Bohr S Model For Hydrogen Atom

The value of in the state of a hydrogen atom is:

In this equation, the wave function describes the ground state (1s state) of the hydrogen atom. The Bohr radius is a fundamental constant and represents the most probable distance of the electron from the nucleus in the ground state.

Correct Answer:

To solve for , observe the exponential form of the wave function. The term represents a decaying function with respect to , and this function will approach 0 as becomes very large. The Bohr radius is the scale factor that defines the characteristic size of the hydrogen atom. Therefore, the inverse of the radial coordinate in this state is given by , which is consistent with the ground state of the hydrogen atom.

Here, is a spherical harmonic function which is dependent on the angular variables and . The spherical harmonic can be written as: where are the associated Legendre polynomials, and is a normalization constant. For the given function , we have .

Step 1: Apply the operator to the eigenfunction

We are tasked with finding the eigenvalue of the operator acting on the eigenfunction . Since the operator involves the derivative with respect to , the action of the operator on the angular part of the wavefunction is of primary interest.

Step 2: Effect of the derivative operator

The derivative operator acting on the spherical harmonic gives a factor of , the magnetic quantum number. Specifically, for , we have .

Step 3: Apply the sine operator

Now we apply the sine operator. The sine operator is a simple function that will act on the result of the derivative. Thus, we get the following result:

Conclusion: Eigenvalue of the operator

From the above result, we see that the operator acting on the eigenfunction gives the eigenvalue . Thus, the eigenvalue is: