The binding energy of a H-atom considering an electron moving around a fixed nuclei (proton), is
`B = - (me^(4))/(8n^(2)epsi_(0)^(2)h^(2))` (m= electron mass)
If one decides to work in a frame of refrence where the electron is at rest, the proton would be movig around it. By similar arguments, the binding energy would be :
`B = - (me^(4))/(8n^(2)epsi_(0)^(2)h^(2))` (M = proton mass)
This last expression is not correct, because

Options

A. n would not be integral
B. Bohr-quantisation applies only two electron
C. the fram in which the electron is at rest is not inertial
D. the motion of the proton would not bein circular orbits, even approximately.

Explanation

To understand why the expression for the binding energy of the hydrogen atom is not correct when considering the electron at rest and the proton moving around it, we can break down the reasoning step by step:

### Step 1: Understanding Binding Energy
The binding energy of an atom is defined as the energy required to separate the electron from the nucleus. For a hydrogen atom, when the electron is considered to be moving around a fixed proton, the binding energy is given by the formula:
\[ B = - \frac{me^4}{8n^2 \epsilon_0^2 h^2} \]
where \( m \) is the mass of the electron, \( e \) is the charge of the electron, \( \epsilon_0 \) is the permittivity of free space, \( n \) is the principal quantum number, and \( h \) is Planck's constant.

### Step 2: Changing the Frame of Reference
If we switch to a frame of reference where the electron is at rest, the proton will be moving around the electron. The binding energy in this new frame is incorrectly assumed to be:
\[ B = - \frac{Me^4}{8n^2 \epsilon_0^2 h^2} \]
where \( M \) is the mass of the proton.

### Step 3: Analyzing the Non-Inertial Frame
In the frame where the electron is at rest, we are dealing with a non-inertial frame because the proton is moving. In non-inertial frames, we must account for fictitious forces, such as centrifugal force, which acts outward from the center of rotation.

### Step 4: Centripetal Force and Acceleration
In the original frame, the centripetal force acting on the electron is provided by the electrostatic attraction between the electron and proton. This force can be expressed as:
\[ F = \frac{ke^2}{r^2} \]
where \( k \) is Coulomb's constant and \( r \) is the distance between the electron and proton.

In the non-inertial frame (electron at rest), the proton experiences a centrifugal force that must be considered. The fictitious centrifugal force will be equal to the centripetal force required to keep the proton in circular motion around the electron.

### Step 5: Mass Comparison
Since the mass of the proton \( M \) is much larger than the mass of the electron \( m \), the centripetal acceleration experienced by the proton will be very small:
\[ a_c = \frac{F}{M} \]
This means that the proton will not accelerate significantly, leading to an incorrect assumption about the binding energy.

### Conclusion
The assumption that the binding energy can be calculated using the mass of the proton in the non-inertial frame is flawed because the dynamics of the system change significantly due to the mass difference and the introduction of fictitious forces. Therefore, the expression for the binding energy derived in this frame is not valid.