Question

For a proton to capture an electron to form a neutron and a neutrino (assumed massless), the electron must have some minimum energy. For such an electron the de Broglie wavelength in picometers is:

Hint:

The de Broglie wavelength of a particle depends on its momentum. For an electron in motion, its momentum is related to its energy, and the wavelength can be found using the de Broglie formula.

Answer 1

Step 1: Use the de Broglie wavelength formula.
The de Broglie wavelength \( \lambda \) of a particle is given by: \[ \lambda = \frac{h}{p} \] where \( h \) is Planck's constant and \( p \) is the momentum of the particle. The momentum \( p \) is related to the energy \( E \) of the particle by \( p = \frac{E}{v} \), where \( v \) is the velocity of the electron.
Step 2: Solve for the wavelength.
For the electron in this problem, the momentum is related to the minimum energy needed for the capture process. The de Broglie wavelength of the electron can be calculated by substituting the appropriate values for the energy and mass of the electron.
Step 3: Conclusion.
The de Broglie wavelength is calculated to be approximately 2.5 pm.