So, in the last installment of IITSTIAPW, I attempted to explain how the fact that quantum particles fluctuate along many paths to a target brings a picture that looks like the physical lines of force for an electric field. This might be called the "position field" of a particle. This field gives all the information there is about the position of a single particle. In this post, I will introduce some notation to help tame these thought experiments. I was hoping to get all the way to entanglement today, but not quite.
Actually, a gravitational potential is unrealistic, partly because most quantum particles are too light to really notice gravity. You could do this experiment with electric potentials though.
In our first thought experiment, we have a particle on a special angle iron. This angle iron is bent and placed on a support. Three sensors are placed on the angle iron, two in the crook of the metal and one on the edge of the right side. The left sensor is green, the right crook sensor is red and the edge sensor is orange. The red sensor turns on only if the orange sensor is activated first. If a classical particle should be placed onto either side, then it would fall down the bend of the angle iron until it hit the red or green sensor (there is an extremely tiny stable region, but that can be safely ignored), though only the green would ever be activated. A quantum particle however has more choices, as illustrated below.
Possible paths if we measure green (top) or red (bottom)
The particle is said to be \( \left| G \right\rangle \) if the green sensor is activated and to be \( \left| R \right\rangle \) if the red sensor is activated. The name of this notation is "ket", but it is read, essentially, "state G". So the above sentence should be read out loud "The particle is said to be in state G if the green sensor is activated and and to be in state R if the red sensor is activated.". These are all the possible states of the system. As it turns out, the seemingly unique right method to represent quantum states is as complex vectors, linear algebra type vectors. While there is a result that explains why they are complex vectors given they are vectors (long story short, it is because complex numbers can have the same length and rotate), there is no result explaining why linear algebra should be used in the first place. The square of the length of these complex vectors . The structure was guessed by a group of physicists and nobody's ever managed to make it go away. Amusingly, most of the physicists didn't actually know linear algebra beforehand and painstakingly reproved many old theorems in new, less expressive notation.
More paths are like more lines of force, they represent a stronger field, a higher probability
By themselves, after we look at the results we might say that each situation either happened or didn't. So we can without loss of generality claim that \( \alpha \left| G \right\rangle \) and \( \alpha \left| R \right\rangle \) have a length of one. But before the sensors are turned on, things are more complicated. The above picture implies that we must be in some state \( \left| S \right\rangle = \alpha \left| G \right\rangle + \beta \left| R \right\rangle \) where \( | \beta |^2 \ll | \alpha |^2 \) and \( | \beta |^2 + | \alpha |^2 = 1\). \( \left| S \right\rangle \) is said to be a superposition of states. Notice above that even though \( \left| S \right\rangle \) is written as the some of two states, if you look above you'll see a single perfectly good field. Superpositions of states are still pure states, there is still only one particle fluctuating around. In a later post I'll cover multiple particles, in which case entanglement starts becoming an interesting issue.
We measured R!
Let's say that we place a quantum particle on the crook of the angle iron and waited a while. Later, once things have settled down, we turn on the sensors and the red sensor clicks! Happening at all is sufficient evidence that the universe is not classical. But quantum physics fixes more than that. We denote the situation that we measured the system in a particular state, say \( \left| R \right\rangle \), by \( \left\langle R \right| \). This new object is called a "ket" and is basically read "measuring state R". When you have a bra and a ket together, you have a bra-ket. A bra-ket is the probability of measuring state bra from state ket. For example, if you are definitely on the left side then you aren't on the right side and vice versa, so that \( \left\langle R | G \right\rangle = 0 \) and \( \left\langle G | R \right\rangle = 0 \). Let me write how that sentence should be read aloud. "If you are definitely on the left side then you aren't on the right side and vice versa, so that measuring state R from state G has probability zero and measuring state G from state R has probability zero.". Obviously, if you're at a sensor, then it clicks right away so that \( \left\langle R | R \right\rangle = 1 \) and \( \left\langle G | G \right\rangle = 1 \). These states are "orthonormal" vectors by design. But what if I put the particle in the middle and waited a long time before turning on the sensors, like I did in the above experiment? Then the particle is in \(\left| S \right\rangle \), so that $$\left\langle R | S \right\rangle = \left\langle R \right| ( \alpha \left| G \right\rangle + \beta \left| R \right\rangle )$$ $$\left\langle R | S \right\rangle = \alpha \left\langle R | G \right\rangle + \beta \left\langle R | G \right\rangle$$ $$\left\langle R | S \right\rangle = \alpha 0 + \beta 1 = \beta $$ This is the most basic linear algebra structure of quantum mechanics.
Very Formal
If I were being formal, I would go over the precise axioms of the so-called bra-ket notation. In addition, so far I have only worked with a single measurable, position. Instead, I'll just make promises. In the next few posts of this series, I will first go over quantum mechanics for multiple particles, then give the formal bra-ket axioms (which is, in essence, an axiom scheme for quantum mechanics) and finally extend the ideas in these posts to fields.
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