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Jamister
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- proving the Heisenberg uncertainty relations for mixed states
How do you prove Heisenberg uncertainty relations for mixed states (density matrix), only from knowing the relation is true for pure states?
but you proved it from the beginning. how do you prove it by using the assumption from pure states, and not repeating the all process againvanhees71 said:You can prove it for any state. Let ##\hat{A}## and ##\hat{B}## be the self-adjoint operators representing observables and assume for simplicity that ##\langle A \rangle=\langle B \rangle=0##, where ##\langle \hat{A} \rangle=\mathrm{Tr}(\hat \rho \hat{A})##. Since ##\hat{\rho}## is a positive semi-definite self-adjoint operator there's a complete orthonormalized eigenbasis ##|u_n \rangle##, i.e.,
$$\hat{\rho}=\sum_n p_n |u_n \rangle \langle u_n |.$$
Thus any expectation value reads
$$\langle A \rangle = \mathrm{Tr} (\hat{\rho} \hat{A}) = \sum_{n} \langle u_n| \rho_n \hat{A} |u_n \rangle = \sum_n P_n \langle u_n |\hat{A}| u_n \rangle.$$
Now apply this to the self-adjoint operator
$$\hat{C}=(\hat{A}-\mathrm{i} \lambda \hat{B})(\hat{A}+\mathrm{i} \lambda \hat{B}), \quad \lambda \in \mathbb{R}.$$
First of all
$$\langle u_n|\hat{C}| u_n \rangle=\langle (\hat{A}+\mathrm{i} \lambda \hat{B}) u_n|(\hat{A}+\mathrm{i} \lambda \hat{B}) u_n \rangle \geq 0.$$
And since ##P_n \geq 0## for all ##n## you also have
$$P(\lambda)=\langle (\hat{A}-\mathrm{i} \lambda \hat{B})(\hat{A}+\mathrm{i} \lambda \hat{B}) \rangle \geq 0.$$
Multiplying this out gives
$$P(\lambda) = \langle A^2 \rangle + \langle \mathrm{i} [\hat{A},\hat{B}] \rangle \lambda + \lambda^2 \langle \hat{B}^2 \rangle.$$
Now ##\langle A^2 \rangle=\Delta A^2## and ##\langle B^2 \rangle=\Delta B^2## are the variances of ##A## and ##B## (because we assumed ##\langle A \rangle=\langle B \rangle=0##), i.e.,
$$P(\lambda)=\lambda^2 \Delta B^2 + \lambda \langle \mathrm{i} [\hat{A},\hat{B}]\rangle + \Delta A^2 \geq 0$$
für ##\lambda \geq 0##. Now ##P(\lambda)## is a real quadratic polynomial and thus its discriminant fulfills,
$$\frac{1}{4} \langle \mathrm{i} [\hat{A},\hat{B}] \rangle^2 - \Delta A^2 \Delta B^2 \leq 0,$$
and from this finally
$$\Delta A \Delta B \geq \frac{1}{2} |\langle \mathrm{i} [\hat{A},\hat{B}] \rangle|,$$
which is the Heisenberg-Robertson uncertainty relation.
I don't think it's obvioushilbert2 said:I wouldn't expect the uncertainty in any observable to get smaller by adding statistical uncertainty to the already existing quantum uncertainty in the position and momentum. It can only increase.
There's something said about this on page 16 here: https://www.univie.ac.at/nuhag-php/dateien/talks/1073_NuHAG_WPI_2008.pdf
You only get one proof for free! If you want another one, you have to show some effort yourself.Jamister said:but you proved it from the beginning. how do you prove it by using the assumption from pure states, and not repeating the all process again
For pure states the proof is almost the same. You just set one of the ##p_n=1## and all others to 0.Jamister said:but you proved it from the beginning. how do you prove it by using the assumption from pure states, and not repeating the all process again
The Heisenberg Uncertainty Relation for mixed states is a principle in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. This is because the more accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa.
The original Heisenberg Uncertainty Principle only applies to pure states, where the state of a particle is known with certainty. The Heisenberg Uncertainty Relation for mixed states takes into account the uncertainty in the state of a particle, which can be a combination of multiple pure states.
The mathematical expression for the Heisenberg Uncertainty Relation for mixed states is: ΔxΔp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant.
The Heisenberg Uncertainty Relation for mixed states is a fundamental principle of quantum mechanics that highlights the inherent uncertainty in the behavior of particles at the quantum level. It challenges our classical intuition and forces us to accept that there are limits to what we can know about the behavior of particles in the quantum world.
No, the Heisenberg Uncertainty Relation for mixed states is a fundamental principle in quantum mechanics and cannot be violated. It is a consequence of the wave-particle duality of quantum particles and is supported by numerous experimental observations.