Question

What is the wavelength of the matter wave associated with a proton moving at 341 m/s?... with a 159-kg astronaut (including her spacesuit) moving at the same speed? ... with Earth moving along its orbit around the Sun?

Answer 1

The de Broglie wavelentgh (λ) of a massive particle is related to its momentum (p) by:

λ = h/p

where h is Planck's constant (6.626*10^(-34) kg*m^2/s)

The mass of a proton is about 1.673*10^-27 kg. The classical momentum of a proton moving at 391 m/s is p = v*m = (341 m/s)*(1.673*10^-27 kg). The de Broglie wavelength of that proton is then:

λ = (6.626*10^(-34) kg*m^2/s)/((341 m/s)*(1.673*10^-27 kg))

λ = 1.16*10^(-9) m = 1.16 nm

The wavelength of a 159 kg astronaut moving at the same speed would be:

λ = (6.626*10^(-34) kg*m^2/s)/((341 m/s)*(159kg))

λ = 1.22*10^(-38) m, which is a fantastically small number

The mass of the Earth is about 6*10^24 kg, and its average orbital speed is about 2*10^5 m/s). The de Broglie wavelength of the Earth would be:

λ = (6.626*10^(-34) kg*m^2/s)/((2*10^5 m/s)*(6*10^24 kg))

λ = 5.5*10^(-64) m