Final answer:
The de Broglie wavelength of a proton divided by the wavelength of a photon is directly proportional to the product of Planck's constant, the wavelength of the photon, and the reciprocal of the proton's velocity.
Step-by-step explanation:
The energy of a photon is given by the equation E = hf, where E is the energy, h is Planck's constant, and f is the frequency of the wave. The kinetic energy of a proton is given by the equation KE = 1/2mv^2, where m is the mass of the proton and v is its velocity.
To compare the energy of a photon with the kinetic energy of a proton, we can equate the two equations:
E = KE => hf = 1/2mv^2
Since we have the equation for the energy of a photon, we can express the wavelength of the photon as λ2 = c/f, where c is the speed of light.
Similarly, the de Broglie wavelength of a proton is given by the equation λ1 = h/mv.
If we divide the de Broglie wavelength of the proton by the wavelength of the photon, we get:
λ1/λ2 = (h/mv)/(c/f)
Since f = c/λ2 and v = c/λ1, we can substitute these values:
λ1/λ2 = (h/mv)/(c/(c/λ2)) = (h/mv)/(1/λ2) = (hλ2)/(mv)
Therefore, λ1/λ2 is proportional to hλ2/mv. This means that the de Broglie wavelength of the proton divided by the wavelength of the photon is directly proportional to the product of Planck's constant, the wavelength of the photon, and the reciprocal of the proton's velocity.