Saturday, 14 March 2015

quantum mechanics - Energy cost of the measurement without perturbing the system?


Background


Let's say I have a Hamiltonian ˆH (assume the Schrodinger equation) and it be in an arbitrary eigen-energy state:


ˆHsystem|m=Em|m


And I want to measure the momentum of the system without perturbing the Hamiltonian. We know this will force the wave-function into a momentum eigenket:


ˆp|pj=pj|pj


I thought what would be the average energy cost of this measurement:


pj|ˆH|pj=pj|ˆH(n|nn|)|pj=nEnn|pjpj|n=nEn|n|pj|2


Hence, the difference in energies for the 2 states are:



ΔE=nEn|n|pj|2Em


I think this energy comes from another system (that of the experimenters?)


Question


Is the above derivation correct? (If I am wrong what is the proper energy cost for a measurement on average?) Can I use this to say "the measurement only makes sense if there is more than one system?" (otherwise I'm not sure where the energy for the measurement would come from)




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