Okay, let me break in down in terms of actual states, and this time, let’s add in the actual detection mechanism, say an electron in a potential well. Say the detector is in the ground state energy, E=0, and the absorption of a photon will bump it up to the next highest state, E=1. We will place this detector in path A, but no detector in path B.
At time t = 0, our toy wavefunction is:
1/sqrt2 |photon in path A, detector E=0> + 1/sqrt2 |photon in path B, detector E=0>
If the photon in A collides with the detector at time t =1, then at time t=2, our evolved wavefunction is:
Within the context of world A, a photon was found by the detector. This is a completely normal way to think and talk about this.
I think it’s straight up wrong to say “the photon is in the detector and in path B”. Nature doesn’t label photons, and it doesn’t distinguish between them. And what is actually in world A is an electron in a higher energy state: it would be weird to say it “contains” a photon inside of it.
Quantum mechanics does not keep track of individual objects, it keeps track of configurations of possible worlds, and assigns amplitudes to each possible way of arranging everything.
Here’s a crude Google Drawing of t = 0 to illustrate what I mean:
Both the concept of a photon and the concept of a world are abstractions on top of what is ultimately just a big pile of complex amplitudes; illusory in some sense.
I agree that talking in terms of many worlds (“within the context of world A...”) is normal and natural. But sometimes it makes sense to refer to and name concepts which span across multiple (conceptual) worlds.
I’m not claiming the conceptual boundaries I’ve drawn or terminology I’ve used in the diagram above are standard or objective or the most natural or anything like that. But I still think introducing probabilities and using terminology like “if you now put a detector in path A , it will find a photon with probability 0.5” is blurring these concepts together somewhat, in part by placing too much emphasis on the Born probabilities as fundamental / central.
But as a test, may I ask what you think the x-axis of the graph you drew is? Ie: what are the amplitudes attached to?
I’m not claiming the conceptual boundaries I’ve drawn or terminology I’ve used in the diagram above are standard or objective or the most natural or anything like that. But I still think introducing probabilities and using terminology like “if you now put a detector in path A , it will find a photon with probability 0.5” is blurring these concepts together somewhat, in part by placing too much emphasis on the Born probabilities as fundamental / central.
I think you’ve already agreed (or at least not objected to) saying that the detector “found the photon” is fine within the context of world A. I assume you don’t object to me saying that I will find the detector flashing with probability 0.5. And I assume you don’t think me and the detector should be treated differently. So I don’t think there’s any actual objection left here, you just seem vaguely annoyed that I mentioned the empirical fact that amplitudes can be linked to probabilities of outcomes. I’m not gonna apologise for that.
But as a test, may I ask what you think the x-axis of the graph you drew is? Ie: what are the amplitudes attached to?
Position, but it’s not meant to be an actual graph of a wavefunction pdf; just a way to depict how the concepts can be sliced up in a way I can actually draw in 2 dimensions.
If you do treat it as a pdf over position, a more accurate way to depict the “world” concept might be as a line which connects points on the diagram for each time step. So for a fixed time step, a world is a single point on the diagram, representing a sample from the pdf defined by the wavefunction at that time.
“position” is nearly right. The more correct answer would be “position of one photon”.
If you had two electrons, say, you would have to consider their joint configuration. For example, one possible wavefunction would look like the following, where the blobs represent high amplitude areas:
This is still only one dimensional: the two electrons are at different points along a line. I’ve entangled them, so if electron 1 is at position P, electron 2 can’t be.
Now, try and point me to where electron 1 is on the graph above.
You see, I’m not graphing electrons here, and neither were you. I’m graphing the wavefunction. This is where your phrasing seems a little weird: you say the electron is the collection of amplitudes you circled: but those amplitudes are attached to configurations saying “the electron is at position x1” or “the electron is at position x2″. It seems circular to me. Why not describe that lump as “a collection of worlds where the electron is in a similar place”?
If you have N electrons in a 3d space, the wavefunction is not a vector in 3d space (god I wish, it would make my job a lot easier). It’s a vector in 3N+1 dimensions, like the following:
where r1, r2, etc are pointing to the location of electron 1, 2, 3, etc, and each possible configuration of electron 1 here, electron 2 there, etc, has an amplitude attached, with configurations that are more often encountered experimentally empirically having higher amplitudes.
Okay, let me break in down in terms of actual states, and this time, let’s add in the actual detection mechanism, say an electron in a potential well. Say the detector is in the ground state energy, E=0, and the absorption of a photon will bump it up to the next highest state, E=1. We will place this detector in path A, but no detector in path B.
At time t = 0, our toy wavefunction is:
1/sqrt2 |photon in path A, detector E=0> + 1/sqrt2 |photon in path B, detector E=0>
If the photon in A collides with the detector at time t =1, then at time t=2, our evolved wavefunction is:
1/sqrt2 |no free photon, detector E=1> + 1/sqrt2 |photon in path B, detector E=0>
Within the context of world A, a photon was found by the detector. This is a completely normal way to think and talk about this.
I think it’s straight up wrong to say “the photon is in the detector and in path B”. Nature doesn’t label photons, and it doesn’t distinguish between them. And what is actually in world A is an electron in a higher energy state: it would be weird to say it “contains” a photon inside of it.
Quantum mechanics does not keep track of individual objects, it keeps track of configurations of possible worlds, and assigns amplitudes to each possible way of arranging everything.
Here’s a crude Google Drawing of t = 0 to illustrate what I mean:
Both the concept of a photon and the concept of a world are abstractions on top of what is ultimately just a big pile of complex amplitudes; illusory in some sense.
I agree that talking in terms of many worlds (“within the context of world A...”) is normal and natural. But sometimes it makes sense to refer to and name concepts which span across multiple (conceptual) worlds.
I’m not claiming the conceptual boundaries I’ve drawn or terminology I’ve used in the diagram above are standard or objective or the most natural or anything like that. But I still think introducing probabilities and using terminology like “if you now put a detector in path A , it will find a photon with probability 0.5” is blurring these concepts together somewhat, in part by placing too much emphasis on the Born probabilities as fundamental / central.
Nice graph!
But as a test, may I ask what you think the x-axis of the graph you drew is? Ie: what are the amplitudes attached to?
I think you’ve already agreed (or at least not objected to) saying that the detector “found the photon” is fine within the context of world A. I assume you don’t object to me saying that I will find the detector flashing with probability 0.5. And I assume you don’t think me and the detector should be treated differently. So I don’t think there’s any actual objection left here, you just seem vaguely annoyed that I mentioned the empirical fact that amplitudes can be linked to probabilities of outcomes. I’m not gonna apologise for that.
Position, but it’s not meant to be an actual graph of a wavefunction pdf; just a way to depict how the concepts can be sliced up in a way I can actually draw in 2 dimensions.
If you do treat it as a pdf over position, a more accurate way to depict the “world” concept might be as a line which connects points on the diagram for each time step. So for a fixed time step, a world is a single point on the diagram, representing a sample from the pdf defined by the wavefunction at that time.
“position” is nearly right. The more correct answer would be “position of one photon”.
If you had two electrons, say, you would have to consider their joint configuration. For example, one possible wavefunction would look like the following, where the blobs represent high amplitude areas:
This is still only one dimensional: the two electrons are at different points along a line. I’ve entangled them, so if electron 1 is at position P, electron 2 can’t be.
Now, try and point me to where electron 1 is on the graph above.
You see, I’m not graphing electrons here, and neither were you. I’m graphing the wavefunction. This is where your phrasing seems a little weird: you say the electron is the collection of amplitudes you circled: but those amplitudes are attached to configurations saying “the electron is at position x1” or “the electron is at position x2″. It seems circular to me. Why not describe that lump as “a collection of worlds where the electron is in a similar place”?
If you have N electrons in a 3d space, the wavefunction is not a vector in 3d space (god I wish, it would make my job a lot easier). It’s a vector in 3N+1 dimensions, like the following:
where r1, r2, etc are pointing to the location of electron 1, 2, 3, etc, and each possible configuration of electron 1 here, electron 2 there, etc, has an amplitude attached, with configurations that are more often encountered experimentally empirically having higher amplitudes.