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nomadreid
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There are two kinds of uncertainty in QM: one of operators (Uncertainty Principles), and one of states (superposition). Is there any direct connection between them?
I presume you mean that you cannot see the superposition from a single measurement, since you could presumably see the superposition by doing an experiment on lots of identical states.
Is there any way that one of these uncertainties can be derived from the other? I see the Uncertainty Principles derived from a straightforward derivation on non-commuting Hermitian operators, but I only come across the mechanism of superposition as a postulate.
A question about your example: it uses a situation with a single operator, but uncertainty principles involve two operators, so upon reflection I am unclear as to how your example would work.
Yes, precisely, but the uncertainty principles I was referring to were the kind such as the Heisenberg Uncertainty Principle, that do refer to two operators. So your example was explaining the "uncertainty" of the superposition principle, but not explaining the link between that kind of uncertainty (superposition) and the Heisenberg Uncertainty Principle (or similar ones).A question about your example: it uses a situation with a single operator, but uncertainty principles involve two operators, so upon reflection I am unclear as to how your example would work.
"Uncertainty" from superposition principle is not about 2 operators. Only about the measured result of the observable in which eigenstate basis you are writing the state vector.
Ravi Mohan said:Consider the system with two dimensional Hilbert space, say spin 1/2. Now we have three operators
[itex]\hat{M}_x[/itex], [itex]\hat{M}_y[/itex] and [itex]\hat{M}_z[/itex] and consider that they don't commute. Consider the system in the eigen state of [itex]\hat{M}_z[/itex] and let it be [itex]|\uparrow_z\rangle[/itex] (spin up in [itex]\hat{a}_z[/itex] direction).
Now consider the relation
[itex]\hat{M}_z[/itex][itex]\hat{M}_x [/itex][itex]\neq[/itex][itex] \hat{M}_x[/itex][itex]\hat{M}_z[/itex]
[itex]\hat{M}_z[/itex][itex]\hat{M}_x [/itex][itex]|\uparrow_z\rangle[/itex][itex]\neq[/itex][itex] \hat{M}_x[/itex][itex]\hat{M}_z[/itex][itex]|\uparrow_z\rangle[/itex]
[itex]\hat{M}_z[/itex][itex]\hat{M}_x [/itex][itex]|\uparrow_z\rangle[/itex][itex]\neq[/itex][itex]\lambda[/itex][itex] \hat{M}_x[/itex][itex]|\uparrow_z\rangle[/itex] (where [itex]\lambda[/itex] is some number)
Very interesting. Now, I am surely being dense, but I would be grateful if you could spell it out for me: why does this conclusion lead to the postulate of superposition?Ravi Mohan said:or
[itex]\hat{M}_z[/itex][itex]\big([/itex][itex]\hat{M}_x [/itex][itex]|\uparrow_z\rangle[/itex][itex]\big)[/itex] [itex]\neq[/itex][itex]\lambda[/itex][itex]\big([/itex][itex] \hat{M}_x[/itex][itex]|\uparrow_z\rangle[/itex][itex]\big)[/itex]
It means [itex] \hat{M}_x[/itex][itex]|\uparrow_z\rangle[/itex] is not an eigen state of [itex] \hat{M}_z[/itex]
Let [itex]|\uparrow_x\rangle[/itex] be the eigen state of
[itex] \hat{M}_x[/itex]. Now since [itex] \hat{M}_x[/itex] is a measurement operator (thus self adjoint), it should be like [itex]|\uparrow_x\rangle[/itex][itex]\langle\uparrow_x|[/itex].
[itex] \hat{M}_x[/itex][itex]|\uparrow_z\rangle[/itex] = [itex]|\uparrow_x\rangle[/itex][itex]\langle\uparrow_x|[/itex][itex]|\uparrow_z\rangle[/itex] = [itex]c_1[/itex][itex]|\uparrow_x\rangle[/itex]
Now [itex]c_1[/itex][itex]|\uparrow_x\rangle[/itex] [itex]\neq[/itex] [itex]|\uparrow_z\rangle[/itex] and this is due to the reason that [itex]\hat{M}_z[/itex] and[itex]\hat{M}_x [/itex] don't commute.
No it doesn't mean "for all λ" at all. It depends on the definition of operator [itex]\hat{M}_z[/itex] and vector [itex]|\uparrow_z\rangle[/itex]. I never said it is normalised.nomadreid said:I presume you mean "for all λ ..."
Ok so I have shown that the eigenvectors of non-commuting aperators are not the same. In this case [itex]c_1[/itex][itex]|\uparrow_x\rangle[/itex] [itex]\neq[/itex] [itex]|\uparrow_z\rangle[/itex] and thus to represent [itex]|\uparrow_z\rangle[/itex] you will need a basis set. Now the eigenvectors of spin along x-axis do form a basis. Hence you can represent [itex]|\uparrow_z\rangle[/itex] as superposition!nomadreid said:Very interesting. Now, I am surely being dense, but I would be grateful if you could spell it out for me: why does this conclusion lead to the postulate of superposition?
The uncertainty principle in quantum mechanics states that it is impossible to simultaneously determine the exact position and momentum of a subatomic particle with absolute precision. This is due to the inherent uncertainty and randomness of quantum systems.
The uncertainty principle challenges our traditional understanding of the physical world by showing that the behavior of subatomic particles cannot be predicted with certainty. It also highlights the limitations of our current scientific methods and encourages us to think about the universe in a probabilistic way.
Superposition is a phenomenon in quantum mechanics where a particle can exist in multiple states simultaneously. This means that a particle can have multiple properties, such as position, momentum, and spin, all at the same time.
Superposition is closely related to the uncertainty principle because it is the reason why we cannot determine the exact state of a particle. Since a particle can exist in multiple states at once, it is impossible to measure its exact position or momentum without affecting its other properties.
While the uncertainty principle and superposition are fundamental principles of quantum mechanics, their effects are not noticeable in everyday life. They only become significant on the microscopic scale of subatomic particles. However, advancements in technology have allowed scientists to observe these principles in controlled laboratory settings.