Consider a one dimensional harmonic oscillator.(adsbygoogle = window.adsbygoogle || []).push({});

We have:

$$\hat{n} = \hat{a}^{\dagger} \hat{a} = \frac{m \omega}{2 \hbar} \hat{x}^2 + \frac{1}{2 \hbar m \omega} \hat{p}^2 - \frac{1}{2}$$

And:

$$\hat{H} = \hbar \omega (\hat{n} + \frac{1}{2})$$

Let's say we want to measure the total energy. We can, using the number operator, but,apparently, this requires (by the definition above) that we measure location and momentum twice consecutively in two separate identical systems.

This raises questions:

1. After measuring one observable, immediately measuring it again should not change the value measured nor the function. So, what is the meaning of this in practice?

2. Theapparentrequirement of interacting with two independent systems in order to calculate/measure the number of quanta inasystem has to be wrong. ?

So finally: How does one measure total energy or number of quanta in reality?

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# Squared operators and sums of operators in practice

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