What does the Copenhagen interpretation say about the position of a particle before measurement?

Question

Suppose there is a particle in space. When we measure the position of that particle, we get a particular value with a probability that can be calculated from the wave function. But, according to the Copenhagen interpretation, where was the particle before the measurement? Was it in a superposition of all possible positions?

Answer 1

Before the measurement the particle's quantum state is given by the wave function, normalised to some volume $V$ of space

$\psi({\bf r},t)=\frac{1}{\sqrt{V}}\exp[({\bf p.r}-Et)/\hbar]$,

This means that, according to the Copenhagen Interpretation of QM, before the measurement, the particle is everywhere, occupying every point inside the volume $V$.

This equation shows that the particle has well defined momentum (sharply defined,) but the position of the particle is completely undetermined, from the point of view that the particle can be found, with probability $P=1/V$, in any one position within the volume $V$. Given that the wave function is the only source of information we can have about the position of the particle, the particle is everywhere inside the volume V. In other words, no point ($x,y,z$) at any time $t$ has extra probability to be occupied by the particle.

If a free particle occupies some region, in the $x$-axis say, having width $\Delta x_0$ at $t=0$, then the particle is described by a wave-packet. The width $\Delta x_0$ of the wave-packet determines the width $\Delta p_0$ of all possible values of momentum the particle can have. This is an expression of Heisenberg's uncertainty principle. Since the particle is free, as time goes on the width expands $\Delta x(t)\rightarrow\infty$ while the momentum of the particle acquires a well defined value. I.e. in momentum space the wave packet reduces to a delta function. Conversely, as $\Delta x(t)\rightarrow 0$ the momentum of the particle is totally undetermined. This interplay is at the heart of quantum mechanics, and we "witness" it in the double slit (or the diffraction grating) experiment.

Answer 2

Was it in a superpostion of all possible positions?

I cut out some (cat) bits from Wikipedia and got this:

In the Copenhagen interpretation, a system stops being a superposition of states and becomes either one or [an]other when an observation takes place. [...] [T]he nature of measurement, or observation, is not well-defined in this interpretation. [An] experiment can be interpreted to mean that [until an observation takes place] the system simultaneously exists in a superposition of the states [...] and that only when [...] an observation [is] performed does the wave function collapse into one of the [...] states.

Source (note the "However"-part that follows)

Answer 3

The Copenhagen interpretation is that the particle's position can only be determined by a measurement. The particle has an associated mathematical function- its wave function- that predicts the outcomes of measurements in a probabilistic sense, and can itself be changed by a measurement.

The particle's wave function can be represented as a superposition of other 'eigenfunctions' which can be the consequence of a given type of measurement. For example, there is a set of eigenfunctions that can represent the results of a measurement of the particle's momentum, and another set that represents the possible results of a measurement of the particle's position.

Loosely speaking, if a particle's wave function is represented as a superposition of momentum eigenfunctions, then the probability of measuring a particular momentum value is related to the extent to which its associated eigenfunction contributes to the superposition.

Generally a particle's wave function will be a superposition of position eigenfunctions (which are idealised delta functions), so when you perform a measurement of the particle's position, the result will reflect the mix of position eigenfunctions in the superposition.

That interpretation was developed because it was consistent with experimental evidence. However, the CI does not say that the particle is the wave function, so the fact that a wave function is a superposition of all possible position eigenstates does not mean that the particle itself is somehow smeared out- it just means that the particle's position before a measurement is not known.