An electron of mass $$m_e$$ and a proton of mass $$m_p = 1836 m_e$$ are moving with the same speed. The ratio of their de Broglie wavelength $$\frac{\lambda_{electron}}{\lambda_{proton}}$$ will be:

The de Broglie wavelength of a particle with mass $$m$$ moving with speed $$v$$ is given by $$\lambda = \frac{h}{mv}$$, where $$h$$ is Planck's constant.

For the electron: $$\lambda_{electron} = \frac{h}{m_e v}$$.

For the proton: $$\lambda_{proton} = \frac{h}{m_p v}$$.

Since both particles are moving with the same speed $$v$$, the ratio of their de Broglie wavelengths is:

$$\frac{\lambda_{electron}}{\lambda_{proton}} = \frac{h/(m_e v)}{h/(m_p v)} = \frac{m_p}{m_e}$$

Given that $$m_p = 1836 \, m_e$$, we get:

$$\frac{\lambda_{electron}}{\lambda_{proton}} = \frac{1836 \, m_e}{m_e} = 1836$$

Therefore, the ratio of the de Broglie wavelengths is 1836.