If a proton and an electron have the same kinetic energy, which has the longer de Broglie wavelength?

Options

A. Electron
B. Proton
C. Both have the same wavelength
D. None of the above

Answer

To determine which particle, a proton or an electron, has a longer de Broglie wavelength when both have the same kinetic energy, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (\( \lambda \)) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

### Step 2: Express momentum in terms of kinetic energy
The momentum \( p \) of a particle can be expressed in terms of its kinetic energy (\( KE \)):
\[ p = mv \]
where \( m \) is the mass of the particle and \( v \) is its velocity. The kinetic energy is given by:
\[ KE = \frac{1}{2} mv^2 \]
From this, we can express \( v \) as:
\[ v = \sqrt{\frac{2 KE}{m}} \]
Substituting this into the momentum equation gives:
\[ p = m \sqrt{\frac{2 KE}{m}} = \sqrt{2m KE} \]

### Step 3: Substitute momentum into the de Broglie wavelength formula
Now we can substitute this expression for momentum into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{p} = \frac{h}{\sqrt{2m KE}} \]

### Step 4: Compare the de Broglie wavelengths of the proton and electron
Let’s denote the mass of the proton as \( m_p \) and the mass of the electron as \( m_e \). Since both particles have the same kinetic energy (\( KE \)), we can write:
\[ \lambda_p = \frac{h}{\sqrt{2m_p KE}} \quad \text{(for proton)} \]
\[ \lambda_e = \frac{h}{\sqrt{2m_e KE}} \quad \text{(for electron)} \]

### Step 5: Analyze the relationship between the wavelengths
To compare the two wavelengths, we can look at the ratio:
\[ \frac{\lambda_p}{\lambda_e} = \frac{\sqrt{2m_e KE}}{\sqrt{2m_p KE}} = \sqrt{\frac{m_e}{m_p}} \]
Since the mass of the electron (\( m_e \)) is much smaller than the mass of the proton (\( m_p \)), the ratio \( \frac{m_e}{m_p} \) is less than 1. Therefore:
\[ \lambda_p < \lambda_e \]
This means that the de Broglie wavelength of the electron is longer than that of the proton.

### Conclusion
Thus, if a proton and an electron have the same kinetic energy, the electron has the longer de Broglie wavelength.

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