Number of Particles in Left Half of a Confined Box with Varying Energies

[phoenix95]

Homework Statement

1000 neutral spinless particles are confined in a one-dimensional box of length 100 nm. At a given instant of time, if 100 of these particles have energy 4ε₀ and the remaining 900 have energy 225ε₀, then the number of particles in the left half of the box will be approximately

(a) 441

(b) 100

(c) 500

(d) 625

Homework Equations

The Attempt at a Solution

Energies 4ε₀ and 225ε₀ correspond to E₂ and E₁₅ respectively, so I tried finding the expectation value for a single particle at these energy levels which turns out to be L/2 for both. But then I thought it is irrelevant for the problem, because I can't assign a single wave function to each of the particle and carry out the calculations.
Can anyone please help me?

Thanks

 

[Xu Shuang]

I think this is a misguided problem. The left side of the box is not in a particular number state, which means if you measure the number of particles of the left half of the box, the result would be probabilistic.

 

[phoenix95]

I think this is a misguided problem. The left side of the box is not in a particular number state, which means if you measure the number of particles of the left half of the box, the result would be probabilistic.

In that case, how should I start?