Einstein famously expressed his discomfort with the concept of observation-dependent reality by asking Bohr, "Does the moon exist when we are not looking at it?" Another, perhaps more provocative idea, is known as "Wigner's Friend." Eugene Wigner expanded upon Schrödinger's cat thought experiment by introducing a second observer. In Wigner's scenario, the first observer measures the system by opening the box, but a second observer, unaware of the first observer's result, then observes the system. This raises a critical question: is the cat still in a superposition after the first observer's measurement, as long as the second observer remains ignorant of the outcome?

Erwin Schrödinger did not fundamentally change his mind about quantum mechanics, but he did have reservations about certain interpretations of it, particularly the Copenhagen interpretation suggested by Neils Bohr. Schrödinger's cat thought experiment was actually a critique of the idea that a quantum system remains in superposition until it is observed. While Schrödinger contributed significantly to the development of quantum mechanics, he was uncomfortable with the notion of collapse of the wave function and the role of the observer. He preferred to think of quantum mechanics as a purely deterministic theory, rather than one that involved inherent randomness.

Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics which is as follows; $$i\hbar \frac{d}{dt}|\psi> = \hat{H}|\psi>$$

Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system.

The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. In simpler terms the equation is; $$E\psi\ = \hat{H}\psi$$ One of the concepts that we must address is that of functions and operators. The difference in real terms is pure mathematical semantics both functions and operators operate upon values whether as a parameter or directly. So what we have is; $$iE_{kinetic} \frac{d}{dt} = E_{total}$$

Schrödinger's wave equation can be expressed in three dimensions, where the general form of the time-independent Schrödinger equation in three dimensions is: $$-\frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}) + V(\mathbf{r}) \psi(\mathbf{r}) = E \psi(\mathbf{r})$$ Where:
\(\nabla\) is the Laplacian operator, the spatial derivatives in three dimensions.
\(\psi\)(r) is the wave function, which depends on the position vector.
\(r\) represents the coordinates \((x,y,z)\).
\(V\)(r) is the potential energy as a function of position.
\(E\) is the energy of the system.
\(\hbar\) is Plancks reduced constant.
\(m\) is the mass of the particle.

In three dimensions, the wave equation describes how the wave function \(\psi(r)\) varies with position in space. The solution to this equation provides information about the probability distribution of a particle's position in three-dimensional space.

So what does this all mean, is it all just Greek? To understand the Schrödinger equation a little better, what can be done is to analyze Neils Bohrs Interpretation of the Hydrogen atom with its orbiting electron, essentially beginning in reverse.

The first thing to do is to plot the hydrogen atom with its orbiting electron onto a graph. It can be seen that the y-axis is named Imaginary and the x-axis Real. This is the result of an unfortunate historic naming convention, however its value is just as real as any value on the x-axis. As the the electron orbits the proton the angle shown on the image between the x-axis and the radius changes which we will call \(\varphi \) (phi). In the interests of simplicity it is assumed that the electron is rotating clockwise around the nucleus.

It can be seen that there is no time axis, the question may then be asked what would the rotation of the electron around the proton look like if it was plotted against a time axis rather than the x-axis. To the more astute, the answer would be a sine wave. In agreement with convention the imaginary axis and the real axis on the graph will be named \(\psi\) (psi).

When looking at this sine wave it is asked what does the Bohr radius represent in ths scenario. Obviously, it is the amplitude of the wave. It can also be seen that the distance from the origin to \(\psi\) on the Real axis represents the wavelength of the sine wave and for each rotation of the electron in its orbit the waveform progresses. When compared to standard quantum mechanics the area under the waveform is representative of the probability. When referring back to the orginal orbiting model it can be seen that the area under the waveform is 50%.

Why this is the case we need to refer back to the original orbiting model and it can be seen that quite simply it is one half of the area of a circle. $$ A=\frac{\pi r^2}{2} $$