# Effects of measurement on particle energy

According to quantum mechanics, once you measure a particle's energy, its wave function collapse into some state, an eigenfunction with some eigenvalue (which is the particle energy). But if a particle is at some stationary state, the Schrödinger equation tells us the wave function at any time t, will be the same eigenfunction multiplied by some phase. From this, we can derive the particle will have constant energy. But it means, if I take a particle with some complex wave function and measure its energy at some time, I made his energy constant ? How does it make sense ?

• If you prepare a state in an eigenstate of the energy operator, then it won't change over time. If you measure the energy of an arbitrary state then the state will collapse to an energy eigenstate. I can't see any contradiction here. – user183962 May 21 at 8:10
• So if you take some particle which is affected by some forces and has a very complex wave function, by measuring it, did you make its energy constant from now on? Real objects don't remain invariant in time, That is sort of my question. I know there isn't a contradiction here, but I still can't understand why it makes sense. – Michael May 21 at 8:31
• Wave function complexity doesn't have any importance. When you measure the energy it collapse to an eigenstate and if the energy operator is the same for the system then there will be only a phase as you mentioned. – user183962 May 21 at 8:35
• I think part of the confusion is that you're thinking the system is not isolated. If there are external forces etc, then the Hamiltonian will be time dependent and energy eigenstates won't be stationary. – jacob1729 May 21 at 9:50
• You are narrating a straightforward sequence of facts and are then asking as to why they make sense. I feel clueless as to how to answer if I don't know as to why you think they shouldn't make sense. – Dvij Mankad May 21 at 12:52

## 1 Answer

The simple picture is that quantum mechanical differential equations have to be solved in order to have quantum mechanical wavefunctions, and boundary conditions have to be imposed so as to pick for the particular potential the particular wavefunction describing the system.

Take the hydrogen atom at its ground state : that is one solution of the Schrodinger equation, and it will stay there forever, if no new boundary conditions/interactions are imposed. A photon hitting the hydrogen atom with the ionization energy will be a different solution of the differential equation, and the wave function will no longer be of the ground state but of the free ion.That is the meaning of "collapse", a change in the wavefunction due to an interaction.

With many body problems one uses quantum field theory based on plane wave solutions of the differential equations, but the logic is the same: the boundary conditions have to be taken into account for the solution and any change in the boundary conditions, changes the wavefunction, which is what "collapse" is about. So a system in an energy eigenstate will stay there until there is an interaction, i.e. a change in the boundary conditions of the problem, which a scattering, for example , will induce. Then a different wavefunction will describe the system.

But it means, if I take a particle with some complex wave function and measure its energy at some time, I made his energy constant.

The energy is constant only for this point measurement in time, i.e you have the information that it was in that measured energy eigenstate. In the hydrogen example above, the kicked out electron signals that the atom + electron are now in a different solution of the problem at hand, a different wavefunction, where the proton is alone and the electron just outside it, with zero momentum , a different wavefunction; and then it is ready to fall back into the original mathematically wave function by emitting a photon,by a new interaction.

Collapse means "change of wavefunction describing the system".