuously decreasing as time goes on.

This makes perfect intuitive sense. As the electric field constantly accelerates the electron, its velocity and momentum increase. Because wavelength is inversely proportional to momentum ($\lambda \propto 1/p$), a growing momentum inevitably leads to a shrinking wavelength. The wave "bunches up" as the particle speeds up.

Finally, it is worth noting the problem's explicit instruction: "ignoring relativistic effects." Why is this important? If the electron were allowed to accelerate for a very long time, its speed would approach the speed of light $c$. In that relativistic regime, its mass would no longer be the constant rest mass $m_0$, but would increase according to $m=\frac{m_0}{\sqrt{1-v^2/c^2}}$. This would make the momentum calculation non-linear, and the rate of change of the wavelength would follow a much more complex, non-trivial curve. By ignoring relativity, we keep the physics focused on the beautiful intersection of basic kinematics and fundamental quantum theory.