How does the wave mechanical model of the atom differ from the bohr model?

In the Bohr atom electrons are assumed to be fairly discrete, fairly physical particles, like very very small negatively charged balls which travel in circular motion (like the planets) around the positively charged nucleus at special radii, a result of “quantizing” the angular momentum (restricting it to list of allowed values), via ## m_{e} v r = n h/{2 pi} ##. This means that only particular energy are allowed, ##E_n =- {Z^2 R_e}/n^2 ##, where {E_n} is the energy of the nth orbit, Z is the charge on the nucleus (atomic number) and ##R_e## is the Rydberg energy, which is 13.6 eV.
The wave model is the full quantum mechanical treatment of the atom and essentially stands today. The electron is NOT discrete, instead in imagined a “smear” of probability.
The Bohr atom (sometimes called the Bohr-Rutherford model) was the result of two results of early 20th century science : the gold foil experiment preformed at Rutherford’s lab, by his minions, Hans Geiger and Ernest Marsden; and the developing quantum theory.
The gold foil experiment found that the atom consisted of a very small and heavy piece of positive charge, now called the nucleus, and smaller electrons which existed around it, stuck by electrostatic forces (negative charges like to hang out with things that are positively charges). The ONLY way this could be understood at the time was that the electrons go around the nucleus like planets around the sun. This is sometimes called the Rutherford model.
The quantum theory of light had fixed the ultraviolet catastrophe which occurred when modelling heat emission (called a Blackbody ) and was used by Einstein to explain the photoelectric effect. It involved treating the energy of light, which had previously been considered to be continuous (of any value), as now only occurring in discrete indivisible pieces called “quanta,” a piece of light, which we now call a photon, energy was equal to frequency times a constant, ##E_{ph}=h f## and it worked great.
This logic was applied to the atom, confining the electrons to special radii, by limiting the angular momentum ## m_{e} v r = n h/{2 pi} ##, and only particular energies and radii were allowed, ##E_n =- {Z^2 R_e}/n^2 ##, where {E_n} is the energy of the nth orbit, Z is the charge on the nucleus (atomic number) and ##R_e## is the Rydberg energy, which is 13.6 eV.
This model for the first time explained the spectra of the hydrogen atom,a special pattern of light. It was caused by electrons rising and falling between these special radii, called orbits and emitting or absorbing light equal to the difference in energy required. This was HUGE. Scientists had been measuring spectra for decades, but had had no explanation for the patterns of light atoms and molecules produced. Now we had hydrogen done. With some tweaking it also allowed from some explanation of the valences. However, it could not explain the spectra of any element other then hydrogen or the subtleties of valences or the “blocking” in the periodic table.
So a semi-quantum treatment of electrons moving about near a nucleus was a great step forward, but not far enough. The wave mechanical model goes further, a full quantum treatment, it had to wait for quantum mechanics to exist. The missing pieces were the development of the Pauli exclusion principal, wave-particle duality, due mostly to Louis de Broglie, that all particles exist in a blurry wave of probability and the equation that governs them is the Schrödinger Equation, both developed in the mid 1920’s.
The Wave model of the atom come from building, then solving the Schrödinger Equation for electrons bond by a nucleus, while there are may may refinements to this, it essentially stand today as how we model matter. The details can be found in a 3rd year QM course, but you care about the results! The wave model explains atomic shell filling, the solving gives several types of orbitals, each with different allowed electrons, the s shell with 2, the p shell with 6, the shell with 10 and the f shell with 14. This explains the
“blocks” in the periodic table, ie each row of transitional metals are filling a d shell, the first 3d, second 4d and the third fills 5d. Orbitals are probability maps of where the electron tends to be and bonds are two atomic orbitals overlapping and joining.
It also explain ALL atomic spectra, in extreme detail and molecular spectra of what we’ve had time to compute and when applied to crystals explains the properties of solids. . It is WILDLY successful and but does come with a draw back. In the Bohr model electron were easier to understand, they were charged balls, now we have blurry probability distributions. You brain was designed to picture things on the scale of basket balls, you can understand how they how and …are. Electron DO NOT BEHAVE LIKE BASKET BALLS. Quantum results can be hard to get you hard around, but that’s ok, it’s very very well tested, this is how the world is.