A particle, at time $t=T$, has the following wavefunction:
$\left| \psi \right> =\frac { 1 }{ \sqrt { 4 } } \left( \left| A \right> \left| - \right> +\left| B \right> \left| + \right> +\left| C \right> \left| - \right> +\left| D \right> \left| + \right> \right) $
where $A$,$B$,$C$,$D$ are the (normalised) position wavefunctions and $+$,$-$ are the spin wavefunctions. The energy of the particle depends on its position, but not its spin. $A$ has the lowest energy (ground state), $B$ and $C$ have the same energy, $D$ has the highest energy.
Imagine we make a measurement at time $t=T$. We only measure its energy, but not its spin. Also, we don't measure the particle's exact energy, instead we check whether its energy is equal to its ground state $A$. The result is that the particle is not in its ground state. What is the wavefunction of the particle after the measurement?
In essence, I would like to understand what does a wavefunction collapse into, when:
- the measurement is not exact but it covers a range of eigenstates (such as a measurement that determines whether the energy is within a certain range, rather than a measurement that determines the exact energy).
- the measurement does not measure other properties (such as only measuring the particle's position but not measuring its spin)
- some states are degenerate (such as state $B$ and state $C$ having the same energy)
Note that I'm not talking about the uncertainty of our apparatus here; I'm considering the situation where our measurements are perfect and we simply choose not to measure everything.
