spiffomatic64 said:
as of now, am I correct in chalking a particles lack of position and velocity to the fact that particles are another form of energy? being that we can measure them using certain tools and what not, but we can only measure them using other particles which are also manifestations of energy as well?
"are particles a form of energy" is a statement that comes up regularly, but I fail to see what it could even mean. Because of course, that begs the question: what is "energy" then, so that it can "make up" particles ? Some jelly-like all-pervading stuff ? Or a number ?
Whatever is a particle (in a specific model!) is given by the model's definition. In classical mechanics, the theatre is set up (3-dim Euclidean space for starters - it gets actually more complicated than that because of Galilean relativity). We say that particles are "matter points in space, which move on a continuous trajectory".
Now, would it make sense to say, in classical mechanics: "are particles a form of position" ?
This question sounds to me as gibberish. Are particles "made up" of "position" ?
Do particles HAVE a position ? In classical mechanics, definitely yes. THAT makes sense. But ARE particles made up of position ? Is hot water made up of temperature ?
In classical mechanics, "energy" is a number that we can calculate for a certain dynamical state, and in many respects it is an *interesting* number, because it has certain properties (is conserved over time, for instance).
In quantum mechanics, we can do 2 things: define a quantum system "from scratch", or "be inspired" by a classical system. Many people think that we need a classical system and then "quantize" it - even Landau says so in his books. But that isn't true. Quantum mechanics can be entirely defined "from scratch". For instance, particle spin is "quantum mechanics from scratch". There's no classical model for it, which has been quantized.
But most quantum mechanical systems are "quantized classical models". The classical model then serves as a guiding principle for the setup of the state space (and its "interpretation").
A quantum theory starts out by giving a "complete state space basis". That is, you have to give a list of possible states your "system" is going to be in, which you will be able to measure. That can be a discrete list, or a continuous list.
The simplest non-trivial discrete list is a list of 2 states, call it "up" and "down". It is up to you to link these states to some operational procedure to "measure" these states, in other words, to give a physical interpretation to these states. That's YOU who puts this in. It's part of your model building. It is not quantum theory that is going to tell you.
This means that your state space is going to be 2-dimensional, and that all possible states are going to be of the form u |up> + v |down>
You might also want to define "other" measurements, which have another basis. That's entirely up to you, and it will depend on properties you want your system to have. You might for instance define a measurement procedure which gives you |left> and |right>. |left> is then equal to |up> + |down> and |right> is equal to |up> - |down> for instance. Whether that is a good idea, and whether that corresponds to one or other genuine laboratory measurement on your system, is again entirely your responsability. You might define things that way, but it might not work out that way in the lab: it simply means that you didn't build a good quantum model of your system and your measurements.
Next, you will have to define a time evolution of each of these states in your list. If I have "up" at a moment t0, what state (of the form u |up> + v |down> ) will my system be in at time t1 We can do that in any way we like, but the time evolution needs to obey certain properties (like being unitary).
Now, there's a big help, because it turns out that in most useful systems, there are eigenstates to the time evolution operator: that is: states that remain themselves up to a factor. States that don't mix with others. We call such states: energy eigenstates. Usually, this is the first thing we look for when setting up a quantum system: what are the energy eigenstates.
This is very analogous to the "energy" concept in classical mechanics, which had its main usefulness in conservative systems where energy was conserved in time.
So we see that quantum theory allows for a huge number of possibilities of models. We wouldn't know where to start, so to say. That's why there's a way to derive quantum models from classical systems.
If you have a classical system, you have a so-called "configuration space". For instance, for a single point particle, that configuration space is just Euclidean space. For a 2-particle system, it is a 6-dimensional space made up of the 6 coordinates of the 2 particles (3 + 3).
Well, the configuration space is going to serve as a "complete state space basis". To each POINT in the configuration space corresponds one quantum state of our state space basis. That already means that our statespace is going to be infinite-dimensional. Let us stick to the 1 particle classical system. Our basis is now the list of all possible positions of the particle (that list is nothing else but 3-dim Euclidean space). So we have now not 2 states (up and down), but a state for each point in space ! There's infinity of them.
We interpret these states of course as "precise position states".
In our classical system, we also have momenta. Well, it turns out that we have to take momentum states as special combinations of position states. Momentum states are like the "left" and "right" states earlier. In fact, a momentum state with momentum p is a sum over all position states, with a coefficient in the sum which is nothing else but exp(i p x).
That didn't have to be so, but it turns out that if we do this that way, that the quantum models that we build do work.
We find our time evolution by looking at the classical hamiltonian of the system. It tells us how to find an operator which is the time derivative of the time evolution operator (this time derivative operator is usually called "the hamiltonian" operator). This is nothing else but the Schroedinger equation.
Voila.
Now, note that there is something strangely perturbing to quantum theory: the time evolution of the quantum state is entirely deterministic ! Where's the randomness ? The randomness in quantum theory comes from the following rule:
If you are going to do a measurement, to that measurement corresponds a certain set of base states. For instance, if you are going to do a position measurement, then our initial defining base (inspired by the configuration space of the classical system) is the set of measurement base states. The actual quantum state can be a sum of those base states. It doesn't have to be exactly one of those states. Well, quantum mechanics tells us that the probability of obtaining a certain result (which, in the case of position measurement, is a single, precise, position), is then given by the value squared of the coefficient of the corresponding base state in the sum.
For instance, if our state at a certain moment is 1/sqrt(2) |up> - 1/sqrt(2) |down>, and we are going to do an up/down measurement, then there is 1/2 chance to find "up" and 1/2 chance to find "down".
After the measurement, the quantum state has abruptly changed, in agreement with the outcome of the measurement.
back to the main question though, even if its not position or velocity, the names for the variables are irrelovent. If one had all of the information/variables to a system (universe or closed off or w/e) wouldn't the system be deterministic from that point on?
Well, again, in order to even be able to answer that question, you should tell me what model you use. If it is standard quantum mechanics, and we accept that the quantum state is "all there is to know" to a system, then there's no way to know what measurements will give us beyond the probabilities given by quantum theory. If you have another model, then depending on your model, you will answer that question this or so.
For instance, there's a very simple model, which is old as the world, and which is entirely deterministic: "it is written in god's book". All events, past, present and future, are part of a big catalog. No "dynamics", no "time evolution", no "state of the system": just a huge catalog of everything that happened and will happen. Useless model, yes. But thinkable model.
Our "laws of nature" are then nothing else but funny correlations between those events in that big catalog. It is a universal model, which can apply in just all circumstances.
You can't get more deterministic.
Im basically fishing for whether or not random is ever truly random, or random can only be an inverse value to the amount of information we have.
As I said, that's model dependent. It is always possible to consider that everything is deterministic, given that there exists a universal model of nature that works that way "god's list".
