Question

Find the ratio `(lambda_a/lambda_p)` of the de Broglie wavelength λ_a associated with an alpha particle to de Broglie wavelength λ_p associated with a proton if both are moving with

1. same velocity
2. same kinetic energy.

Answer 1

For an alpha particle (α) and a proton (p), we use the standard mass ratio where the mass of an alpha particle is approximately four times the mass of a proton (m_α = 4m_p).

The ratio of their de Broglie wavelengths `(lambda_alpha/lambda_p)` is:

a. For the same velocity:

λ = `h/(mv)`

So,

`lambda_a/lambda_p = m_p/m_alpha`

= `m_p/(4m_p)`

= `1/4`

b. For the same kinetic energy:

Momentum (p) = `sqrt(2 m K)`

So,

λ = `h/(sqrt(2 m K)) prop 1/sqrt m`

Thus,

`lambda_alpha/lambda_p = sqrt(m_p/m_alpha)`

= `sqrt(1/4)`

= `1/2`