Questions tagged [hilbert-space]
This tag is for questions relating to Hilbert Space, a vector space equipped with an inner product, an operation that allows defining lengths and angles, and the space is complete. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.
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How is a quantum superposition different from a mixed state?
According to Wikipedia, if a system has $50\%$ chance to be in state $\left|\psi_1\right>$ and $50\%$ to be in state $\left|\psi_2\right>$, then this is a mixed state. Now, consider the state $...
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Understanding the Bloch sphere
It is usually said that the points on the surface of the Bloch sphere represent the pure states of a single 2-level quantum system. A pure state being of the form: $$ |\psi\rangle = a |0\rangle+b |1\...
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Hilbert space of harmonic oscillator: Countable vs uncountable?
Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of ...
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What is the difference between general measurement and projective measurement?
Nielsen and Chuang mention in Quantum Computation and Information that there are two kinds of measurement : general and projective ( and also POVM but that's not what I'm worried about ). General ...
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Differences between pure/mixed/entangled/separable/superposed states
I am currently trying to establish a clear picture of pure/mixed/entangled/separable/superposed states. In the following I will always assume a basis of $|1\rangle$ and $|0\rangle$ for my quantum ...
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Why do we need infinite-dimensional Hilbert spaces in physics?
I am looking for a simple way to understand why do we need infinite-dimensional Hilbert spaces in physics, and when exactly do they become neccessary: in classical, quantum, or relativistic quantum ...
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Rigged Hilbert space and QM
Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.
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Why are Only Real Things Measurable?
Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
user24082
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Density matrix formalism
The density matrix $\hat{\rho}$ is often introduced in textbooks as a mathematical convenience that allows us to describe quantum systems in which there is some level of missing information. $\hat{\...
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What is the issue with interactions in QFT?
I've started studying QFT this year and in trying to find a more rigorous approach to the subject I ended up find out lots of people saying that "there is no way known yet to make QFT rigorous when ...
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Not all self-adjoint operators are observables?
The WP article on the density matrix has this remark: It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable.[...
user4552
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What's wrong with this derivation that $i\hbar = 0$?
Let $\hat{x} = x$ and $\hat{p} = -i \hbar \frac {\partial} {\partial x}$ be the position and momentum operators, respectively, and $|\psi_p\rangle$ be the eigenfunction of $\hat{p}$ and therefore $$\...
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Why ket and bra notation?
So, I've been trying to teach myself about quantum computing, and I found a great YouTube series called Quantum Computing for the Determined. However. Why do we use ket/bra notation? Normal vector ...
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A "Hermitian" operator with imaginary eigenvalues
Let $${\bf H}=\hat{x}^3\hat{p}+\hat{p}\hat{x}^3$$ where $\hat{p}=-id/dx$. Clearly ${\bf H}^{\dagger}={\bf H}$, because ${\bf H}={\bf T} + {\bf T}^{\dagger}$, where ${\bf T}=\hat{x}^3\hat{p}$. In this ...
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Physical interpretation of different selfadjoint extensions
Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. ...
