QuGUI V2.4 "QuPhiGUI" Abstract

Modified and extented version of the application. Includes 3D model of qubit orientation probability.

To introduce this new way of studying the qubit's dynamics, we have to consider what qualities can be described once the ' phase is known. Let's consider the following questions:

1. Vertical motion (up/down)
2. Motion along the axis (if there is, is it negative/positive?)
3. Motion along the axis (if there is, is it negative/positive?)

Now, let's visualise a group, or better a beam of atoms, that are magnetic (iron, perhaps). Let's make it go through the gap in between the two ends of a horseshoe magnet. Can you imagine it?

In a classical situation the atoms would spread out and the range of the beam would 'enlarge'- due to their magnetism. However, this is not the case when it's done experimen- tally. Instead, the beam splits in two: the 'projection' along the axis is either 'up' or 'down' [50/50].

If a beam is put through a horseshoe magnet with an orientation towards the right [a C shape], a part of the beam will go "all the way up". If then, the same beam is consequently put through another horseshoe magnet with this time an upwards orientation [a U shape], the question changes, yet regardless of any previous results, the beam will go either right or left.

However, there's always a contact with classical mechanics, even if, in practice, the questions are always the same three: does the atom have a vertical orientation (up or down), one along the axis (positive or negative) or one along the axis (positive or negative).

The orientation of an object is always 'written' somewhere within it- the question is, how do we 'read' it? The answer lies within the state of the atom; if the state is completely unde- ned, it still has a probability of being up/down. The new value of , could perhaps have, 'up' orientation and 'down' orientation. This means the state is inclined, depending on how is changed. If needs to be measured, the ratio of intensity between the up/down values must be measured. Talking about orientation, the angle can be compared to 'altitude', whilst the angle with 'longitude'.

Let's take a case where and . Due to the fact that it is on the horizontal plane, if the magnet splits the beam in two, and if is different from , there will always be a different distribution caused by the magnet on the plane, even if remains the same. This is always the mean value of the orientation.

As it can be seen, orientation may be 'classically' well dened, but it is not the case in quantum mechanics. As some call it, there is not the same 'sharpness'. However, if there are many and you take the mean values, then, on average, you get 'an orientation'. This comes from a certain process, roughly it looks like this:

But what is the 'procedure'? It is a set of questions. The type of questions depend on the mathematics of the problem.

Let's take a step back. A vector is determined by its components. In classical mechanics, the measurements (of the components) are that of the state of the vector - there is no ambiguity. The difference between the way in which a problem is described in quantum mechanics, is that the state and the values change when the system is described.

If there is a system, for instance our , what is the mean value of the component? The 'procedure' is needed to find out. It's based on the state and on a set of observable factors, put on the state. These observable factors are the , , and components.

The observable factors, given a state, become matrices (that are general operators), more specifically three, one for each direction. After a series of mathematical operations, the mean value of the component along , , and is obtained. The possible values that can be obtained by measuring may be all 'up', 'down', 'left', 'right', 'forwards', and 'back'.

You might be asking yourself, what's in the system? What can be measured? From the procedure, given the state, the two values are 'close relatives' of the angles used to describe the position on an object (take note of the square brackets). Usually, the state is written as a function of similar angles to and , that correspond with angles that would be measured after the procedure.

In order to describe an object's orientation, there are 'standards. The most used conven- tion for the orientation, is to start from an axis. 2 angles are needed. The first is the angle the object has in relation to the axis, an angle similar (but not the same!) as our angle .

On the horizontal,

This corresponds with the 'latitude'. (Note that this new version of QuGUI, V2.4, the is at at default.) The second angle, which corresponds with the 'longitude', taking the projection on the horizontal plane, corresponds to how much it 'turns'. If it turns counterclockwise:

Now, the equation has changed and become:

With the previous and , , instead of being at its usual, positive angle, it was oriented 'down', remaining as .

If the (overall) mean value of the needs to be calculated, and we have a wide range of average values, a magnet can be aligned on the axis and, depending on the proportion of how the beam goes 'up'/'down', the result is given. They can be 'reconstructed' as an mean value, which are, on average, those given by the and values.

Let's move on to talking in more detail about the 'procedure'. On a side, let's also keep in mind that vectors are the 'quantic amplitudes', and that the 'operators' are:

As well as 3, 2x2, matrices:

The mean of the state is given by the product of the . Let's look at a very simple example: If we get