# What happens to the Hamiltonian of the wave function after measurement?

## Main Question or Discussion Point

As I understand it, the Hamiltonian is the kinetic plus the potential energy of the wave function. When a measurement is done what happens to the kinetic and potential energy?

Does it dissipate? Is it conserved in the measured state? Does it decrease?

Does the Hamilton or kinetic+Potential energy of the system(wave function) change when it is measured? Does the energy decrease and is transferred to the measuring device, does the energy increase and then the energy of the measuring device would decrease or does everything stay the same and there's no change of energy for the system or the measuring device?

Thanks

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bhobba
Mentor
Energy is conserved in classical and quantum physics. There is a powerful theorem that is in fact the modern conception of energy, called Noethers Theorem, that guarantees this:
http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

Measurement in no way changes this. Most of the time in QM measurement, what you are measuring is destroyed - but energy is still conserved. Even if it isn't destroyed the total energy (what is observed and the observing apparatus) is not affected.

Specifically in QM operators have no average rate of change if they commute with the Hamiltonian. Since, obviously, the Hamiltonian commutes with itself, there is no average gain or loss of the Hamiltonian.

The twist of course in QM is this on the average thing - which is all you can predict in QM.

Thanks
Bill

@bhobba, thanks for the response.

This is what I thought, the energy is conserved but does it change or a better word would be does it shift? Does the total energy of the system shift after a measurement occurs. Is the energy dispersed in the environment of the measuring apparatus? Does the total energy shift to the particle after measurement?

Where does the total energy of the system go after measurement? Does the wave function that evolves after measurement share the same potential and kinetic energy of the wave function prior to measurement?

Thanks

jfizzix
Gold Member
In measuring an atom or any other object, you interact it with a measuring device.

The measurement process is a physical interaction between atom and device, where total energy is conserved.

To describe this process, you need a Hamiltonian that describes the atom, device, and their interaction.

It is this global Hamiltonian that is a constant of motion before and after the measurement.

Thus, the energy of the atom may shift after measurement, but this is accompanied by a corresponding shift in the energy of the measurement device (assuming atom plus device is a closed system).