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We get the value of an observable $A$ for a given state $|\lambda\rangle$ of a system from the eigenequation $\hat{A} |\lambda\rangle = \lambda |\lambda\rangle$ where $\hat{A}$ is the operator corresponding to observable $A$ and $\lambda$ is the required value of the observable. But this is true only when the statevector $|\lambda\rangle$ is an eigenvector of the operator $\hat{A}$. How do we assign a value to observable $A$ when the statevector $|\lambda\rangle$ is not an eigenvector of the operator $\hat{A}$?

I am a beginner in quantum physics with elementary understanding of linear algebra so please try to answer accordingly.

PS: This question is related and its accepted answer is helpful. But from it I could not understand how to assign a value to the observable.

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  • $\begingroup$ Thank you @Ghoster for the edit! $\endgroup$
    – t2m
    Sep 20, 2022 at 17:07

2 Answers 2

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How do we assign a value to observable $A$ when the statevector $|\lambda\rangle$ is not an eigenvector of the operator $\hat{A}$?

We don't.

One of the toughest parts of learning quantum mechanics is learning to let go — in particular, learning to let go of the requirement that everything must have a value. In QM, if you have a system in a given state $|\psi⟩$ and a given observable $\hat A$ representing a physical quantity $A$, the generic setting is that the system cannot be assigned a value for $A$.

In some lucky occasions, of course, it turns out that $A$ does have a well-defined value (and these are, as you'll have guessed, when $|\psi⟩$ is an eigenstate of $\hat A$).

But, in general, observables in quantum mechanics do not have value until and unless a projective measurement of that observable is performed on the system.

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  • $\begingroup$ According to answer of @Arturodonjuan, we can find average value of observable when the state vector is not eigenvector of operator. I assume that both answers spell truth with different perspectives. But then I can't reconcile both answers. Can you please explain on that line? $\endgroup$
    – t2m
    Sep 21, 2022 at 8:05
  • $\begingroup$ The 'expectation value' of $A$ in the state $|\psi⟩$ is what you get if you take the system in the state $|\psi⟩$ and then (i) perform a projective measurement of $A$, which will then (ii) change the state of the system to one of the eigenstates of $\hat A$ and produce the corresponding eigenvalue as the value of $A$, and then (iii) you repeat this procedure multiple times (typically on a large ensemble), which will then typically (iv) give a wide spread of different values of $A$ with different frequency; and then finally you (v) take the average of the value of $A$ over this distribution. $\endgroup$ Sep 21, 2022 at 16:57
  • $\begingroup$ However, this does not mean that the expectation value of $A$ is in any way assignable as "the value of $A$ prior to the measurement". That value does not exist, and cannot be assigned. $\endgroup$ Sep 21, 2022 at 16:57
  • $\begingroup$ I get your point that there is an 'expectation value' but don't get that it is not assignable as a value of $A$ prior to the measurement. I think, even the definite outcome given by the eigenvalues too is what we get after measurement. Isn't it? $\endgroup$
    – t2m
    Sep 21, 2022 at 17:13
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    $\begingroup$ Thank you for helping with insightful answers. $\endgroup$
    – t2m
    Sep 22, 2022 at 12:25
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A pure physical state is comprised of a linear sum of eigenstates:

$$|\psi\rangle = \sum_{i} c_i |\lambda_i\rangle$$

where $|\lambda_i\rangle$ are complete eigenstates of a self-adjoint operator $A$, $\langle \lambda_i |\lambda_j\rangle = \delta_{ij}$ and $\sum_i |c_i|^2=1$.

The probability that we will find $|\psi \rangle$ in the $|\lambda_i\rangle$ state (i.e. having value $\lambda_i$ when acted upon by operator $A$) is given by $|\langle \lambda_i | \psi\rangle|^2=|c_i|^2$.

The average value of the operator $A$ in the state $|\psi\rangle$ is given by $\langle \psi | A |\psi \rangle = \sum_i |c_i|^2\lambda_i$. This is the average value of $A$ you would measure if you prepared an infinite number of identical systems described by the state-vector $|\psi\rangle$.

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  • $\begingroup$ Is there a reason for saying a state as 'pure' physical state? $\endgroup$
    – t2m
    Sep 21, 2022 at 8:54
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    $\begingroup$ Yes because a physical state is in general described by as a 'mixed' state, represented by a density matrix/operator. A special case is when the physical state/system can be represented by a single state vector, i.e. it is a 'pure' state. See the wikipedia article here. $\endgroup$ Sep 21, 2022 at 17:02
  • $\begingroup$ Can you explain in the post, or point me to an easy to understand online resource, why the average value (expectation value) is as you have written? $\endgroup$
    – t2m
    Sep 22, 2022 at 9:39
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    $\begingroup$ @Aarone Do you mean why the expectation value of an observable $\hat A$ in a physical state $|\psi\rangle$ is given by $\langle \psi | \hat A |\psi\rangle$? This is sort of by definition, an axiom of the framework of quantum mechanics. With this, the relation $\langle \psi | \hat A |\psi\rangle = \sum_i |c_i|^2 \lambda_i$ follows by direct substitution of $|\psi\rangle = \sum_i c_i |\lambda_i\rangle$, and using that the set of eigenstates $|\lambda_i\rangle$ is orthonormal, i.e. $\langle \lambda_i |\lambda _j\rangle = \delta_{ij}$. $\endgroup$ Sep 22, 2022 at 18:06
  • $\begingroup$ I was asking about how $\langle \psi | \hat A |\psi\rangle$ is equal to $\sum_i |c_i|^2 \lambda_i$. Your hint is sufficient. Thank you. $\endgroup$
    – t2m
    Sep 23, 2022 at 2:08

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