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Let's consider $\left| \Psi \right> $ some state of quantum system.

Let's also consider some Hermitian Operator $\hat{Q}$, with the discrete specrum: $$ \hat{Q}\left| Q_n \right> = Q_n\left| Q_n \right>. $$

Now, if one try to mearure some physical quantity $\hat{Q}$ in non eigenstate $\left| \Psi \right> $, the result should be one of the eigenstates of $\hat{Q}$, say $\left| Q_n \right> $. $$ \hat{Q}\left| \Psi \right> = Q_n\left| Q_n \right>. $$ Thus, we observe what called wave function collapse: $$ \left| \Psi\right> \xrightarrow{\text{collapse}} \left| Q_n \right>. $$

As described in literature, the definition of an operator is $$ \hat{Q}\left|\Psi\right> = \left|\Phi\right>, $$ the thing that convert one state vector $\left|\Psi\right>$ to another $\left|\Phi\right>$. I had never met that for the Hermitian Operator acting to non eigenstate gives the operator's eigenstate.

So, is the equation $\hat{Q}\left| \Psi \right> = Q_n\left| Q_n \right>$ mathematically correct?

Or, from another point of view, Does it correct to say The Hermitian Operator Reduce the Noneigenstate State to the Self Eigenstate?

Mark Mitchison
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Sergio
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1 Answers1

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The measurement of an observable $\hat{Q}$ on a state $|\Psi\rangle$ is not represented by the equation $$ \hat{Q}|\Psi\rangle = Q_n |Q_n\rangle.\qquad \rm (wrong!)$$ This is a common misconception for those learning QM for the first time.

A Hermitian operator $\hat{Q}$ represents an observable in the sense that:

  1. The eigenvalues $Q_m$ of $\hat{Q}$ represent the physical outcomes of the measurement, e.g. different possible values of charge, momentum, energy, etc.
  2. The probability of obtaining outcome $Q_m$ is given by $p_m = |\langle Q_m|\Psi\rangle|^2$, where $|Q_m\rangle$ is the eigenvector of $\hat{Q}$ associated with eigenvalue $Q_m$, i.e. $\hat{Q}|Q_m\rangle = Q_m|Q_m\rangle$.
  3. The post-measurement state, conditioned on the fact that outcome $Q_m$ was obtained, is $|Q_m\rangle$.

These properties do not imply that measurement is represented by "applying operator $\hat{Q}$ to state $|\Psi\rangle$".

Mark Mitchison
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