A particle has a de Broglie wavelength of 3.30 10-10 m. Then its kinetic energy quadruples. What is the particle's new de Broglie wavelength, assuming that relativistic effects can be ignored?
λ =h/p,
KE = p^2/2•m
p =sqrt(2•m•KE)
λ1 =h/sqrt(2•m•KE1),
λ2 =h/sqrt(2•m•KE2)=
= h/sqrt(2•m•4•KE1)=
= h/sqrt(4•p)= λ1/2 =
=1.65•10^-10 m.
To find the new de Broglie wavelength, we need to first understand the relationship between the de Broglie wavelength and kinetic energy.
According to de Broglie's equation, the de Broglie wavelength (λ) of a particle is given by:
λ = h / p,
where λ is the de Broglie wavelength, h is the Planck constant (6.626 × 10^-34 Js), and p is the momentum of the particle.
Since momentum (p) is related to kinetic energy (K) by the equation:
p = sqrt(2mK),
where m is the mass of the particle, we can substitute this expression for momentum in the de Broglie's equation:
λ = h / (sqrt(2mK)).
Now, let's denote the initial de Broglie wavelength as λ_1 and the final de Broglie wavelength as λ_2. Also, let's denote the initial kinetic energy as K_1 and the final kinetic energy as K_2.
Given that the initial de Broglie wavelength (λ_1) is 3.30 x 10^-10 m, we can rewrite the equation for the initial wavelength as:
λ_1 = h / (sqrt(2mK_1)).
Now, assuming that the relativistic effects can be ignored, we know that the mass of the particle remains constant when the kinetic energy is quadrupled. Therefore, the equation for the final de Broglie wavelength (λ_2) becomes:
λ_2 = h / (sqrt(2mK_2)).
Since we are interested in finding the new de Broglie wavelength after the kinetic energy quadruples, we can express the relationship between initial and final kinetic energy as:
K_2 = 4K_1.
By substituting this expression in the equation for the final wavelength, we get:
λ_2 = h / (sqrt(2m * 4K_1)).
Simplifying this equation further, we have:
λ_2 = h / (sqrt(8mK_1)).
Now, we can substitute the values into the equation to find the new de Broglie wavelength.