Answer:

To find the percentage increase in the de Broglie wavelength of a particle when its energy is reduced to one-fourth, we need to understand the relationship between energy and de Broglie wavelength.

The de Broglie wavelength (λ) is given by the equation:

λ = h / p

where h is the Planck's constant (6.626 x 10^-34 Js) and p is the momentum of the particle.

The momentum of a particle can be expressed as:

p = √(2mE)

where m is the mass of the particle and E is its energy.

Now, let's consider the initial energy of the particle as E1 and its corresponding de Broglie wavelength as λ1. When the energy is reduced to one-fourth, the new energy is E2 = E1/4. We need to find the new de Broglie wavelength λ2.

Step 1: Expressing the initial and final de Broglie wavelengths in terms of energy:

Using the equations mentioned above, we can express the initial and final de Broglie wavelengths as:

λ1 = h / √(2mE1)

λ2 = h / √(2mE2) = h / √(2m(E1/4)) = h / (2√(mE1))

Step 2: Calculating the percentage increase in the de Broglie wavelength:

To find the percentage increase, we need to calculate the difference between λ2 and λ1 and express it as a percentage of λ1:

Δλ = λ2 - λ1 = h / (2√(mE1)) - h / √(2mE1) = h / √(2mE1) * (1/2 - 1) = -h / (2√(mE1))

Now, let's calculate the percentage increase:

Percentage increase = (Δλ / λ1) * 100 = (-h / (2√(mE1)) / (h / √(2mE1))) * 100 = -1/2 * 100 = -50%

Therefore, the percentage increase in the de Broglie wavelength when the energy of a particle is reduced to one-fourth is -50%.

Note: The negative sign indicates a decrease in the de Broglie wavelength, which is expected as the energy decreases.