# Noether's Theorem and the Measurement? (In Copenhagen)

## Background

I'm trying to recently understand the Copenhagen interpretation. We all know due to Noether's theorem energy is translational invariance in time.

However, in quantum mechanics the experimentalist can have the same initial configuration ("physical system" + "Apparatus") and do a measurement and get different outcomes. Hence, while energy (as a number) is conserved Nother's theorem obviously does not apply (during the measurement).

Now, the Copenhagenist seems to say the measuring instrument is some kind of semi-classical apparatus and the system is purely quantum - the measurement is an artefact of this.

## Question

How does the Copenhagenist justify this apparent discrepancy (violation of translational invariance in time)?

• Have you ever heard of Ehrenfest's theorem? – Aaron Stevens Sep 12 '19 at 14:18
• Yes, however, for the Copenhagenist if semi (quantum/)classical mechanics and unitary evolution is there is to everything ... Then :/ – More Anonymous Sep 12 '19 at 14:20
• More explicitly, during unitary evolution atleast if I have the same initial state after time $t$ I have the same final state. In classical mechanics if I have the same initial configuration then after time $t$ I have the same final state. Over here same refers time translational invariance. Given this any semi-classical theory should agree on time translational invariance. – More Anonymous Sep 12 '19 at 14:29
• FWIW, Noether's original theorem applies to a classical Lagrangian system, not quantum mechanics per se. – Qmechanic Sep 12 '19 at 15:48
• @Qmechanic I see. However people do use Noether's theorem in QM and at least during the unitary evolution the original interpretation is correct. – More Anonymous Sep 12 '19 at 15:50

The reason you cannot apply Noether's theorem reasoning to this scenario is because the dynamics, given by the Hamiltonian, are not transitionally invariant in time. If the system is governed by the free Hamiltonian $$H_{sys}$$ that is time independent then it is fine. However, when you are measuring the system you have to apply an external interaction to do the measurement. The external iteration term $$H_{int}$$ in the Hamiltonian is switched on temporarily to do the measurement and then it is switched off. The total Hamiltonain is therefore time dependent $$H(t)=H_{sys}+\alpha (t)H_{int}$$ $$\alpha (t)$$ is a stet function in time.