Can Identical Quantum Particles Have Different Energies?

[touqra]

Let say I have prepared two identical particle, both describable by a wavefunction Psi, whereby,
Psi = a*1 + b*2, where, 1 and 2 are two stationary wavefunctions.

If I perform an experiment to find out the systems' energy, this is equivalent to operating a Hamiltonian on Psi. Operating,
HPsi = H(a*1 + b*2) = a*E1*1 + b*E2*2
where, E1 is eigenvalue with eigenfunction 1,
E2 = eigenvalue with eigenfunction 2.

That means, I might get energy = E1 for the first particle from the experiment, and
energy = E2 for the second particle.

How can we get two different energy value E1 and E2 when I prepared both the particles exactly the same and both have the same wavefunction. So they must give me the same energy.
Otherwise, where does the energy difference E1-E2 come from?"

 

[florianb]

Let say I have prepared two identical particle, both describable by a wavefunction Psi, whereby,
Psi = a*1 + b*2, where, 1 and 2 are two stationary wavefunctions.

If I perform an experiment to find out the systems' energy, this is equivalent to operating a Hamiltonian on Psi. Operating,

Not quite... If you perform an experiment to find out the system energy, you're
measuring the energy. Therefore, you're projecting the state into the the corresponding eigenspace.

You will measure E1 with probability |a|^2 and E2 with prob. |b|^2 and the final state will be either |1> or |2>.

That means, I might get energy = E1 for the first particle from the experiment, and
energy = E2 for the second particle.

How can we get two different energy value E1 and E2 when I prepared both the particles exactly the same and both have the same wavefunction. So they must give me the same energy.
Otherwise, where does the energy difference E1-E2 come from?"

This comes from the way you prepared the state. The fact that the wave function is a superposition of both states means that you're not sure in which energy state it is. For example, you could have shone a laser on an atom trying to put it into an excited state, but you're not sure whether a photon was absorbed...

I hope this will help you understand a little bit more of QM.
Best regards

 

[R0man]

The energy difference between the two particles in this scenario can be attributed to the concept of quantum superposition. In quantum mechanics, particles can exist in multiple states at the same time, which is described by their wavefunction. In this case, the two particles are described by the same wavefunction, but they can still have different energies.

This is because when we perform the experiment to measure the energy, we are essentially collapsing the wavefunction of each particle into a single state. This means that each particle can only have one energy value at a time, even though they were described by a superposition of states before the measurement.

The energy difference between the two particles comes from the fact that their wavefunctions are not exactly the same. While they may be described by the same overall wavefunction, the individual components (1 and 2) may have slightly different properties, such as their location or momentum. This can result in different energy values when the Hamiltonian is applied to the wavefunction.

Additionally, the energy values (E1 and E2) are determined by the specific Hamiltonian used in the experiment. If the particles were prepared in a different way or if a different Hamiltonian was used, the energy values may be different.

In summary, the energy difference between the two particles can be explained by the principles of quantum mechanics, specifically the concept of superposition and the effects of measurement on the wavefunction.