Born rule for degenerate eigenvalue

In summary, the Born rule for measuring a value a for an observable A in a normalized state |\psi\rangle is |\langle a|\psi\rangle|^2, but in the case of degeneracy, one should use the spectral decomposition of the self-adjoint operator and projectors onto subspaces. The probability of finding a degenerate eigenvalue is equal to the expectation of the projector in the state, and for an incomplete measurement, the probabilities for all possible eigenstates should be summed or integrated over.
  • #1
dEdt
288
2
The probability of measuring a value [itex]a[/itex] for an observable [itex]A[/itex] if the system is in the normalized state [itex]|\psi\rangle[/itex] is
[tex]|\langle a|\psi\rangle|^2[/tex]
where [itex]\langle a|[/itex] is the normalized eigenbra with eigenvalue [itex]a[/itex].

This is more-or-less the formulation of the Born rule as it appears in my text. But this seems to only make sense if [itex]\langle a|[/itex] is non-degenerate. So, what's the rule if we have a degeneracy?
 
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  • #2
In the general case one should use the spectral decomposition of the self-adjoint operator which describes the observable A, thus one uses projectors. The projector for a subspace belonging to a degenerate eigenvalue (for simplicity, assume the spectrum to be purely a point spectrum) is the sum of each projector according to the rule

[tex] P_ n = \sum_{i=1}^{g_n} |ni\rangle \langle ni| [/tex]

with g_n the dimension of the subspace which corresponds to the degenerate eigenvalue a_n in which [itex] |ni\rangle [/itex] form an orthonormal subbasis. This P_n goes then in the general Born rule (again pure point spectrum):

[tex] p(a_n)_{|\psi\rangle} = \langle \psi |P_n|\psi\rangle [/tex]
 
  • #3
The general formulation of the Born rule uses projectors onto subspaces. The probability of finding a eigensubspace of the measurement operator is equal to the expectation of the projector in the state: p = <psi|P|psi> You can easily see that if you project onto a 1-dimensional subspace P can be written as P = |n><n| and the probability becomes p = <psi|n><n|psi> = |<psi|n>|^2
 
  • #4
The simple version of the above replies is: if there are multiple eigenstates with eigenvalue a, add up the Born probabilities for all those states to get the probability to measure the value a.
 
  • #5
dextercioby said:
In the general case one should use the spectral decomposition of the self-adjoint operator which describes the observable A, thus one uses projectors. The projector for a subspace belonging to a degenerate eigenvalue (for simplicity, assume the spectrum to be purely a point spectrum) is the sum of each projector according to the rule

I have a question Dexter,

If the projector of a subspace which belongs to a generate eigenvalue, can one say that the projector is a type of generator of those which belong in the subgroup?
 
  • #6
I'd interpret Born's rule for a degenerate eigenvalue from the point of view of statistical mechanics. Given a system to be prepared in some state, represented by the statistical operator [itex]\hat{R}[/itex], we ask the question about the probability (density) to find a specific value [itex]a[/itex] of an observable [itex]A[/itex], represented by a self-adjoint operator [itex]\hat{A}[/itex]. If [itex]|a,\beta \rangle[/itex] is a complete set of (generalized) eigenvectors of [itex]\hat{A}[/itex] for the eigenvalue [itex]a[/itex] normalized to unity (or to the [itex]\delta[/itex] distribution), then the probability (density) that the system is found in a specific state given by one of these eigenvalues is
[tex]P(a,\beta)=\langle a,\beta|\hat{R}|a,\beta \rangle.[/tex]
This probability can be found experimentally by measuring a complete set of compatible observables (including [itex]A[/itex]).

If you know only measure [itex]A[/itex] you have to sum (integrate) over all the non-measured observables since, because the basis vectors are orthonormalized, the outcomes are mutually exclusive, i.e., you have
[tex]P(a)=\sum_{\beta} P(a,\beta) \quad \text{or} \quad \int \mathrm{d} \beta \; P(a,\beta).[/tex]
So the Born rule for an incomplete measurement in the case of degenerate eigenvalues follows directly from the Born rule for a complete measurement and basic rules of probability theory.
 

Related to Born rule for degenerate eigenvalue

What is the Born rule for degenerate eigenvalues?

The Born rule for degenerate eigenvalues is a fundamental principle in quantum mechanics that describes the probability of obtaining a certain measurement result when a system is in a state described by a degenerate eigenvalue. It states that the probability of obtaining a particular measurement result is proportional to the squared magnitude of the projection of the state vector onto the corresponding eigenspace.

What does it mean for an eigenvalue to be degenerate?

An eigenvalue is considered degenerate if there are multiple linearly independent eigenvectors associated with it. This means that the same eigenvalue can correspond to different physical states of a system, making it impossible to uniquely determine the state of the system based on the eigenvalue alone.

How does the Born rule apply to degenerate eigenvalues?

The Born rule for degenerate eigenvalues takes into account the degeneracy of the eigenvalue by considering the projection of the state vector onto the entire eigenspace, rather than just a single eigenvector. This allows for a more accurate calculation of the probability of obtaining a particular measurement result.

Can the Born rule be applied to non-degenerate eigenvalues?

Yes, the Born rule can be applied to both degenerate and non-degenerate eigenvalues. In the case of non-degenerate eigenvalues, the probability of obtaining a particular measurement result is simply proportional to the squared magnitude of the projection of the state vector onto the corresponding eigenvector.

What is the significance of the Born rule for degenerate eigenvalues in quantum mechanics?

The Born rule for degenerate eigenvalues is a fundamental principle in quantum mechanics that allows for the calculation of probabilities for measurement outcomes. It is essential for understanding the behavior of quantum systems and has been confirmed through numerous experiments. It also has implications for the interpretation of quantum mechanics and the nature of reality at the quantum level.

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