1. Interpret the question

We need to identify which Bohr postulate is not consistent with the modern quantum mechanical model.

2. Recall the key difference between Bohr model and quantum mechanics

• In Bohr's model, the electron is assumed to move in a fixed circular orbit around the nucleus.
• In quantum mechanics, an electron is not described as moving in a definite circular path. Instead, it is described by a wavefunction and is found in an orbital (a probability distribution).

So any statement involving a definite circular trajectory is not in agreement with quantum mechanics.

3. Check each option

Option A

"The electron in a H atom's stationary state moves in a circle around the nucleus."

This is a Bohr-specific classical picture. In quantum mechanics, electrons do not move in definite circular orbits.

So, Option A is not in agreement with quantum mechanical model.

Option B

"An atom can take only certain distinct energies $E_1, E_2, E_3$, etc. These allowed states of constant energy are called the stationary states of atom."

Quantum mechanics also predicts quantized energy levels for bound electrons.

So, Option B is in agreement.

Option C

"An atom in a stationary state does not emit electromagnetic radiation as long as it stays in the same state."

In quantum mechanics, a stationary state has a definite energy and does not radiate simply by remaining in that state.

So, Option C is in agreement.

Option D

"When an electron makes a transition from a higher energy stationary state to a lower energy stationary state, then it emits a photon of light."

Quantum mechanics also supports emission/absorption of photons during transitions between energy levels.

So, Option D is in agreement.

4. Final conclusion

The Bohr postulate that is not consistent with the quantum mechanical model is:

$\boxed{\text{A}}$