Second quantization question: one particle or n particle?

[mings6]

For the simple harmonic oscillator case, the energy is E=(n+1/2)hw, and N|n>=n|n>.

It seems second quantization explain it as there are n bosons with each particle has energy homework plus vacuum 1/2hw. But we know before second quantization, there is only one particle with energy nhw plus vacuum 1/2hw.

Though we think those two different pictures have same total energy, the wave function of (N particle at ground particle) and (one particle at Nth state) are not same. So where is my mistake?

 

[A. Neumaier]

For the simple harmonic oscillator case, the energy is E=(n+1/2)hw, and N|n>=n|n>.

It seems second quantization explain it as there are n bosons with each particle has energy homework plus vacuum 1/2hw. But we know before second quantization, there is only one particle with energy nhw plus vacuum 1/2hw.

Though we think those two different pictures have same total energy, the wave function of (N particle at ground particle) and (one particle at Nth state) are not same. So where is my mistake?

In second quantization you have arbitrarily many identical particles rather than a single one.