Quantum Mirror – Quantum Mirror

We use candles and mirrors here to represent states and operators. Particularly symmetrical states are their own reflection. They are located right in the middle, dividing the mirror plane. All other states are not their own mirror image; instead, they come in pairs. In this case, there can only be an odd number of states: three, five, and so on.

Let’s have a closer look at the scenario where there are seven states. Here, we have a total of l=3 azimuthal nodal lines. In the most symmetric case, the three nodal lines are horizontal. The nodal rotation operators rotate one nodal line from horizontal to vertical. They thus generate the state m=+1 from the state m=0, with one nodal line rotating clockwise. This is reflected as m=-1, with one nodal line rotating anti-clockwise.

If we apply the nodal rotation operator again, it will rotate another nodal line from horizontal to vertical.

If the nodal rotation operator is applied once more, it will generate states in which all nodal lines are rotating about the z-axis, clockwise or anti-clockwise.

There are no more horizontal nodal lines. Applying the nodal rotation operator again will result in zero. The state will be destroyed.

We have thus discussed all potential vibrational states on the spherical surface in three dimensions. All the states shown here are eigenstates, when classified with regards to the D_z operator. The nodal rotation operators d₊ and d_– rotate nodal lines from horizontal to vertical, generating the states m₊ and m_– from the state m.

We start from the most symmetric eigenstates of the D_z operator on the mirror plane. We can then obtain all other eigenstates by applying the nodal rotation operators.

There is one free parameter in the figures of both operators and states on the spherical surface, which we have yet to analyse: If we change the distance between the states and their mirror images, everything else will remain the same. We can choose the distance arbitrarily without destroying the symmetry between the eigenstates.

In case of classical operators on the spherical surface, this distance delta has no greater significance. It varies depending on the application. In quantum physics, however, this is the crux of the matter. This distance is a universal physical constant ℏ, that is, 10^-34 Js. This value is absolutely immutable. It is equally valid on the Earth, in the Sun’s core, and inside a black hole. Therefore, in the transition to quantum physics, the operators are “only” scaled. The main difficulty lies not in the mathematical structure, but in interpreting this scaling.

Let’s compare operators and states on the spherical surface in quantum physics with those in classical physics. In quantum physics, the distance between the states is a universal physical constant. In classical physics, it is random.

But how do things stand in case of states? In both cases, the eigenstates can be classified as vibrations on a spherical surface, using the number and location of nodal lines. In this example, we have l=2, m=0. There is one crucial difference, though. In classical physics, vibrational states are directly observable. They are real vibrations on a spherical surface, such as a soap bubble.

In quantum physics, we have a vibration that is not directly observable. We could interpret it as a square root of a probability, or as a wave function.

When we cut the sphere in half, we obtain a vibration on a circular line. In the quantum dimension, however, the result is a section of a complex wave function.

For historical reasons, in quantum mechanics, the operators L_z, L₊, and L_– are called angular momentum operators. However, the only thing these operators have in common with the classical angular momentum is the physical unit Js. They have more similarities only in very few special cases.