# 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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### What is the dimension considered in the Schmidt Decomposition?

In the Schmidt decomposition, is the dimension considered of each Hilbert space the complex or real one? Meaning the complex dimension of $\mathbb C^2$ has dimension $2$, not $4$. If so when you ...
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### Quantum: why linear combination of vectors (superposition) is described as "both at the same time"?

I want to get a better understanding of quantum phenomena and out world in general. Before long I've thought of Schrödinger cat as being both alive and dead (or spin both up and down). Now after some ...
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### Asymptotic states and physical states in QFT scattering theory

Context In the scattering theory of QFT, one may impose the asymptotic conditions on the field: \begin{align} \lim_{t\to\pm\infty} \langle \alpha | \hat{\phi}(t,\mathbf{x}) | \beta \rangle = \sqrt{Z} \...
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### Can the Parity Operator in polar coordinates be defined as $\hat\Pi\psi(r,\theta,\phi) = \psi(r,\theta+\pi,\phi).$?

I was reading about Symmetries & Conservation Laws from Introduction to Quantum Mechanics, David J. Griffiths when I came across this question about the parity operator in three dimensions: ...
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### Can I use any linearly independent, orthogonal, eigenkets as starting basis to construct $S_x$, $S_y$ and $S_z$? [closed]

I know how to construct $S_z$ using $|\uparrow\rangle$=$\left(\begin{matrix}1\\0\end{matrix}\right)$ and $|\downarrow\rangle$=$\left(\begin{matrix}0\\1\end{matrix}\right)$ as starting basis. And I can ...
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### Completeness meaning (complete basis vs complete metric space) [migrated]

Today my professor started talking about the formalism of QM. We talked about that that eigenvectors of a Hermitian operator (over Hilbert space) is a "complete set". He also mentioned ...
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### Why we use trace-class operators and bounded operators in quantum mechanics?

The set of trace-class operators $\mathcal{B_1(H)}$ on the Hilbert space $\mathcal{H}$ is like the Banach space $l^1$, while the set of bounded operators $\mathcal{B_\infty(H)}$ is like the Banach ...
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### Does this double well potential contradict the fact that there is no degeneracy for one-dimensional bound states?

This example is taken from Modern Quantum Mechanics by Sakurai. Consider a symmetric double well potential in one-dimension with a barrier of height $V_0$ and width $a$ at the middle. The eigenstates ...
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### 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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### Why the Slavnov operator is self-adjoint? [duplicate]

In the context of BRST we can define the Slavnov operator $\Delta_{BRST}$ which generates BRST transformations. My lecture notes claim that $\Delta_{BRST}$ is self-adjoint, but I don't see why.
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### What exactly does it mean for two bosons to be in the same state?

If I understand QM correctly, it's a fact that two bosons can have the same wave function in principle. What I'm wondering is if the particles governed by the wave functions can also be in the same ...
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### How are quantum states of particles represented in particle processes?

For example, lets say we have an electron-positron annihilation scenario. What will be the density matrix representing the quantum state of the electron and the positron? What will be the density ...
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### How is the quantum harmonic oscillator related to Fock states?

The question is basically in the title. From what I understand, in the Fock state there is a certain number of particles in each energy level. The creation/annihilation operators create or destroy a ...
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### In the path integral formulation of QFT, how do we get quantized particles out of a field?

Every QFT textbook starts by basically postulating that we have discrete states connected by creation and annihilation operators. In Quantum Mechanics, we started from a differential equation and ...
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### Is the tensor product injective on pure quantum state vectors?

I am reading lecture notes on quantum information/computing, and the tensor product of two pure qubit states $|b_1\rangle\otimes |b_2\rangle\in\mathbb{C}^{2\times2}$ was introduced as the "...
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### Quantum sensing using Ion penning traps[[ODF, Dicke states and pi/2 pulse]]

I am reading the article https://www.science.org/doi/full/10.1126/science.abi5226 It's about Quantum sensing in ion penning traps. Question 1. Does the axial mode of the penning trap have nothing to ...
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