Atomic Models – States, Operations and the Hydrogen Atom

Quantum physics has advanced since Bohr developed his atomic model over 100 years ago. An important step in developing the Bohr atomic model was introducing operators and states to describe and manipulate an electron in the quantum dimension. How do these two models relate to each other? The Bohr atomic model was first refined by De Brogli, who assigned standing waves to electron orbits. In this example, we have l=2. Accordingly, there are electron positions with two nodal lines in the quantum dimension.

In quantum physics, we interpret the postulate L=n·ℏ differently. Angular momentum becomes a rotation operator. In order to analyse electron states, let us start with eigenstates with regard to the rotation operator about the z-axis.

In quantum physics, L=n·ℏ does not hold. General superposition states do not have a specified angular momentum. They have only their respective eigenstates. Therefore, ℏ is the smallest possible measurable difference of the angular momentum.

Analysing classical vibrational states and operators will help us to understand quantum states. Between all possible vibrational states on a glass pane, let us consider the eigenstates of the energy operator. These eigenstates have two types of nodal lines: radial or azimuthal, or combinations of both. These fundamental frequencies will help us understand the respective quantum states, even though we cannot know what these states are really like.

In a hydrogen atom, vibrational states have roughly the same binding energy on each orbit n=1, 2, 3, and so on. That means they have the same number of nodal lines. The innermost orbit n=1 does not have any nodal lines. The next orbit, n=2, has one nodal line, either radial or azimuthal. The third orbit, n=3, has two nodal lines. Those can be two radial lines, or one radial and one azimuthal, or two azimuthal lines; thus l=0, 1, 2.

If the electron were a two-dimensional object, this is how its spectrum would look like in the quantum dimension. In three-dimensional space, the spherical symmetry rules out other options for radial nodal lines. In case of one azimuthal nodal line in three dimensions, there are more possibilities, because that nodal line can be horizontal or vertical to the z-axis. Each new azimuthal nodal line will result in two additional states.

If the electron were a three-dimensional object, this is how its spectrum would look like in the quantum dimension. In fact, the quantum dimension has more to offer. A spin is an additional vibrational mode of the electron state. In three dimensions, it has two possible eigenstates with regards to the rotation operator: “up” and “down”. This additional degree of freedom of the electron is visualized here using another mirror plane. The number of eigenstates doubles. The electron state is thus a product of position and spin state in the quantum dimension.

The Bohr atomic model could therefore be developed as follows, so that it can be applied in the quantum dimension:

First, the angular momentum becomes the rotation operator. h-bar thus represents the smallest possible measurable difference of the angular momentum.

Secondly, by counting nodal lines, we can explain why there are exactly 2 n squared eigenstates per orbit or energy level n, where n is equal to 1, 2, 3, and so on.