Deduce the condition when the De-Broglie wavelength associated with an electron would be equal to that associated with a proton if a proton is `1836` times heavier than an electron.

Explanation

To deduce the condition when the de Broglie wavelength associated with an electron is equal to that associated with a proton, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

### Step 2: Set up the equation for both particles
For the electron (e) and the proton (p), we can write:
\[ \lambda_e = \frac{h}{m_e v_e} \]
\[ \lambda_p = \frac{h}{m_p v_p} \]
We want to find the condition when these two wavelengths are equal:
\[ \lambda_e = \lambda_p \]

### Step 3: Equate the two expressions
Setting the two expressions for the de Broglie wavelengths equal gives us:
\[ \frac{h}{m_e v_e} = \frac{h}{m_p v_p} \]
Since \( h \) is common on both sides, we can cancel it out:
\[ \frac{1}{m_e v_e} = \frac{1}{m_p v_p} \]

### Step 4: Rearrange the equation
Rearranging the equation leads to:
\[ m_p v_p = m_e v_e \]

### Step 5: Introduce the mass ratio
Given that the mass of the proton \( m_p \) is 1836 times the mass of the electron \( m_e \):
\[ m_p = 1836 m_e \]
Substituting this into the equation gives:
\[ 1836 m_e v_p = m_e v_e \]

### Step 6: Simplify the equation
Dividing both sides by \( m_e \) (assuming \( m_e \neq 0 \)):
\[ 1836 v_p = v_e \]

### Step 7: Express the condition
Thus, we find the condition when the de Broglie wavelengths are equal:
\[ v_e = 1836 v_p \]

### Final Result
The velocity of the electron must be 1836 times that of the velocity of the proton for their de Broglie wavelengths to be equal.

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