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I don't know if this will be an "explanation", but hopefully it is a useful "description".

More generally than projective measurements, one always measures an operator. (A projector is a special case of this.) So what does it mean to "measure an operator"?

Well, operators often correspond to 'observable' physical quantities. The most important in quantum mechanics, for instance, is energy; but one can also (sometimes inderiectly) measure other quantities, such as angular momentum, z-components of magnetic fields, etc. What is being measured always gives real-valued results --- in principle, some definite result (e.g. an electron is in the 'spin +1/2' state as opposed to 'spin −1/2', or in the first excited energy level as opposed to the ground-state in a hydrogen atom, etc.), albeit each a priori possible result is realized with some probability.

We assign each of the real-valued outcomes of a measurement to a subspace. The way we do this is to describe a Hermitian operator --- i.e. an operator which associates a real eigenvalue to different subspaces, with the subspaces summing up to the whole Hilbert space. A projector is such an operator, where the real values are 0 and 1; i.e. describing that a vector belongs to a designated subspace (yielding a value of 1), or its orthocomplement (yielding a value of 0). These Hermitian operators are observables, and the eigenspaces are those for which the observable has a "definite" value.

But what about those vectors which are not eigenvectors, and do not have "definite" values for these observables? Here is the non-explaining part of the description: we project them into one of the eigenspaces, to obtain an eigenvector with a well-defined value. Which projection we apply is determined at random. The probability distribution is given by the familiar Born rule:

$$ \Pr\limits_{|\psi\rangle}\bigl( E = c \bigr) \;=\; \langle \psi | \Pi_c | \psi \rangle \;, $$

where $\Pi_c$ is the projector onto the c-eigenspace of an 'observable quantity' E (represented by a Hermitian operator $A = \sum_c \; c \cdot \Pi_c$). The post-measured state is some projection of the state $|\psi\rangle$ onto some eigenspace of the observable A. And so if $| \psi_0 \rangle$ is the pre-measurement state, $| \psi_1 \rangle$ is the post-measurement state, and $\Pi_c$ is the 'actual result' measured (i.e. the eigenspace onto which the pre-measurement state was actually projected), we have the proportionality result

$$ | \psi_1 \rangle \;\propto\; \Pi_c | \psi_0 \rangle $$

by the projection rule just described. This is why there is the projector in your formula.

In general, the vector $| \psi_1' \rangle = \Pi_c | \psi_0 \rangle$ is not a unit vector; because we wish to describe the post-measurement state by another unit vector, we must rescale it by

$$ \|\;|\psi_1'\rangle\;\| = \sqrt{\langle \psi'_1 | \psi'_1 \rangle} = \sqrt{\langle \psi_0 | \Pi_c | \psi_0 \rangle} \;,$$

which is the square-root of the probability with which the result would occur a priori. And so, we recover the formula in your question,

$$ | \psi_1 \rangle \;=\; \frac{\Pi_c | \psi_0 \rangle}{ \sqrt{ \langle \psi_0 | \Pi_c | \psi_0 \rangle }} \;.$$

(If this formula seems slightly clumsy, take heart that it looks and feels a little bit better if you represent quantum states by density operators.)

Edited to add: the above should not be construed as a description of POVMs. A "positive operator valued measurement" is better seen as describing the expectation value of various measurable observables E_c in a collection { E_c }_c ∈ C .

I don't know if this is an "explanation", but hopefully it is a useful "description".

More generally than projective measurements, one always measures an operator. (A projector is a special case of this.) So what does it mean to "measure an operator"?

Well, operators often correspond to 'observable' physical quantities. The most important in quantum mechanics, for instance, is energy; but one can also (sometimes indirectly) measure other quantities, such as angular momentum, z-components of magnetic fields, etc. What is being measured always gives real-valued results --- in principle, some definite result (e.g. an electron is in the 'spin +1/2' state as opposed to 'spin −1/2', or in the first excited energy level as opposed to the ground-state in a hydrogen atom, etc.), albeit each a priori possible result is realized with some probability.

We assign each of the real-valued outcomes of a measurement to a subspace. The way we do this is to describe a Hermitian operator --- i.e. an operator which associates a real eigenvalue to different subspaces, with the subspaces summing up to the whole Hilbert space. A projector is such an operator, where the real values are 0 and 1; i.e. describing that a vector belongs to a designated subspace (yielding a value of 1), or its orthocomplement (yielding a value of 0). These Hermitian operators are observables, and the eigenspaces are those for which the observable has a "definite" value.

But what about those vectors which are not eigenvectors, and do not have "definite" values for these observables? Here is the non-explaining part of the description: we project them into one of the eigenspaces, to obtain an eigenvector with a well-defined value. Which projection we apply is determined at random. The probability distribution is given by the familiar Born rule:

$$ \Pr\limits_{|\psi\rangle}\bigl( E = c \bigr) \;=\; \langle \psi | \Pi_c | \psi \rangle \;, $$

where $\Pi_c$ is the projector onto the c-eigenspace of an 'observable quantity' E (represented by a Hermitian operator $A = \sum_c \; c \cdot \Pi_c$). The post-measured state is some projection of the state $|\psi\rangle$ onto some eigenspace of the observable A. And so if $| \psi_0 \rangle$ is the pre-measurement state, $| \psi_1 \rangle$ is the post-measurement state, and $\Pi_c$ is the 'actual result' measured (i.e. the eigenspace onto which the pre-measurement state was actually projected), we have the proportionality result

$$ | \psi_1 \rangle \;\propto\; \Pi_c | \psi_0 \rangle $$

by the projection rule just described. This is why there is the projector in your formula.

In general, the vector $| \psi'_1 \rangle = \Pi_c | \psi_0 \rangle$ is not a unit vector; because we wish to describe the post-measurement state by another unit vector, we must rescale it by

$$ \|\;|\psi'_1\rangle\;\| = \sqrt{\langle \psi'_1 | \psi'_1 \rangle} = \sqrt{\langle \psi_0 | \Pi_c | \psi_0 \rangle} \;,$$

which is the square-root of the probability with which the result would occur a priori. And so, we recover the formula in your question,

$$ | \psi_1 \rangle \;=\; \frac{\Pi_c | \psi_0 \rangle}{ \sqrt{ \langle \psi_0 | \Pi_c | \psi_0 \rangle }} \;.$$

(If this formula seems slightly clumsy, take heart that it looks and feels a little bit better if you represent quantum states by density operators.)

Edited to add: the above should not be construed as a description of POVMs. A "positive operator valued measurement" is better seen as describing the expectation value of various measurable observables E_c in a collection { E_c }_c ∈ C .

I don't know if this will be an "explanation", but hopefully it is a useful "description".

More generally than projective measurements, one always measures an operator. (A projector is a special case of this.) So what does it mean to "measure an operator"?

Well, operators often correspond to 'observable' physical quantities. The most important in quantum mechanics, for instance, is energy; but one can also (sometimes inderiectly) measure other quantities, such as angular momentum, z-components of magnetic fields, etc. What is being measured always gives real-valued results --- in principle, some definite result (e.g. an electron is in the 'spin +1/2' state as opposed to 'spin −1/2', or in the first excited energy level as opposed to the ground-state in a hydrogen atom, etc.), albeit each a priori possible result is realized with some probability.

We assign each of the real-valued outcomes of a measurement to a subspace. The way we do this is to describe a Hermitian operator --- i.e. an operator which associates a real eigenvalue to different subspaces, with the subspaces summing up to the whole Hilbert space. A projector is such an operator, where the real values are 0 and 1; i.e. describing that a vector belongs to a designated subspace (yielding a value of 1), or its orthocomplement (yielding a value of 0). These Hermitian operators are observables, and the eigenspaces are those for which the observable has a "definite" value.

But what about those vectors which are not eigenvectors, and do not have "definite" values for these observables? Here is the non-explaining part of the description: we project them into one of the eigenspaces, to obtain an eigenvector with a well-defined value. Which projection we apply is determined at random. The probability distribution is given by the familiar Born rule:

$$ \Pr\limits_{|\psi\rangle}\bigl( E = c \bigr) \;=\; \langle \psi | \Pi_c | \psi \rangle \;, $$

where $\Pi_c$ is the projector onto the c-eigenspace of an 'observable quantity' E (represented by a Hermitian operator $A = \sum_c \; c \cdot \Pi_c$). The post-measured state is some projection of the state $|\psi\rangle$ onto some eigenspace of the observable A. And so if $| \psi_0 \rangle$ is the pre-measurement state, $| \psi_1 \rangle$ is the post-measurement state, and $\Pi_c$ is the 'actual result' measured (i.e. the eigenspace onto which the pre-measurement state was actually projected), we have the proportionality result

$$ | \psi_1 \rangle \;\propto\; \Pi_c | \psi_0 \rangle $$

by the projection rule just described. This is why there is the projector in your formula.

In general, the vector $| \psi_1' \rangle = \Pi_c | \psi_0 \rangle$ is not a unit vector; because we wish to describe the post-measurement state by another unit vector, we must rescale it by

$$ \|\;|\psi_1'\rangle\;\| = \sqrt{\langle \psi'_1 | \psi'_1 \rangle} = \sqrt{\langle \psi_0 | \Pi_c | \psi_0 \rangle} \;,$$

which is the square-root of the probability with which the result would occur a priori. And so, we recover the formula in your question,

$$ | \psi_1 \rangle \;=\; \frac{\Pi_c | \psi_0 \rangle}{ \sqrt{ \langle \psi_0 | \Pi_c | \psi_0 \rangle }} \;.$$

(If this formula seems slightly clumsy, take heart that it looks and feels a little bit better if you represent quantum states by density operators.)

Edited to add: the above should not be construed as a description of POVMs. A "positive operator valued measurement" is better described as a description of measuring the expectation value of the associated operators E_c in a collection { E_c }_c ∈ C .

I don't know if this will be an "explanation", but hopefully it is a useful "description".

More generally than projective measurements, one always measures an operator. (A projector is a special case of this.) So what does it mean to "measure an operator"?

Well, operators often correspond to 'observable' physical quantities. The most important in quantum mechanics, for instance, is energy; but one can also (sometimes inderiectly) measure other quantities, such as angular momentum, z-components of magnetic fields, etc. What is being measured always gives real-valued results --- in principle, some definite result (e.g. an electron is in the 'spin +1/2' state as opposed to 'spin −1/2', or in the first excited energy level as opposed to the ground-state in a hydrogen atom, etc.), albeit each a priori possible result is realized with some probability.

We assign each of the real-valued outcomes of a measurement to a subspace. The way we do this is to describe a Hermitian operator --- i.e. an operator which associates a real eigenvalue to different subspaces, with the subspaces summing up to the whole Hilbert space. A projector is such an operator, where the real values are 0 and 1; i.e. describing that a vector belongs to a designated subspace (yielding a value of 1), or its orthocomplement (yielding a value of 0). These Hermitian operators are observables, and the eigenspaces are those for which the observable has a "definite" value.

But what about those vectors which are not eigenvectors, and do not have "definite" values for these observables? Here is the non-explaining part of the description: we project them into one of the eigenspaces, to obtain an eigenvector with a well-defined value. Which projection we apply is determined at random. The probability distribution is given by the familiar Born rule:

$$ \Pr\limits_{|\psi\rangle}\bigl( E = c \bigr) \;=\; \langle \psi | \Pi_c | \psi \rangle \;, $$

where $\Pi_c$ is the projector onto the c-eigenspace of an 'observable quantity' E (represented by a Hermitian operator $A = \sum_c \; c \cdot \Pi_c$). The post-measured state is some projection of the state $|\psi\rangle$ onto some eigenspace of the observable A. And so if $| \psi_0 \rangle$ is the pre-measurement state, $| \psi_1 \rangle$ is the post-measurement state, and $\Pi_c$ is the 'actual result' measured (i.e. the eigenspace onto which the pre-measurement state was actually projected), we have the proportionality result

$$ | \psi_1 \rangle \;\propto\; \Pi_c | \psi_0 \rangle $$

by the projection rule just described. This is why there is the projector in your formula.

In general, the vector $| \psi_1' \rangle = \Pi_c | \psi_0 \rangle$ is not a unit vector; because we wish to describe the post-measurement state by another unit vector, we must rescale it by

$$ \|\;|\psi_1'\rangle\;\| = \sqrt{\langle \psi'_1 | \psi'_1 \rangle} = \sqrt{\langle \psi_0 | \Pi_c | \psi_0 \rangle} \;,$$

which is the square-root of the probability with which the result would occur a priori. And so, we recover the formula in your question,

$$ | \psi_1 \rangle \;=\; \frac{\Pi_c | \psi_0 \rangle}{ \sqrt{ \langle \psi_0 | \Pi_c | \psi_0 \rangle }} \;.$$

(If this formula seems slightly clumsy, take heart that it looks and feels a little bit better if you represent quantum states by density operators.)

Edited to add: the above should not be construed as a description of POVMs. A "positive operator valued measurement" is better seen as describing the expectation value of various measurable observables E_c in a collection { E_c }_c ∈ C .

I don't know if this will be an "explanation", but hopefully it is a useful "description".

More generally than projective measurements, one always measures an operator. (A projector is a special case of this.) So what does it mean to "measure an operator"?

Well, operators often correspond to 'observable' physical quantities. The most important in quantum mechanics, for instance, is energy; but one can also (sometimes inderiectly) measure other quantities, such as angular momentum, z-components of magnetic fields, etc. What is being measured always gives real-valued results --- in principle, some definite result (e.g. an electron is in the 'spin +1/2' state as opposed to 'spin −1/2', or in the first excited energy level as opposed to the ground-state in a hydrogen atom, etc.), albeit each a priori possible result is realized with some probability.

We assign each of the real-valued outcomes of a measurement to a subspace. The way we do this is to describe a Hermitian operator --- i.e. an operator which associates a real eigenvalue to different subspaces, with the subspaces summing up to the whole Hilbert space. A projector is such an operator, where the real values are 0 and 1; i.e. describing that a vector belongs to a designated subspace (yielding a value of 1), or its orthocomplement (yielding a value of 0). These Hermitian operators are observables, and the eigenspaces are those for which the observable has a "definite" value.

But what about those vectors which are not eigenvectors, and do not have "definite" values for these observables? Here is the non-explaining part of the description: we project them into one of the eigenspaces, to obtain an eigenvector with a well-defined value. The probability distribution is given by the familiar Born rule:

$$ \Pr\limits_{E = c}\bigl( |\psi\rangle \bigr) = \langle \psi | \Pi_c | \psi \rangle \;, $$

where $\Pi_c$ is the projector onto the c-eigenspace of an 'observable quantity' E (represented by a Hermitian operator $A = \sum_c \; c \cdot \Pi_c$). The post-measured state is some projection of the state $|\psi\rangle$ onto some eigenspace of the observable A. And so if $| \psi_0 \rangle$ is the pre-measurement state, $| \psi_1 \rangle$ is the post-measurement state, and $\Pi_c$ is the 'actual result' measured (i.e. the eigenspace onto which the pre-measurement state was actually projected), we have the proportionality result

$$ | \psi_1 \rangle \;\propto\; \Pi_c | \psi_0 \rangle $$

by the projection rule just described. This is why there is the projector in your formula.

In general, the vector $| \psi_1' \rangle = \Pi_c | \psi_0 \rangle$ is not a unit vector; because we wish to describe the post-measurement state by another unit vector, we must rescale it by

$$ \|\;|\psi_1'\rangle\;\| = \sqrt{\langle \psi'_1 | \psi'_1 \rangle} = \sqrt{\langle \psi_0 | \Pi_c | \psi_0 \rangle} \;,$$

which is the square-root of the probability with which the result would occur a priori. And so, we recover the formula in your question,

$$ | \psi_1 \rangle \;=\; \frac{\Pi_c | \psi_0 \rangle}{ \sqrt{ \langle \psi_0 | \Pi_c | \psi_0 \rangle }} \;.$$

(If this formula seems slightly clumsy, take heart that it looks and feels a little bit better if you represent quantum states by density operators.)

I don't know if this will be an "explanation", but hopefully it is a useful "description".

More generally than projective measurements, one always measures an operator. (A projector is a special case of this.) So what does it mean to "measure an operator"?

Well, operators often correspond to 'observable' physical quantities. The most important in quantum mechanics, for instance, is energy; but one can also (sometimes inderiectly) measure other quantities, such as angular momentum, z-components of magnetic fields, etc. What is being measured always gives real-valued results --- in principle, some definite result (e.g. an electron is in the 'spin +1/2' state as opposed to 'spin −1/2', or in the first excited energy level as opposed to the ground-state in a hydrogen atom, etc.), albeit each a priori possible result is realized with some probability.

We assign each of the real-valued outcomes of a measurement to a subspace. The way we do this is to describe a Hermitian operator --- i.e. an operator which associates a real eigenvalue to different subspaces, with the subspaces summing up to the whole Hilbert space. A projector is such an operator, where the real values are 0 and 1; i.e. describing that a vector belongs to a designated subspace (yielding a value of 1), or its orthocomplement (yielding a value of 0). These Hermitian operators are observables, and the eigenspaces are those for which the observable has a "definite" value.

But what about those vectors which are not eigenvectors, and do not have "definite" values for these observables? Here is the non-explaining part of the description: we project them into one of the eigenspaces, to obtain an eigenvector with a well-defined value. Which projection we apply is determined at random. The probability distribution is given by the familiar Born rule:

$$ \Pr\limits_{|\psi\rangle}\bigl( E = c \bigr) \;=\; \langle \psi | \Pi_c | \psi \rangle \;, $$

where $\Pi_c$ is the projector onto the c-eigenspace of an 'observable quantity' E (represented by a Hermitian operator $A = \sum_c \; c \cdot \Pi_c$). The post-measured state is some projection of the state $|\psi\rangle$ onto some eigenspace of the observable A. And so if $| \psi_0 \rangle$ is the pre-measurement state, $| \psi_1 \rangle$ is the post-measurement state, and $\Pi_c$ is the 'actual result' measured (i.e. the eigenspace onto which the pre-measurement state was actually projected), we have the proportionality result

$$ | \psi_1 \rangle \;\propto\; \Pi_c | \psi_0 \rangle $$

by the projection rule just described. This is why there is the projector in your formula.

In general, the vector $| \psi_1' \rangle = \Pi_c | \psi_0 \rangle$ is not a unit vector; because we wish to describe the post-measurement state by another unit vector, we must rescale it by

$$ \|\;|\psi_1'\rangle\;\| = \sqrt{\langle \psi'_1 | \psi'_1 \rangle} = \sqrt{\langle \psi_0 | \Pi_c | \psi_0 \rangle} \;,$$

which is the square-root of the probability with which the result would occur a priori. And so, we recover the formula in your question,

$$ | \psi_1 \rangle \;=\; \frac{\Pi_c | \psi_0 \rangle}{ \sqrt{ \langle \psi_0 | \Pi_c | \psi_0 \rangle }} \;.$$

(If this formula seems slightly clumsy, take heart that it looks and feels a little bit better if you represent quantum states by density operators.)

Edited to add: the above should not be construed as a description of POVMs. A "positive operator valued measurement" is better described as a description of measuring the expectation value of the associated operators E_c in a collection { E_c }_c ∈ C .