70
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Why is it impossible to measure position and momentum at the same time with arbitrary precision?
When someone asks "Is it really impossible to simultaneously measure position and momentum with arbitrary precision in quantum theory?", the best preliminary answer one can give is another ...
- 3,446
40
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The importance of the phase in quantum mechanics
When people say that the phase doesn't matter, they mean the overall, "global" phase. In other words, the state $|0 \rangle$ is equivalent to $e^{i \theta} |0 \rangle$, the state $|1\rangle$ is ...
- 99.9k
36
votes
Am I in a superposition?
This is a deceptively simple question, and I would like to answer it in the context of Condensed Matter Theory. We will start with the question: why do objects in our everyday life appear to be ...
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36
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Is my interpretation of the underlying idea behind this 2022 Nobel Prize story “How physicists proved the universe isn’t real” more or less accurate?
First, it's important to clarify two definitions.
The principle of locality loosely says that interactions occur at individual points in space. If you say that one charged particle at a point $\...
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29
votes
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Why do eigenvalues correspond to observable quantities?
Suppose we don't know quantum mechanics yet and we want to calculate the expecatation value of an observable $A$. Could be momentum, spin whatever. It is given by
$$\mathbb E(A)=\sum_i a_i\,p(a_i)$$
...
23
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Why are wavefunction collapses instantaneous?
It’s important to remember that quantum mechanics is a tool that we use to describe the world — it is not the same as the world. For all that we love to talk about wavefunctions, it’s not clear at ...
rob♦
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21
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Why do we use Eigenvalues to represent Observed Values in Quantum Mechanics?
One of the postulates of quantum mechanics is that for every observable A, there corresponds a linear Hermitian operator A^, and when we measure the observable A, we get an eigenvalue of A^ as the ...
- 66k
20
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Are observables in QFT actually observable?
A quantum mechanical secret kept very well in plain sight is that the word "observable" when defined as "self-adjoint operator on a Hilbert space" does not actually mean "you ...
- 120k
20
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Does the collapse of the wave function depend on the observer?
The measurement problem is one of the most relevant open problems of quantum mechanics. What is a measurement? What constitutes an observer and what doesn't? Is the wavefunction a physical object (...
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19
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Do we or do we not observe (measure) superpositions all the time?
To answer the question, you have to specify what observable’s eigenstates the “superpositions” you’re talking about are superpositions of.
If you measure the energy, you always get one single energy ...
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17
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Confusion about the description of the uncertainty principle
The first one is correct, the second is not.
The second definition1 is actually describing the observer effect. Explanations written by non-experts often mix the two up. But one key difference is ...
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17
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Why is it impossible to measure position and momentum at the same time with arbitrary precision?
You can't measure precise values at the same time because precise values for both don't exist at the same time.
All the properties of, say, an electron and be inferred from the electron's wave ...
- 34.9k
17
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What is the Hamiltonian of a Measurement?
The following argument is very vague and hand-wavey - as is inevitable when discussing measurement apparatus with on the order of $10^{23}$ particles. It is intended only to illustrate the idea. We ...
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16
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The importance of the phase in quantum mechanics
The global phase does not matter. In your example
$\lambda(\vert\psi_1\rangle+\vert\psi_2\rangle)$ has the same physical contents as $\vert\psi_1\rangle+\vert\psi_2\rangle$ but this will be in ...
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16
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Are two states with the same measurement probabilities necessarily equal up to unitary equivalence?
Let $H$ denote a finite-dimensional complex Hilbert space, $\rho$ and $\rho^\prime$ be two density matrices, i.e. positive semi-definite operators with unit trace and $A$ an arbitrary hermitian ...
- 7,063
14
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Proof that you can't disentangle two parties if you only operate on one
If there was some unitary operator factorized as $\mathbb I_A \otimes U_B$ that would send the entangled state $|\psi\rangle$ to a factorized state $|\phi_A\rangle\otimes |\phi_B\rangle$, ie :
$$|\...
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14
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Accepted
Can the collapse of the wave function be modelled as a quantum system on its own?
To model the act of measurement itself as an interaction of the measurement apparatus and the measured system as quantum systems is a perfectly standard thing to do, though you might get disagreements ...
- 120k
12
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Why do eigenvalues correspond to observable quantities?
I think the conceptual issue is that you've got it backwards. Given any set of eigenfunctions, you can have operators with those eigenfunctions, whose corresponding eigenvalues are anything you want. ...
- 99.9k
12
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How is it that a particle's wave function is not a real thing, yet we can still observe it?
Plenty of things in physics are mathematical objects/models that we use to describe the world around us and can also use to guide our measurements and observations. For example, when you are walking ...
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11
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A question regarding the commutators of operators
I think that the misunderstanding follows from the following fact. Given that $[A,B]=0$ then we can build a simultaneous eigenbasis for $A$ and $B$, call it $\mathcal{B}_1$.
On the other hand we have ...
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Are observables in QFT actually observable?
As emphasized in the excellent answer by ACuriousMind, some of the operators that we usually call "observables" might not be feasibly measurable through any physical process that the theory ...
- 53.2k
11
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Which experiments confirm that the Born rule is $|\psi|^2$ rather than $|\psi|$?
I might be missing the depth somewhere, but I would suggest the electron double-slit experiment first: $|\sin(kx)|$ clearly looks different as compared to $\sin^2(kx)$.
The second example is the Malus'...
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10
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Why do we use Eigenvalues to represent Observed Values in Quantum Mechanics?
The Stern-Gerlach experiment and similar experiments show
A system has a state.
The states of a system form a Hilbert space. You can choose a set of basis states and represent the system's current ...
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10
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Am I in a superposition?
TL;DR Your position is not well-defined... but for more mundane reasons.
Quantum mechanical view
You are not in a pure state - that is you are not an object, that can be described by a wave function, ...
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10
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Am I in a superposition?
The question 'where am I', taken at a microscopic level, is hardly meaningful.
A human body, as a physical system, is overwhelmingly complex. It certainly cannot be described as a quantum system in ...
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10
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Are two states with the same measurement probabilities necessarily equal up to unitary equivalence?
Actually $\rho=\rho'$. Indeed, specializing $A= P= |\psi\rangle \langle \psi|$ for $||\psi||=1$, the hypothesis implies $\langle\psi| (\rho-\rho')\psi\rangle =0$. Linearity permits to relax the ...
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10
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What reason/evidence do we have to think that the Planck length is the smallest length possible?
Or are there only amateurs who think so?
Bingo :) The Planck units don’t give sharp limits on the existence of quantities like length, time, etc (or rather, there’s no reason to think that they do). ...
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10
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What is the Hamiltonian of a Measurement?
Measurement is an irreversible process, while the time evolution given by the Schrödinger equation is reversible.
The Schrödinger equation gives you unitary time evolution. But wave function collapse ...
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9
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The importance of the phase in quantum mechanics
While the other answers are correct, this is not a different answer but rather an illustration that the relative phase is indeed important in quantum mechanics. We know that bosons (particles with ...
- 409
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Accepted
Measurement on mixed states
As mentioned by the OP both versions are the same. For an observable $A$ of the form
$$A = \sum\limits_k a_k \, P_k \quad , $$
with the projections $P_k^2 =P_k = P_k^\dagger$ on the eigenspace ...
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