Can a particle in superposition act as x*(energy?)

Question

I can't find an answer anywhere - I even asked my physics teacher, he hasn't a clue. Is superposition an illusion, or can a particle literally act as an $x$ number of particles?

Answer 1

It's clear that you don't really understand what superposition is. Often the concept is verbalized as follows:

A physical system simultaneously exists in all of its possible states [but when a measurement is done, the result corresponds to one particular state].

This shouldn't$^1$ be taken literally because it implies that the system (I'll mostly talk about systems instead of particles because this is more general) actually exists in several forms. Rather you should think of it like this: the state of the system is blurred, mixed. There is only one state of the system, but it is not uniquely defined, there's some uncertainty (that is basically the one word around which the entire theory of quantum mechanics revolves). In other words: the state of the system is probabilistic.

Take for example a system that can exist in two ('pure') states.$^2$ We often denote a general state as $|\Psi\rangle$ in physics, so let's use that notation to refer to the actual state of the system and let's call the theoretically possible states of our system $|1\rangle$ and $|2\rangle$. What these states are physically is not important for this explanation, but you could imagine them being the spin of an electron for example (which can be either up or down). The superposition principle then comes down to the fact that in general the state of the system will be

$$|\Psi\rangle = A_1|1\rangle + A_2|2\rangle$$

where $A_1$ and $A_2$ are measures for how likely it is to find the system in state $|1\rangle$ or state $|2\rangle$, respectively. In particular $P(i) = |A_i|^2$ is the probability to find state $|i\rangle$.

So we don't know what state the system (electron) is in until we look at it (measure the spin), that makes sense right? But superposition is something more profound. Not only can we not know what state the system is in, the system itself doesn't 'know' either! We can conclude this from e.g. Stern-Gerlach experiments. In those experiments, particles which are prepared in the same physical state $|\Psi\rangle$ show different behaviour. One particle will behave as though it is in state $|1\rangle$ and another particle (in the same physical state, remember!) will behave as though it is in state $|2\rangle$. If you do the experiment with enough particles, you find that there's a certain probability to find each particular state.

The superposition principle deals with this strange behaviour by saying that the state of the system (electron) is probabilistic: in the case of two possible states it consists for $x_1$ percent out of state $|1\rangle$ and for $x_2$ percent out of state $|2\rangle$. This is what the above notation means and what the $P(i) = x_i/100$ correspond to.

A sandbox

A visual aid for interpretation could be to think of the system as being a sandbox. Each of its possible states corresponds to a certain grain diameter of the sand. A mixed state, a superposition, then means the sandbox is filled with several of these types of sand, all having a different grain diameter. Imagine we have again two possible states, so two different grain diameters, say $a$ and $b$. Then imagine you have 2 types of sieve (type $A$ and $B$) that are specially designed so they select only one size diameter sand ($A$ selects $a$, $B$ selects $b$).

Say you have 100 of these sieves in total, of which 60 are type $A$ and 40 type $B$. Now a measurement corresponds to you picking one sieve at random and using it to sieve out one particular type of sand, one particular state of the system. Before you measure the system, it exists in a superposition of states. Afterwards it exists in a single ('pure') state. In a real system, the system won't stay in this pure state after the measurement, it will slowly become mixed again. To accomodate for this feature, imagine we deposit all of the sieved out grains into a funnel which slowly allows them to flow back into the sandbox.

This entire analogy obviously isn't perfect, but you can view a a real quantum system's state as the collection of the sandbox and the sieves (you need to include the sieves because they determine the coefficients $A_i$ in the superposition in this analogy).

---

$^1$ I say shouldn't but in principle you can interpret it as you like since we can't ever know. After all we are restricted to measurements and measurement results. However, some of our ideas about how the universe should be get in trouble if you take the superposition principle literally. In particular it leads to a problem with continuity (although there are ideas about noncontinuous space and time as well, but these don't interfere with the idea of continuity at the scales we're discussing now).

$^2$ We call a state 'pure' if it is uniquely determined, there is no [quantum] uncertainty about it. A poor analogy is a die: when it is stationary, the top face is in a pure state (1, 2, 3, 4, 5 or 6). But when it is rolling, the state of the top face is blurred. As I said the analogy is (very) poor, but it should give you some idea of what a pure state is in everyday language. Another (again poor) analogy is the weather: we can define 4 pure states: sunny, cloudy, rainy, foggy. Then a general state is the weather in Belgium.