Energy measurement from position eigenstateIs this a sufficient condition for a state to be an...
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Energy measurement from position eigenstate
Is this a sufficient condition for a state to be an eigenstate?Measurement of energy apparently violating the position-momentum Uncertainty Principle in a potential that does not depend on distance?How do I get observables to calculate uncertainty?Definitions of position operator in QMProbability of quantum transitionProbability of Finding a Particle in the Ground-state after the Size of the Box is DoubledIssue with 1D particle in a box position expectation valueExpectation values of the position operator is equal to zero in case of even potentials?Expectation values of odd operators in case of even potentials are zero?How to find the coordinate representation of the kinetic operator?
$begingroup$
Given that the eigenstates of the position operator can be written as $delta(x-x')$, and suppose we measure a particle in an infinite potential with walls at $x=0$ and $x=L$. I measure the particle to be in the position $x=L/2$, so the particle is in the eigenstate $ |x rangle = delta(x-L/2)$. Suppose now that I want to measure the energy of the particle. The eigenstates of the energy operator are given by:
$$ |psi_nrangle = sqrt{frac{2}{L}}sin left( frac{npi x}{L} right) $$
In order to measure energy I understand that I have to expand the original eigenstate in terms of the new energy eigenstates:
$$
|xrangle = sum|psi_nranglelanglepsi_n|xrangle
$$
where the probability of collapse into the eigenstate is given by:
$$
P_n = |langlepsi_n|xrangle|^2
$$
But now I sort of run into an issue. Sure then, I can say that:
$$
langlepsi_n|xrangle = int sqrt{frac{2}{L}}sin left( frac{npi x}{L} right)delta(x-L/2)dx
$$
and since
$$
int delta(x-x')f(x)dx = f(x')
$$
I can say
$$
langlepsi_n|xrangle = sqrt{frac{2}{L}}sin left( frac{npi }{2} right)
$$
and,
$$
P_n=|langlepsi_n|xrangle|^2 = frac{2}{L}sin^2 left( frac{npi}{2} right)
$$
I know that this means that all odd values of n are equally probably and all even values are not possible, but probability is supposed to be dimensionless, so what's happened here? What rookie error have I made?
quantum-mechanics homework-and-exercises energy potential quantum-measurement
edited 2 hours ago
Aaron Stevens
13.5k42250
asked 5 hours ago
monkeyofsciencemonkeyofscience
574
$endgroup$
|
show 5 more comments
$begingroup$
Given that the eigenstates of the position operator can be written as $delta(x-x')$, and suppose we measure a particle in an infinite potential with walls at $x=0$ and $x=L$. I measure the particle to be in the position $x=L/2$, so the particle is in the eigenstate $ |x rangle = delta(x-L/2)$. Suppose now that I want to measure the energy of the particle. The eigenstates of the energy operator are given by:
$$ |psi_nrangle = sqrt{frac{2}{L}}sin left( frac{npi x}{L} right) $$
In order to measure energy I understand that I have to expand the original eigenstate in terms of the new energy eigenstates:
$$
|xrangle = sum|psi_nranglelanglepsi_n|xrangle
$$
where the probability of collapse into the eigenstate is given by:
$$
P_n = |langlepsi_n|xrangle|^2
$$
But now I sort of run into an issue. Sure then, I can say that:
$$
langlepsi_n|xrangle = int sqrt{frac{2}{L}}sin left( frac{npi x}{L} right)delta(x-L/2)dx
$$
and since
$$
int delta(x-x')f(x)dx = f(x')
$$
I can say
$$
langlepsi_n|xrangle = sqrt{frac{2}{L}}sin left( frac{npi }{2} right)
$$
and,
$$
P_n=|langlepsi_n|xrangle|^2 = frac{2}{L}sin^2 left( frac{npi}{2} right)
$$
I know that this means that all odd values of n are equally probably and all even values are not possible, but probability is supposed to be dimensionless, so what's happened here? What rookie error have I made?
quantum-mechanics homework-and-exercises energy potential quantum-measurement
edited 2 hours ago
Aaron Stevens
13.5k42250
asked 5 hours ago
monkeyofsciencemonkeyofscience
574
$endgroup$
$begingroup$
Rookie. $|psirangle$ is dimensionless, but $|xrangle$ has dimension $L^{-1/2}$, as you may check from $langle x| x'rangle=delta(x-x')$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
That is... confusing kets with wave functions as you do makes the first few formulas you write magnificently meaningless... Recall $psi(x)=langle x| psirangle$, with dimension $L^{-1/2}$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
Gotcha. So would it be correct to say that $psi (x) = langle x|L/2 rangle$? And $|psirangle = |L/2rangle$? So then if my energy eigenstates are represented by $|phirangle$ then $P_n = |langle phi| L/2 rangle|^2$?
$endgroup$
– monkeyofscience
3 hours ago
1
$begingroup$
No, normalized w.f.s have dimension $L^{-1/2}$, but what you wrote has dimensions of 1/L, like a delta function. So that is your dimensional mismatch (your "And ...?"). You may try a truly normalized Gaussian w.f. with labile width, and take the width to zero in a limit, but the expressions you write in this comment are visibly inconsistent.
$endgroup$
– Cosmas Zachos
3 hours ago
$begingroup$
Dang it. So how on earth do I get the position eigenfunctions to a point where I can talk meaningfully about the probability of an energy eigenfunction?
$endgroup$
– monkeyofscience
3 hours ago
|
show 5 more comments
$begingroup$
Given that the eigenstates of the position operator can be written as $delta(x-x')$, and suppose we measure a particle in an infinite potential with walls at $x=0$ and $x=L$. I measure the particle to be in the position $x=L/2$, so the particle is in the eigenstate $ |x rangle = delta(x-L/2)$. Suppose now that I want to measure the energy of the particle. The eigenstates of the energy operator are given by:
$$ |psi_nrangle = sqrt{frac{2}{L}}sin left( frac{npi x}{L} right) $$
In order to measure energy I understand that I have to expand the original eigenstate in terms of the new energy eigenstates:
$$
|xrangle = sum|psi_nranglelanglepsi_n|xrangle
$$
where the probability of collapse into the eigenstate is given by:
$$
P_n = |langlepsi_n|xrangle|^2
$$
But now I sort of run into an issue. Sure then, I can say that:
$$
langlepsi_n|xrangle = int sqrt{frac{2}{L}}sin left( frac{npi x}{L} right)delta(x-L/2)dx
$$
and since
$$
int delta(x-x')f(x)dx = f(x')
$$
I can say
$$
langlepsi_n|xrangle = sqrt{frac{2}{L}}sin left( frac{npi }{2} right)
$$
and,
$$
P_n=|langlepsi_n|xrangle|^2 = frac{2}{L}sin^2 left( frac{npi}{2} right)
$$
I know that this means that all odd values of n are equally probably and all even values are not possible, but probability is supposed to be dimensionless, so what's happened here? What rookie error have I made?
quantum-mechanics homework-and-exercises energy potential quantum-measurement
edited 2 hours ago
Aaron Stevens
13.5k42250
asked 5 hours ago
monkeyofsciencemonkeyofscience
574
$endgroup$
Given that the eigenstates of the position operator can be written as $delta(x-x')$, and suppose we measure a particle in an infinite potential with walls at $x=0$ and $x=L$. I measure the particle to be in the position $x=L/2$, so the particle is in the eigenstate $ |x rangle = delta(x-L/2)$. Suppose now that I want to measure the energy of the particle. The eigenstates of the energy operator are given by:
$$ |psi_nrangle = sqrt{frac{2}{L}}sin left( frac{npi x}{L} right) $$
In order to measure energy I understand that I have to expand the original eigenstate in terms of the new energy eigenstates:
$$
|xrangle = sum|psi_nranglelanglepsi_n|xrangle
$$
where the probability of collapse into the eigenstate is given by:
$$
P_n = |langlepsi_n|xrangle|^2
$$
But now I sort of run into an issue. Sure then, I can say that:
$$
langlepsi_n|xrangle = int sqrt{frac{2}{L}}sin left( frac{npi x}{L} right)delta(x-L/2)dx
$$
and since
$$
int delta(x-x')f(x)dx = f(x')
$$
I can say
$$
langlepsi_n|xrangle = sqrt{frac{2}{L}}sin left( frac{npi }{2} right)
$$
and,
$$
P_n=|langlepsi_n|xrangle|^2 = frac{2}{L}sin^2 left( frac{npi}{2} right)
$$
I know that this means that all odd values of n are equally probably and all even values are not possible, but probability is supposed to be dimensionless, so what's happened here? What rookie error have I made?
quantum-mechanics homework-and-exercises energy potential quantum-measurement
quantum-mechanics homework-and-exercises energy potential quantum-measurement
edited 2 hours ago
Aaron Stevens
13.5k42250
asked 5 hours ago
monkeyofsciencemonkeyofscience
574
edited 2 hours ago
Aaron Stevens
13.5k42250
asked 5 hours ago
monkeyofsciencemonkeyofscience
574
edited 2 hours ago
Aaron Stevens
13.5k42250
edited 2 hours ago
Aaron Stevens
13.5k42250
edited 2 hours ago
Aaron Stevens
13.5k42250
13.5k42250
asked 5 hours ago
monkeyofsciencemonkeyofscience
574
asked 5 hours ago
monkeyofsciencemonkeyofscience
574
asked 5 hours ago
monkeyofsciencemonkeyofscience
574
574
$begingroup$
Rookie. $|psirangle$ is dimensionless, but $|xrangle$ has dimension $L^{-1/2}$, as you may check from $langle x| x'rangle=delta(x-x')$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
That is... confusing kets with wave functions as you do makes the first few formulas you write magnificently meaningless... Recall $psi(x)=langle x| psirangle$, with dimension $L^{-1/2}$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
Gotcha. So would it be correct to say that $psi (x) = langle x|L/2 rangle$? And $|psirangle = |L/2rangle$? So then if my energy eigenstates are represented by $|phirangle$ then $P_n = |langle phi| L/2 rangle|^2$?
$endgroup$
– monkeyofscience
3 hours ago
1
$begingroup$
No, normalized w.f.s have dimension $L^{-1/2}$, but what you wrote has dimensions of 1/L, like a delta function. So that is your dimensional mismatch (your "And ...?"). You may try a truly normalized Gaussian w.f. with labile width, and take the width to zero in a limit, but the expressions you write in this comment are visibly inconsistent.
$endgroup$
– Cosmas Zachos
3 hours ago
$begingroup$
Dang it. So how on earth do I get the position eigenfunctions to a point where I can talk meaningfully about the probability of an energy eigenfunction?
$endgroup$
– monkeyofscience
3 hours ago
|
show 5 more comments
$begingroup$
Rookie. $|psirangle$ is dimensionless, but $|xrangle$ has dimension $L^{-1/2}$, as you may check from $langle x| x'rangle=delta(x-x')$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
That is... confusing kets with wave functions as you do makes the first few formulas you write magnificently meaningless... Recall $psi(x)=langle x| psirangle$, with dimension $L^{-1/2}$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
Gotcha. So would it be correct to say that $psi (x) = langle x|L/2 rangle$? And $|psirangle = |L/2rangle$? So then if my energy eigenstates are represented by $|phirangle$ then $P_n = |langle phi| L/2 rangle|^2$?
$endgroup$
– monkeyofscience
3 hours ago
1
$begingroup$
No, normalized w.f.s have dimension $L^{-1/2}$, but what you wrote has dimensions of 1/L, like a delta function. So that is your dimensional mismatch (your "And ...?"). You may try a truly normalized Gaussian w.f. with labile width, and take the width to zero in a limit, but the expressions you write in this comment are visibly inconsistent.
$endgroup$
– Cosmas Zachos
3 hours ago
$begingroup$
Dang it. So how on earth do I get the position eigenfunctions to a point where I can talk meaningfully about the probability of an energy eigenfunction?
$endgroup$
– monkeyofscience
3 hours ago
$begingroup$
Rookie. $|psirangle$ is dimensionless, but $|xrangle$ has dimension $L^{-1/2}$, as you may check from $langle x| x'rangle=delta(x-x')$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
Rookie. $|psirangle$ is dimensionless, but $|xrangle$ has dimension $L^{-1/2}$, as you may check from $langle x| x'rangle=delta(x-x')$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
That is... confusing kets with wave functions as you do makes the first few formulas you write magnificently meaningless... Recall $psi(x)=langle x| psirangle$, with dimension $L^{-1/2}$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
That is... confusing kets with wave functions as you do makes the first few formulas you write magnificently meaningless... Recall $psi(x)=langle x| psirangle$, with dimension $L^{-1/2}$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
Gotcha. So would it be correct to say that $psi (x) = langle x|L/2 rangle$? And $|psirangle = |L/2rangle$? So then if my energy eigenstates are represented by $|phirangle$ then $P_n = |langle phi| L/2 rangle|^2$?
$endgroup$
– monkeyofscience
3 hours ago
$begingroup$
Gotcha. So would it be correct to say that $psi (x) = langle x|L/2 rangle$? And $|psirangle = |L/2rangle$? So then if my energy eigenstates are represented by $|phirangle$ then $P_n = |langle phi| L/2 rangle|^2$?
$endgroup$
– monkeyofscience
3 hours ago
1
1
$begingroup$
No, normalized w.f.s have dimension $L^{-1/2}$, but what you wrote has dimensions of 1/L, like a delta function. So that is your dimensional mismatch (your "And ...?"). You may try a truly normalized Gaussian w.f. with labile width, and take the width to zero in a limit, but the expressions you write in this comment are visibly inconsistent.
$endgroup$
– Cosmas Zachos
3 hours ago
$begingroup$
No, normalized w.f.s have dimension $L^{-1/2}$, but what you wrote has dimensions of 1/L, like a delta function. So that is your dimensional mismatch (your "And ...?"). You may try a truly normalized Gaussian w.f. with labile width, and take the width to zero in a limit, but the expressions you write in this comment are visibly inconsistent.
$endgroup$
– Cosmas Zachos
3 hours ago
$begingroup$
Dang it. So how on earth do I get the position eigenfunctions to a point where I can talk meaningfully about the probability of an energy eigenfunction?
$endgroup$
– monkeyofscience
3 hours ago
$begingroup$
Dang it. So how on earth do I get the position eigenfunctions to a point where I can talk meaningfully about the probability of an energy eigenfunction?
$endgroup$
– monkeyofscience
3 hours ago
|
show 5 more comments
1 Answer
1
active
oldest
votes
$begingroup$
The formula
$$
P_n = |langlepsi_n|psirangle|^2
$$
assumes that the pre-measurement state $|psirangle$ and the observable's eigenstates $|psi_nrangle$ are both normalized to be unit state-vectors. In other words, the formula for arbitrary non-zero state-vectors is
$$
P_n = frac{|langlepsi_n|psirangle|^2}{
langlepsi_n|psi_nrangle,langlepsi|psirangle}.
$$
Notice that this expression for $P_n$ is dimensionless by construction.
The problem with the case described in the OP is that $|xrangle=delta(x-L/2)$ is not normalizable: it does not belong to the Hilbert space, so it can't be used for the pre-measurement state $|psirangle$.
That's not a problem in principle, because real position-measurments don't have infinite precision, so the state after a real position-measurement would not be $|xrangle$. It would be some normalizable state-vector $|psirangle$ whose corresponding wavefunction is sharply concentrated near a particular position, but with a non-zero width that makes it normalizable.
answered 4 hours ago
Chiral AnomalyChiral Anomaly
12.4k21540
$endgroup$
1
$begingroup$
How does your equation for $P$ work if you switched the two vectors? i.e. for the position probability density we usually see of $|langle x|psirangle|^2$ ?
$endgroup$
– Aaron Stevens
34 mins ago
1
$begingroup$
@AaronStevens That's a great question. (You always ask great questions!) For the position probability density, we would use $P(x)=|langle x|psirangle|^2/langlepsi|psirangle$. The other factor in the denominator isn't included in this case, and the normalization is instead absorbed into the meaning of "density."
$endgroup$
– Chiral Anomaly
28 mins ago
1
$begingroup$
@AaronStevens Another way to think of it is that the possible outcomes of a measurement are represented by projection operators that project the originally-normalizable state onto a new normalizable state. In the case of a position measurement, we can project into a finite interval, but not to a point; but we can define a density that can be integrated over any finite interval to give the same result. Thanks for asking that question -- it needed to be addressed.
$endgroup$
– Chiral Anomaly
27 mins ago
1
$begingroup$
That all makes sense. It just seems odd to me that one is doable and one is not because $|langle x|psirangle|^2=langle psi|xranglelangle x |psirangle=langle x|psiranglelanglepsi|xrangle=|langlepsi|xrangle|^2$ But perhaps the issue is in the interpretations of how these values relate to measurements. Certainly the probability of finding the particle at any specified location is $0$, so the problem assumes an impossible thing already (and we don't have to think about normalization for that).
$endgroup$
– Aaron Stevens
21 mins ago
1
$begingroup$
@AaronStevens Yeah, that's a quirk of the notation. The order in which the inner product is written is not what matters; what matters is the roles of the two state-vectors. One of them is the pre-measurement state (which must be normalizable), and one of them is the post-measurement state (which must be normalizable if we want a probability, but can be non-normalizable if we're content with a density). The asymmetry in the meaning is not reflected in the notation. That's one advantage of expressing things in terms of projection operators; then the notation is more clearly asymmetric.
$endgroup$
– Chiral Anomaly
18 mins ago
|
show 3 more comments
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1 Answer
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active
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1 Answer
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active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
The formula
$$
P_n = |langlepsi_n|psirangle|^2
$$
assumes that the pre-measurement state $|psirangle$ and the observable's eigenstates $|psi_nrangle$ are both normalized to be unit state-vectors. In other words, the formula for arbitrary non-zero state-vectors is
$$
P_n = frac{|langlepsi_n|psirangle|^2}{
langlepsi_n|psi_nrangle,langlepsi|psirangle}.
$$
Notice that this expression for $P_n$ is dimensionless by construction.
The problem with the case described in the OP is that $|xrangle=delta(x-L/2)$ is not normalizable: it does not belong to the Hilbert space, so it can't be used for the pre-measurement state $|psirangle$.
That's not a problem in principle, because real position-measurments don't have infinite precision, so the state after a real position-measurement would not be $|xrangle$. It would be some normalizable state-vector $|psirangle$ whose corresponding wavefunction is sharply concentrated near a particular position, but with a non-zero width that makes it normalizable.
answered 4 hours ago
Chiral AnomalyChiral Anomaly
12.4k21540
$endgroup$
1
$begingroup$
How does your equation for $P$ work if you switched the two vectors? i.e. for the position probability density we usually see of $|langle x|psirangle|^2$ ?
$endgroup$
– Aaron Stevens
34 mins ago
1
$begingroup$
@AaronStevens That's a great question. (You always ask great questions!) For the position probability density, we would use $P(x)=|langle x|psirangle|^2/langlepsi|psirangle$. The other factor in the denominator isn't included in this case, and the normalization is instead absorbed into the meaning of "density."
$endgroup$
– Chiral Anomaly
28 mins ago
1
$begingroup$
@AaronStevens Another way to think of it is that the possible outcomes of a measurement are represented by projection operators that project the originally-normalizable state onto a new normalizable state. In the case of a position measurement, we can project into a finite interval, but not to a point; but we can define a density that can be integrated over any finite interval to give the same result. Thanks for asking that question -- it needed to be addressed.
$endgroup$
– Chiral Anomaly
27 mins ago
1
$begingroup$
That all makes sense. It just seems odd to me that one is doable and one is not because $|langle x|psirangle|^2=langle psi|xranglelangle x |psirangle=langle x|psiranglelanglepsi|xrangle=|langlepsi|xrangle|^2$ But perhaps the issue is in the interpretations of how these values relate to measurements. Certainly the probability of finding the particle at any specified location is $0$, so the problem assumes an impossible thing already (and we don't have to think about normalization for that).
$endgroup$
– Aaron Stevens
21 mins ago
1
$begingroup$
@AaronStevens Yeah, that's a quirk of the notation. The order in which the inner product is written is not what matters; what matters is the roles of the two state-vectors. One of them is the pre-measurement state (which must be normalizable), and one of them is the post-measurement state (which must be normalizable if we want a probability, but can be non-normalizable if we're content with a density). The asymmetry in the meaning is not reflected in the notation. That's one advantage of expressing things in terms of projection operators; then the notation is more clearly asymmetric.
$endgroup$
– Chiral Anomaly
18 mins ago
|
show 3 more comments
$begingroup$
The formula
$$
P_n = |langlepsi_n|psirangle|^2
$$
assumes that the pre-measurement state $|psirangle$ and the observable's eigenstates $|psi_nrangle$ are both normalized to be unit state-vectors. In other words, the formula for arbitrary non-zero state-vectors is
$$
P_n = frac{|langlepsi_n|psirangle|^2}{
langlepsi_n|psi_nrangle,langlepsi|psirangle}.
$$
Notice that this expression for $P_n$ is dimensionless by construction.
The problem with the case described in the OP is that $|xrangle=delta(x-L/2)$ is not normalizable: it does not belong to the Hilbert space, so it can't be used for the pre-measurement state $|psirangle$.
That's not a problem in principle, because real position-measurments don't have infinite precision, so the state after a real position-measurement would not be $|xrangle$. It would be some normalizable state-vector $|psirangle$ whose corresponding wavefunction is sharply concentrated near a particular position, but with a non-zero width that makes it normalizable.
answered 4 hours ago
Chiral AnomalyChiral Anomaly
12.4k21540
$endgroup$
1
$begingroup$
How does your equation for $P$ work if you switched the two vectors? i.e. for the position probability density we usually see of $|langle x|psirangle|^2$ ?
$endgroup$
– Aaron Stevens
34 mins ago
1
$begingroup$
@AaronStevens That's a great question. (You always ask great questions!) For the position probability density, we would use $P(x)=|langle x|psirangle|^2/langlepsi|psirangle$. The other factor in the denominator isn't included in this case, and the normalization is instead absorbed into the meaning of "density."
$endgroup$
– Chiral Anomaly
28 mins ago
1
$begingroup$
@AaronStevens Another way to think of it is that the possible outcomes of a measurement are represented by projection operators that project the originally-normalizable state onto a new normalizable state. In the case of a position measurement, we can project into a finite interval, but not to a point; but we can define a density that can be integrated over any finite interval to give the same result. Thanks for asking that question -- it needed to be addressed.
$endgroup$
– Chiral Anomaly
27 mins ago
1
$begingroup$
That all makes sense. It just seems odd to me that one is doable and one is not because $|langle x|psirangle|^2=langle psi|xranglelangle x |psirangle=langle x|psiranglelanglepsi|xrangle=|langlepsi|xrangle|^2$ But perhaps the issue is in the interpretations of how these values relate to measurements. Certainly the probability of finding the particle at any specified location is $0$, so the problem assumes an impossible thing already (and we don't have to think about normalization for that).
$endgroup$
– Aaron Stevens
21 mins ago
1
$begingroup$
@AaronStevens Yeah, that's a quirk of the notation. The order in which the inner product is written is not what matters; what matters is the roles of the two state-vectors. One of them is the pre-measurement state (which must be normalizable), and one of them is the post-measurement state (which must be normalizable if we want a probability, but can be non-normalizable if we're content with a density). The asymmetry in the meaning is not reflected in the notation. That's one advantage of expressing things in terms of projection operators; then the notation is more clearly asymmetric.
$endgroup$
– Chiral Anomaly
18 mins ago
|
show 3 more comments
$begingroup$
The formula
$$
P_n = |langlepsi_n|psirangle|^2
$$
assumes that the pre-measurement state $|psirangle$ and the observable's eigenstates $|psi_nrangle$ are both normalized to be unit state-vectors. In other words, the formula for arbitrary non-zero state-vectors is
$$
P_n = frac{|langlepsi_n|psirangle|^2}{
langlepsi_n|psi_nrangle,langlepsi|psirangle}.
$$
Notice that this expression for $P_n$ is dimensionless by construction.
The problem with the case described in the OP is that $|xrangle=delta(x-L/2)$ is not normalizable: it does not belong to the Hilbert space, so it can't be used for the pre-measurement state $|psirangle$.
That's not a problem in principle, because real position-measurments don't have infinite precision, so the state after a real position-measurement would not be $|xrangle$. It would be some normalizable state-vector $|psirangle$ whose corresponding wavefunction is sharply concentrated near a particular position, but with a non-zero width that makes it normalizable.
answered 4 hours ago
Chiral AnomalyChiral Anomaly
12.4k21540
$endgroup$
The formula
$$
P_n = |langlepsi_n|psirangle|^2
$$
assumes that the pre-measurement state $|psirangle$ and the observable's eigenstates $|psi_nrangle$ are both normalized to be unit state-vectors. In other words, the formula for arbitrary non-zero state-vectors is
$$
P_n = frac{|langlepsi_n|psirangle|^2}{
langlepsi_n|psi_nrangle,langlepsi|psirangle}.
$$
Notice that this expression for $P_n$ is dimensionless by construction.
The problem with the case described in the OP is that $|xrangle=delta(x-L/2)$ is not normalizable: it does not belong to the Hilbert space, so it can't be used for the pre-measurement state $|psirangle$.
That's not a problem in principle, because real position-measurments don't have infinite precision, so the state after a real position-measurement would not be $|xrangle$. It would be some normalizable state-vector $|psirangle$ whose corresponding wavefunction is sharply concentrated near a particular position, but with a non-zero width that makes it normalizable.
answered 4 hours ago
Chiral AnomalyChiral Anomaly
12.4k21540
answered 4 hours ago
Chiral AnomalyChiral Anomaly
12.4k21540
answered 4 hours ago
Chiral AnomalyChiral Anomaly
12.4k21540
answered 4 hours ago
Chiral AnomalyChiral Anomaly
12.4k21540
12.4k21540
1
$begingroup$
How does your equation for $P$ work if you switched the two vectors? i.e. for the position probability density we usually see of $|langle x|psirangle|^2$ ?
$endgroup$
– Aaron Stevens
34 mins ago
1
$begingroup$
@AaronStevens That's a great question. (You always ask great questions!) For the position probability density, we would use $P(x)=|langle x|psirangle|^2/langlepsi|psirangle$. The other factor in the denominator isn't included in this case, and the normalization is instead absorbed into the meaning of "density."
$endgroup$
– Chiral Anomaly
28 mins ago
1
$begingroup$
@AaronStevens Another way to think of it is that the possible outcomes of a measurement are represented by projection operators that project the originally-normalizable state onto a new normalizable state. In the case of a position measurement, we can project into a finite interval, but not to a point; but we can define a density that can be integrated over any finite interval to give the same result. Thanks for asking that question -- it needed to be addressed.
$endgroup$
– Chiral Anomaly
27 mins ago
1
$begingroup$
That all makes sense. It just seems odd to me that one is doable and one is not because $|langle x|psirangle|^2=langle psi|xranglelangle x |psirangle=langle x|psiranglelanglepsi|xrangle=|langlepsi|xrangle|^2$ But perhaps the issue is in the interpretations of how these values relate to measurements. Certainly the probability of finding the particle at any specified location is $0$, so the problem assumes an impossible thing already (and we don't have to think about normalization for that).
$endgroup$
– Aaron Stevens
21 mins ago
1
$begingroup$
@AaronStevens Yeah, that's a quirk of the notation. The order in which the inner product is written is not what matters; what matters is the roles of the two state-vectors. One of them is the pre-measurement state (which must be normalizable), and one of them is the post-measurement state (which must be normalizable if we want a probability, but can be non-normalizable if we're content with a density). The asymmetry in the meaning is not reflected in the notation. That's one advantage of expressing things in terms of projection operators; then the notation is more clearly asymmetric.
$endgroup$
– Chiral Anomaly
18 mins ago
|
show 3 more comments
1
$begingroup$
How does your equation for $P$ work if you switched the two vectors? i.e. for the position probability density we usually see of $|langle x|psirangle|^2$ ?
$endgroup$
– Aaron Stevens
34 mins ago
1
$begingroup$
@AaronStevens That's a great question. (You always ask great questions!) For the position probability density, we would use $P(x)=|langle x|psirangle|^2/langlepsi|psirangle$. The other factor in the denominator isn't included in this case, and the normalization is instead absorbed into the meaning of "density."
$endgroup$
– Chiral Anomaly
28 mins ago
1
$begingroup$
@AaronStevens Another way to think of it is that the possible outcomes of a measurement are represented by projection operators that project the originally-normalizable state onto a new normalizable state. In the case of a position measurement, we can project into a finite interval, but not to a point; but we can define a density that can be integrated over any finite interval to give the same result. Thanks for asking that question -- it needed to be addressed.
$endgroup$
– Chiral Anomaly
27 mins ago
1
$begingroup$
That all makes sense. It just seems odd to me that one is doable and one is not because $|langle x|psirangle|^2=langle psi|xranglelangle x |psirangle=langle x|psiranglelanglepsi|xrangle=|langlepsi|xrangle|^2$ But perhaps the issue is in the interpretations of how these values relate to measurements. Certainly the probability of finding the particle at any specified location is $0$, so the problem assumes an impossible thing already (and we don't have to think about normalization for that).
$endgroup$
– Aaron Stevens
21 mins ago
1
$begingroup$
@AaronStevens Yeah, that's a quirk of the notation. The order in which the inner product is written is not what matters; what matters is the roles of the two state-vectors. One of them is the pre-measurement state (which must be normalizable), and one of them is the post-measurement state (which must be normalizable if we want a probability, but can be non-normalizable if we're content with a density). The asymmetry in the meaning is not reflected in the notation. That's one advantage of expressing things in terms of projection operators; then the notation is more clearly asymmetric.
$endgroup$
– Chiral Anomaly
18 mins ago
1
1
$begingroup$
How does your equation for $P$ work if you switched the two vectors? i.e. for the position probability density we usually see of $|langle x|psirangle|^2$ ?
$endgroup$
– Aaron Stevens
34 mins ago
$begingroup$
How does your equation for $P$ work if you switched the two vectors? i.e. for the position probability density we usually see of $|langle x|psirangle|^2$ ?
$endgroup$
– Aaron Stevens
34 mins ago
1
1
$begingroup$
@AaronStevens That's a great question. (You always ask great questions!) For the position probability density, we would use $P(x)=|langle x|psirangle|^2/langlepsi|psirangle$. The other factor in the denominator isn't included in this case, and the normalization is instead absorbed into the meaning of "density."
$endgroup$
– Chiral Anomaly
28 mins ago
$begingroup$
@AaronStevens That's a great question. (You always ask great questions!) For the position probability density, we would use $P(x)=|langle x|psirangle|^2/langlepsi|psirangle$. The other factor in the denominator isn't included in this case, and the normalization is instead absorbed into the meaning of "density."
$endgroup$
– Chiral Anomaly
28 mins ago
1
1
$begingroup$
@AaronStevens Another way to think of it is that the possible outcomes of a measurement are represented by projection operators that project the originally-normalizable state onto a new normalizable state. In the case of a position measurement, we can project into a finite interval, but not to a point; but we can define a density that can be integrated over any finite interval to give the same result. Thanks for asking that question -- it needed to be addressed.
$endgroup$
– Chiral Anomaly
27 mins ago
$begingroup$
@AaronStevens Another way to think of it is that the possible outcomes of a measurement are represented by projection operators that project the originally-normalizable state onto a new normalizable state. In the case of a position measurement, we can project into a finite interval, but not to a point; but we can define a density that can be integrated over any finite interval to give the same result. Thanks for asking that question -- it needed to be addressed.
$endgroup$
– Chiral Anomaly
27 mins ago
1
1
$begingroup$
That all makes sense. It just seems odd to me that one is doable and one is not because $|langle x|psirangle|^2=langle psi|xranglelangle x |psirangle=langle x|psiranglelanglepsi|xrangle=|langlepsi|xrangle|^2$ But perhaps the issue is in the interpretations of how these values relate to measurements. Certainly the probability of finding the particle at any specified location is $0$, so the problem assumes an impossible thing already (and we don't have to think about normalization for that).
$endgroup$
– Aaron Stevens
21 mins ago
$begingroup$
That all makes sense. It just seems odd to me that one is doable and one is not because $|langle x|psirangle|^2=langle psi|xranglelangle x |psirangle=langle x|psiranglelanglepsi|xrangle=|langlepsi|xrangle|^2$ But perhaps the issue is in the interpretations of how these values relate to measurements. Certainly the probability of finding the particle at any specified location is $0$, so the problem assumes an impossible thing already (and we don't have to think about normalization for that).
$endgroup$
– Aaron Stevens
21 mins ago
1
1
$begingroup$
@AaronStevens Yeah, that's a quirk of the notation. The order in which the inner product is written is not what matters; what matters is the roles of the two state-vectors. One of them is the pre-measurement state (which must be normalizable), and one of them is the post-measurement state (which must be normalizable if we want a probability, but can be non-normalizable if we're content with a density). The asymmetry in the meaning is not reflected in the notation. That's one advantage of expressing things in terms of projection operators; then the notation is more clearly asymmetric.
$endgroup$
– Chiral Anomaly
18 mins ago
$begingroup$
@AaronStevens Yeah, that's a quirk of the notation. The order in which the inner product is written is not what matters; what matters is the roles of the two state-vectors. One of them is the pre-measurement state (which must be normalizable), and one of them is the post-measurement state (which must be normalizable if we want a probability, but can be non-normalizable if we're content with a density). The asymmetry in the meaning is not reflected in the notation. That's one advantage of expressing things in terms of projection operators; then the notation is more clearly asymmetric.
$endgroup$
– Chiral Anomaly
18 mins ago
|
show 3 more comments
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$begingroup$
Rookie. $|psirangle$ is dimensionless, but $|xrangle$ has dimension $L^{-1/2}$, as you may check from $langle x| x'rangle=delta(x-x')$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
That is... confusing kets with wave functions as you do makes the first few formulas you write magnificently meaningless... Recall $psi(x)=langle x| psirangle$, with dimension $L^{-1/2}$.
$endgroup$
– Cosmas Zachos
4 hours ago
$begingroup$
Gotcha. So would it be correct to say that $psi (x) = langle x|L/2 rangle$? And $|psirangle = |L/2rangle$? So then if my energy eigenstates are represented by $|phirangle$ then $P_n = |langle phi| L/2 rangle|^2$?
$endgroup$
– monkeyofscience
3 hours ago
1
$begingroup$
No, normalized w.f.s have dimension $L^{-1/2}$, but what you wrote has dimensions of 1/L, like a delta function. So that is your dimensional mismatch (your "And ...?"). You may try a truly normalized Gaussian w.f. with labile width, and take the width to zero in a limit, but the expressions you write in this comment are visibly inconsistent.
$endgroup$
– Cosmas Zachos
3 hours ago
$begingroup$
Dang it. So how on earth do I get the position eigenfunctions to a point where I can talk meaningfully about the probability of an energy eigenfunction?
$endgroup$
– monkeyofscience
3 hours ago