The significance of the Dirac notation

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entropy1
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If we have the wavefunction ##|ab \rangle##, what do the a and b stand for? In particular, do a and b signify an outcome of some pending or possible measurement, or do they signify some aspect of the wavefunction, and if so, which aspect?
 

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PeroK
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If we have the wavefunction ##|ab \rangle##, what do the a and b stand for? In particular, do a and b signify an outcome of some pending or possible measurement, or do they signify some aspect of the wavefunction, and if so, which aspect?
Where are you learning Dirac notation?
 
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entropy1
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Where are you learning Dirac notation?
I learned it from Susskind. Perhaps he explained it in his book but I can't really remember.
 
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PeroK
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I learned it from Susskind.
What does Susskind say is meant by ##|ab \rangle##?
 
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entropy1
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What does Susskind say is meant by ##|ab \rangle##?
I figured it was equivalent with ##|↑↓\rangle##.
 
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PeroK
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I figured it was equivalent with ##|↑↓\rangle##.
##a## and ##b## are variables and ##\uparrow## and ##\downarrow## represent specific states. Do you understand the concept of a state?
 
  • #7
entropy1
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Do you understand the concept of a state?
No, not really. I am confused about it. I suppose it is a vector in Hilbert Space.
 
  • #9
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If I understand you correctly, a and b represent states. My question is: which states do they represent? The states of an unmeasured wavefunction or the states of measured outcomes (eigenvectors)?
 
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dextercioby
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They are generic ones, even though in most situations are eigenstates of particular observables, case in which a and b are replaced by other letters or signs.

More precisely, mathematicians are fond of using the letter A for a generic (not necessarily self-adjoint) operator in a Hilbert space. In Dirac notation its spectral equation then reads [tex]|a\rangle =a |a\rangle [/tex].
 
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If I understand you correctly, a and b represent states. My question is: which states do they represent?
They are variables. They could be any states, given the context they are presented in. Similar to ##ax^2 + bx + c## being a general quadratic equation.
 
  • #12
entropy1
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Maybe to be more precise: if we have ##c_1|↑↓\rangle + c_2|↓↑\rangle## representing an entangled state, what do each of the four states signify?
 
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Maybe to be more precise: if we have ##c_1|↑↓\rangle + c_2|↓↑\rangle## representing an entangled state, what do each of the four states signify?
The state ##|\uparrow \downarrow \rangle## is a composite (two particle) state where (loosely) the first particle is in the up state and the second particle is in the down state. The state ##|\downarrow \uparrow \rangle## is similarly defined. The whole thing is a linear combination of these two states.
 
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entropy1
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The state ##|\uparrow \downarrow \rangle## is a composite (two particle) state where (loosely) the first particle is in the up state and the second particle is in the down state. The state ##|\downarrow \uparrow \rangle## is similarly defined. The whole thing is a linear combination of these two states.
They can't however mean the states of the particles before measurement, for, say, the spin has no value. So it must be the state after measurement, then, or, the eigenvector that represents the measurement outcome?
 
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They can't however mean the states of the particles before measurement, for, say, the spin has no value. So it must be the state after measurement, then, or, the eigenvector that represents the measurement outcome?
I suspect you are missing something fundamental in your understanding of the nature of QM states. So, I'm not sure how to answer that. The state you quoted does not imply a measurement of spin on the system or on either particle.

You could say that the system has been prepared in that entangled state.
 
  • #16
entropy1
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I suspect you are missing something fundamental in your understanding of the nature of QM states. So, I'm not sure how to answer that. The state you quoted does not imply a measurement of spin on the system or on either particle.

You could say that the system has been prepared in that entangled state.
But the states (the four arrows) must represent something right? If the spins (let's assume) of the entangled pair are not defined before measurement, this entangled state does not represent those spins, right? Yet.
 
  • #17
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But the states (the four arrows) must represent something right? If the spins (let's assume) of the entangled pair are not defined before measurement, this entangled state does not represent those spins, right? Yet.
You don't have to assume that the spins are not well-defined, the entangled composite state tells you that. Like all states it contains the information about the likelihood of any measurement outcome on the system.
 
  • #18
entropy1
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Ok. Suppose particle A measures spin up. The entangled state "collapses" to ##|↑↓\rangle##. So the ↓ state can now signify the state of particle B (before it is measured). So now ↑ signifies a measurement outcome and ↓ a state before measurement, not a measurement outcome.

But you could also measure particle B first, in which case ↓ signifies the measurement outcome (particle B), and ↑ the state of particle A (before measurement). So in this matter, what is really going on with respect to the ontology of the arrow states?

You could say that ##|↑↓\rangle## means that you can measure particle A up and particle B down. But if the measurement basises are not the same, we won't measure these states simultaneously.
 
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PeroK
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So in this matter, what is really going on?
What do you mean by that? This is the way QM works.
 
  • #20
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If we have the wavefunction ##|ab \rangle##, what do the a and b stand for? In particular, do a and b signify an outcome of some pending or possible measurement, or do they signify some aspect of the wavefunction, and if so, which aspect?
The symbols within a bra or a ket are just labels.

We start with some Hilbert space that is appropriate for whatever problem we're considering. Then we think of a convenient scheme for labeling the vectors of that Hilbert space, and use these labels as we please. A two-dimensional Hilbert space will be spanned by two basis vectors, and if our problem involves the z-axis spin of the particle it's natural to choose the spin-up-z and spin-down-z vectors as the basis. Once we've done that, we can call these vectors ##|\uparrow\rangle## and ##|\downarrow\rangle##, or ##|z_+\rangle## and ##|z_-\rangle##, or ##|+\rangle## and ##|-\rangle##, or ##|sheep\rangle## and ##|goat\rangle##, or ......
You will always need the context (possibly implied) to know which vectors in which Hilbert space are represented by any particular kets.

There is one fairly common convention that you'll want to be aware of. Suppose the operator ##A## acts on the vectors of one Hilbert space, and we've decided to write the eigenvector of ##A## that has eigenvalue ##a## as ##|a\rangle##; and the operator ##B## acts on the vectors of another Hilbert space, and we've decided to write the eigenvector of ##B## that has eigenvalue ##b## as ##|b\rangle##. In this case the vector ##|a\rangle\otimes|b\rangle## is a vector in yet a third Hilbert space, the tensor product of the first two, and this will often be written as ##|ab\rangle##. (An example would be a two particle system in which ##A## measures the spin of the one particle on some axis, and ##B## measures the spin of the other particle - that's what's going on with the ##|\uparrow\downarrow\rangle## notation).

This convention is common enough that you can assume that it is the answer to your question unless something else in the surrounding context says otherwise.
 
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  • #21
entropy1
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What do you mean by that? This is the way QM works.
I just mean the four arrows in the entangled state are signifying different things depending on your point of view.
 
  • #22
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I just mean the four arrows in the entangled state are signifying different things depending on your point of view.
There's no point of view involved. There is a well-defined (entangled) state for a composite system of two particles.
 
  • #23
entropy1
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There's no point of view involved. There is a well-defined (entangled) state for a composite system of two particles.
Ok so we have a different POV :oldbiggrin:

If you feel like it, read #18 :wink:

I don't know why I said I didn't understand the state concept. Well, at least not fully.
 
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  • #24
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I just mean the four arrows in the entangled state are signifying different things depending on your point of view.
No, they mean whatever they mean in whatever notation we decided to use.
 
  • #25
entropy1
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No, they mean whatever they mean in whatever notation we decided to use.
Sure. And it works if you mean that. But I find it confusing. It appears to me as if it has no clearly defined ontology. Anyway, not everyone shares this concern of course.
 
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