A particle has a de Broglie wavelength of 3.30 x 10^-10 m. Then its kinetic energy quadruples. What is the particle's new de Broglie wavelength, assuming that relativistic effects can be ignored?
L = h/p = h/mv
v = h/mL
v^2 = h^2/(m^2 L^2)
m v^2 = h^2/(m L^2) (forget the 1/2, cancels)
new energy
m Vnew^2 = h^2/(m Lnew^2) = 4 h^2/(m L^2)
so
1/Lnew^2 = 4/L^2
4 Lnew^2 = L^2
2 Lnew = L
Lnew = (1/2) L
1.65*10^-10 m
To determine the particle's new de Broglie wavelength, we need to consider the relationship between the de Broglie wavelength and the kinetic energy of a particle. The de Broglie wavelength (λ) is given by the equation:
λ = h / p
where h is the Planck constant (h ≈ 6.626 x 10^-34 J·s) and p is the momentum of the particle.
The momentum of a particle can be calculated using the equation:
p = √(2mE)
where m is the mass of the particle and E is its kinetic energy.
Given that the initial de Broglie wavelength (λ1) is 3.30 x 10^-10 m, we can rearrange the first equation to solve for the momentum (p1):
p1 = h / λ1
Substituting the given values, we find:
p1 = (6.626 x 10^-34 J·s) / (3.30 x 10^-10 m)
Now that we have the momentum, we can calculate the initial kinetic energy (E1) using the second equation:
E1 = p1^2 / (2m)
Next, we need to find the new kinetic energy (E2) when the initial kinetic energy quadruples. Given that the relation between the initial and new kinetic energies is:
E2 = 4E1
We can then substitute the initial kinetic energy into the equation to find E2:
E2 = 4 x E1
With the new kinetic energy, we can calculate the new momentum (p2) using the same equation:
p2 = √(2mE2)
Finally, we can use the new momentum to find the new de Broglie wavelength (λ2) using the first equation:
λ2 = h / p2
By following these steps, we can calculate the new de Broglie wavelength of the particle.