Animated Solution for Physics - Dual Nature of Matter and Radiation: The de-Broglie wavelength of a particle having kinetic energy $E$ is $\lambda$. How much extra energy must be given to this particle, so that the de-Broglie wavelength reduces to 75% of the initial value ?

Imagine a particle zooming through space. According to quantum mechanics, this particle isn't just a solid chunk of matter; it also behaves like a wave. The wavelength of this "matter wave" is called the de-Broglie wavelength, denoted by $\lambda$. The fundamental equation that bridges the particle's kinetic energy $E$ and its wave nature is:

Here, $h$ is Planck's constant and $m$ is the mass of the particle. Since we are dealing with the same particle throughout the problem, both $h$ and $m$ remain constant. This allows us to strip away the constants and focus purely on the relationship that matters:

This inverse square root proportionality tells us a beautiful physical truth: if you want to compress the wavelength (make it smaller), you must pump more kinetic energy into the particle.

We are given two distinct states for our particle.

In the Initial State, the particle has an energy $E_1=E$ and a wavelength ${\lambda }_1=\lambda$.

In the Final State, the wavelength is reduced to $75\%$ of its original value. Mathematically, $75\%$ is $0.75$, or the fraction $\frac{3}{4}$. So, ${\lambda }_2=0.75\lambda =\frac{3}{4}\lambda$. The question asks for the extra energy required, which means the final energy isn't just a new value; it's the original energy plus some added amount: $E_2=E+\Delta E$.

Using our proportionality, we can set up a ratio comparing the two states:

Now, we carefully substitute our known values into the ratio:

The $\lambda$ terms on the left side cancel out perfectly. The fraction $\frac{1}{0.75}$ is equivalent to $\frac{1}{3/4}$, which flips to become $\frac{4}{3}$. Our equation now looks much cleaner:

To liberate the energy terms from the square root, we must square both sides of the equation. Squaring $\frac{4}{3}$ gives us $\frac{16}{9}$:

We are now in the home stretch. Cross-multiplying to solve for $\Delta E$:

Distributing the $9$ on the right side:

Subtracting $9E$ from both sides isolates our target variable:

Finally, dividing by $9$ gives us the exact amount of extra energy required:

This perfectly matches option (b). The beauty of this problem lies in carefully tracking the difference between the total final energy and the extra added energy. Always read the wording carefully!