Quantum Mechanical Model Of Atom

The angular momentum of an electron in the hydrogen atom is given as: This represents the third allowed orbit in the Bohr model. According to Bohr's model, the kinetic energy of an electron in a hydrogen atom is related to its angular momentum by the formula: But we can also use the formula for the energy levels of the hydrogen atom, which are given by: For (since the given angular momentum corresponds to the third orbit), we have: The kinetic energy is the negative of the total energy in this case (since the potential energy is double the kinetic energy and both energies are negative), so:

The correct answer is (B) : 1.51 eV.

Where:

• = de Broglie wavelength
• h = Planck's constant
• p = momentum of the particle

The momentum of the particle can be related to its kinetic energy ( ) as:

Where:

• m = mass of the particle
• K = kinetic energy of the particle

Now, consider the situation where the kinetic energy of the particle is:

We can calculate the new wavelength ( ) using the same equation:

Substitute the expression for momentum ( ) in terms of the new kinetic energy ( ):

Substitute into the above equation:

Simplifying the expression:

We can see that the new wavelength is equal to:

Therefore, the correct option is: (D)