The kinetic energy of a particle is equal to the energy of a photon. The particle moves at 6.9% of the speed of light. Find the ratio of the photon wavelength to the de Broglie wavelength of the particle. Take the speed to be non-relativistic.

To find the ratio of the photon wavelength to the de Broglie wavelength of the particle, we can use the formulas for the kinetic energy and the energy of a photon.

The kinetic energy of a non-relativistic particle can be calculated using the formula:

KE = (1/2) * m * v^2

where KE is the kinetic energy, m is the mass of the particle, and v is its velocity.

The energy of a photon can be calculated using the formula:

E = h * c / λ

where E is the energy of the photon, h is the Planck constant, c is the speed of light, and λ is the wavelength of the photon.

Given that the kinetic energy of the particle is equal to the energy of the photon, we can equate these two formulas:

(1/2) * m * v^2 = h * c / λ

We are asked to find the ratio of the photon wavelength (λ) to the de Broglie wavelength of the particle (λ_dB). The de Broglie wavelength is given by the formula:

λ_dB = h / (m * v)

To find the ratio, we can substitute the value of λ_dB in the given equation:

(1/2) * m * v^2 = h * c / λ_dB

Now, we can solve this equation to find the ratio.

Given that the particle moves at 6.9% of the speed of light, we have v = 0.069 * c, where c is the speed of light.

Substituting this value in the above equation, we get:

(1/2) * m * (0.069 * c)^2 = h * c / (h / (m * (0.069 * c)))

Simplifying, we find:

(1/2) * m * (0.069 * c)^2 = m * (0.069 * c)

Canceling out the mass and rearranging the terms, we get:

(0.069 * c)^2 = 2 * (0.069 * c)

Dividing both sides by (0.069 * c), we get:

0.069 * c = 2

Dividing both sides by 0.069, we find:

c ≈ 2 / 0.069

c ≈ 28.99

Therefore, the speed of light is approximately 28.99 times greater than the given velocity of the particle.

Now, to find the wavelength ratio, we can substitute the values back into the equations.

For the wavelength of the photon (λ), we use:

λ = h * c / E

For the de Broglie wavelength of the particle (λ_dB), we use:

λ_dB = h / (m * v)

Now, we can calculate the ratio of λ to λ_dB:

λ / λ_dB = (h * c / E) / (h / (m * v))

Canceling out the h terms and simplifying, we get:

λ / λ_dB = (c / E) * (m * v)

Substituting the values we found earlier for c and v:

λ / λ_dB = (28.99 / E) * (0.069 * m)

We are not given the specific values of the mass (m) of the particle or the energy (E) of the photon. Therefore, we cannot calculate the numerical value of the wavelength ratio. However, we have derived the general formula for the ratio, and once we have the values for mass and energy, we can use the formula to calculate the ratio.