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Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

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Creation and annihilation operators in complex projective space

Note 1: The Hilbert space $\mathcal{H}$ of a quantum system cannot possibly be the genuine space of states in quantum mechanics. In particular, $\vec{0} \in \mathcal{H}$, which is not a valid state. H …
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Identical particles and the fundamental group of classical configuration space

I have read through Leinaas and Myrheim's paper. I will first present my understanding, then ask my questions. My Understanding We consider some classical configuration space$^1$ $X$. Let $h$ be a Hil …
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3 votes
6 answers
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Why do people say the dynamics of quantum mechanics is always linear?

This statement seems false. An example of a non-linear equation governing the dynamics of a quantum system is the Gross-Pitaevskii equation.
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3 votes
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Is there a no-cloning theorem for open quantum dynamics?

There are numerous papers that prove a no-cloning theorem (or more generally a no-broadcasting theorem) at various levels of generality. However, it is unclear to me if Cloning is proven to be imposs …
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Some questions on extraction of quantum probabilities from vectors and Hilbert space?

In the framework of textbook quantum mechanics, states are abstract vectors $\lvert \psi \rangle$ of a Hilbert space $\mathcal{H}$. More technically, a state is an equivalence class of vectors defined …
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Non-finite expectation values in quantum mechanics [duplicate]

In textbook quantum mechanics, one deals with expectation values of the form $$\langle O \rangle = \text{tr}(\rho O)$$ where $\rho$ is assumed to be trace-class (in particular, $\text{tr}\rho = 1$). H …
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What remains unchanged after a process in Quantum Mechanics?

Actually, energy should not be conserved. In the situation you are considering, you initially have a particle in box of side length $L$. Then, you increase the side length of the box, instantaneously, …
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1 vote
3 answers
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What is the mathematically precise definition of raising and lowering operators?

As a finite dimensional example, we have spin raising and lowering operators with the defining property $$[S_z, S_+] = \omega S_+,$$ $$\quad \quad \quad \iff [S_z, S_-] = -\omega S_-$$ for some consta …
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2 votes
1 answer
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Can one encode topology into the position operator in quantum mechanics?

We work with the concrete example of electromagnetism, but we intend to ask a question in broader scope at the end. In classical field theory, the electromagnetic potential $\mathcal{A}$ lives on a pr …
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2 votes
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Quantum mechanics with a classical Chern-Simons term

In this post, quantum mechanics falls under what is traditionally called "first quantization". This is in contrast to quantum field theory which traditionally falls under "second quantization". In tex …
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Defining the geometry of Bell inequalities

Bell inequalities can be discussed in the language of geometry. In papers such as [1], there is a general flow of definitions leading to the geometric picture of Bell inequalities: $$\text{Behaviors} …
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Are states partially ordered in the same way via entanglement and Bell violations?

Recall what a partially ordered set is. Let $E(\rho)$ be an entanglement measure. Let $B(\rho) \leq 0$ be a Bell inequality. Define the Bell violation measure with respect to $B$ as $$\tilde{B}(\rho) …
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1 vote
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How can the Lindblad master equation model decoherence?

In some references that I have read, a crucial assumption in deriving the Lindblad master equation is that the system and environment remain separable for all time. Hence, the system and environment c …
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3 votes
1 answer
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What is the importance of $SU(2)$ being the double cover of $SO(3)$?

To my understanding, it is important that $SU(2)$ is (isomorphic to) the universal cover of $SO(3)$. This is important because $SU(2)$ is then simply-connected and has a Lie algebra isomorphic to $\ma …
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4 votes
3 answers
886 views

Is the zero vector necessary to do quantum mechanics?

Textbook quantum mechanics describes systems as Hilbert spaces $\mathcal{H}$, states as unit vectors $\psi \in \mathcal{H}$, and observables as operators $O: \mathcal{H} \to \mathcal{H}$. Ultimately, …
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