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Questions tagged [hilbert-space]

Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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Symmetry acting on a complex fermion operator

Suppose $S$ is a $\mathbb{Z}_2$ symmetry operator, i.e. $S^2=1$, acting on the fermion $c_{n}$ via $$S c_{n} S^{-1} = \sum_{m} U_{nm} c_{m}$$ and I am interested in $S$ is both linear or anti linear, ...
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What is a singular continuous spectrum?

I read some answers about this and the wikipedia page that basically always say that a spectrum can be decomposed into: $$\mu = \mu_{ac} + \mu_{sc} + \mu_{pp}, $$ where $\mu_{ac}$ is absolutely ...
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1answer
70 views

Matrix representation of spin-2 system? [on hold]

I am surprised no one has asked this before, but what is the matrix representation of a spin-2 system? Also, what are the equivalent of the Pauli matrices for the system?
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3answers
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Spin states in hilbert space

Do spin states (for ex: $\langle u| $ & $\langle d| $ and $\langle l| $ & $\langle r| $) along different axes (x-, y-, z- axis) of a quantum object belong to the same Hilbert space (where $\...
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Total angular momentum operator on $|m\rangle$ and $|m-1\rangle$ results in different eigenvalue [duplicate]

In the lectures by Prof. Leonard Susskind, he mentioned that the total angular momentum squared operator can be represented by $$ L^2 = L_z^2 + L_z + L_- L_+ $$ ($L_+, L_-$ being ladder operators). ...
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1answer
74 views

How to simplify this expression in Dirac notation

An expression cropped up in a homework problem that I'm not sure how to simplify. Consider the following, where $|x\rangle $ is a position eigenstate and $|p_1\rangle, |p_2\rangle$ are momentum ...
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1answer
132 views

Why is the wave-function defined in abstract configuration space, as opposed to real space $\mathbb{R}^3$?

I've just began studying QM so I apologise in advance if this question is silly or badly put. My initial understanding of the wave-function $\psi(x,t)$ was that it exists in real 3-dimensional space $...
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2answers
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Mathematical formulation of quantum mechanics

I am reading a book on quantum mechanics, but it is difficult to understand. Quantum mechanics is roughly formulated as follows: Physicsl state is a normalized ray $\{e^{i\theta}\psi|\theta \in \...
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1answer
34 views

Perelomov coherent states for an arbitrary Hamiltonian

I'm reading about Perelomov coherent states, but I'm not sure if I'm getting it right. From this question and some Perelomov papers I understand the following: The Perelomov coherent states are ...
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1answer
46 views

Eigenstates of position in Schrödinger picture

Hallo I'm trying to understand the concept of representation in the position space. I read that $|x\rangle$ are the eigenstates of the position operator, but I think this states should evolve in time ...
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1answer
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Say you run qubit $A = (|0⟩ + |1⟩) / √2$ and qubit $B = |0⟩$ through a CNOT gate. What is the state of qubit $B$ afterwards?

I am new to the weeds of quantum computing and this question is probably pretty elementary. Say you run qubit A = (|0⟩ + |1⟩) / √2 and qubit B = |0⟩ through a CNOT gate. What is the state of qubit B ...
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About superposition of states

In quantum computing, we can always create an arbitrary superposition of states by rotation of $|0\rangle$ state for one qubit. This raises a question: for arbitrary superposition of states, is there ...
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1answer
62 views

Does $[L_z,H] = 0$ imply the state is also an Eigenstate of $H$ is also an eigenstate of $L_z$?

Given that the Hamiltonian $\mathcal{H}$ is rotationally invariant then we know $[L_z,\mathcal{H}] = 0$. Does that imply that an eigenstate of H is also an eigenstate of $\mathcal{H}$? More ...
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1answer
141 views

What does $|$ mean in the Schrödinger Equation?

I saw the $|$ symbol in the Schrödinger Equation $$i\hbar\frac{\partial}{\partial{t}}|\Psi(r,t)\rangle=\hat{H}|\Psi(r,t)\rangle$$ But I don't know what the $|$ means. What does $|$ mean in the ...
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2answers
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Relation between giving the form of an operator in a given representation, and bra ket notation [on hold]

So I understand that kets are abstract objects that are the elemnets of a Hilberts space. Say $|\psi \rangle$. We can write this ket in a position representation $\langle r|\psi \rangle = \psi(r)$, ...
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Doubt regarding the angular momentum in quantum mechanics

What does $L^2|l,m\rangle$ indicate? Can anyone specify the 2 states in a ket vector of angular momentum $|l,m\rangle$? $$L^2|l,m\rangle=\hbar^2(l+1)l|l,m\rangle$$ and how to differentiate between $\...
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1answer
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Does total $\hat{S}^2$ always commute with total $\hat{S}_z$ even for interacting spins?

I was given the following operator $\hat{f}$ describing the interaction of two spin-$\frac12$ particles: $$\hat{f}=a+b{\hat{\bf S}_1}\cdot{\hat{\bf S}_2}.$$ I was told that I can prove that $\hat{f}$...
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2answers
79 views

Magnitude of the cross product of two bra-kets?

From the mathematical perspective, one can take the magnitude of a cross product: $$ |a\times b|=|a| |b| \sin{\theta}\cdot n, $$ where $\theta$ is the angle between a and b in the plane containing ...
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0answers
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Why do we assume wavefunctions to be finite and continuous everywhere? [duplicate]

Why do we assume the wavefunctions to be everywhere finite and continuous? Finiteness maybe required due to square integrability but why continuous? Such a restriction is not imposed on its derivative....
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2answers
132 views

Quantum mechanics and Group theory

Vectors are representations transform under $SO(3)$ Group, 4-vectors are representations transform under $SO(1,3)$ Group, Like wave function in discrete but infinite basis (hilbert space) are some ...
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2answers
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Demonstration of the completness of an orthonormal set of functions

I find this concept of completness a little bit dense when it comes to prove this property of some set of orthonormal functions. In one of my classes, my professor proved this for the orthonormal set ...
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1answer
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How is expectation of $x^2$ at time $t$ calculated?

Ehrenfest theorem for position operator states $$\frac{d\left<x \right>}{dt} = \left<[H,x]\right> + \left<\frac{\partial x}{\partial t}\right>$$ where $H = \frac{p^2}{2m}$ and $\...
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Bras, kets, a Hilbert space and its dual

So I’m trying to get this all straightened out in my head. In Quantum mechanics we use a Hilbert Space $\mathcal{H}$ as our vector space and we say that its elements is the set of kets $\left|\psi\...
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1answer
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Why can't we superpose two quantum vacuum states?

i read in this paper (Spontaneous Symmetry Breaking as the Mechanism of Quantum Measurement by Michael Grady) that we are not allowed to consider the superposition of two vacuum states. i do not ...
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2answers
33 views

Different global phase shifts of Pauli-$z$ Matrix eigenstates from rotations around $z$-axis

I understand the pauli matrix $\sigma_z = \bigl( \begin{smallmatrix}1 & 0\\ 0 & -1\end{smallmatrix}\bigr)$ rotates a state around $z$-axis by angle $\pi$ in $SO(3)$. We can see it works by ...
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0answers
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QFT Hilbert space and analysis of quantum black holes

QFT Hilbert space is infinite dimensional and it is known that given a region $A$ and its complement $A^c$ of the spacetime, the QFT Hilbert space $\textbf{does not}$ decompose into $\mathcal{H}_A \...
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1answer
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Definition of QFT in Vertex Operator Algebra by Kac

QFT is composed of the following data with some axioms(I omitted them here). (1) Hilbert space $H$. (2) Vacuum belongs to $H$. (3) There is unitary representation of Poincare group. (4) A ...
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1answer
95 views

Representation of operators and wavefunctions as matrices and vectors

I remember reading somewhere that in quantum mechanics you can always set up your Hilbert space to be finite or countably infinite. However, I don't see how it's possible to to that we want to ...
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1answer
39 views

The charge given by a commutator

I saw in the text that $[Q,X]=cX$ and says the operator $X$ has charge $c$ under the generator $Q$. I tried to understand why the coefficient $c$ means the charge. So I used this relation to get the ...
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1answer
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Symmetry in Fock-space 2-body interaction

The simplest two body interaction term for fermions is $$H = \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l$$ and I'm trying to determine the symmetries on $U$. Unfortunately I keep getting ...
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1answer
67 views

Hermitian operator

When we say that an operator is Hermitian in QM, does it depend on the Hilbert space under consideration, or not? Are there operators that are Hermitian in one Hilbert space but not in another?
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Infinite Series vs Integral Representation of State Vectors in QM? [duplicate]

Shankar's Principles of Quantum Mechanics, pg. 57-59 subsection Generalisation to Infinite Dimensions states that state vectors defined as, $$|f_n\rangle = \sum_{i} f_n(x_i) |x_i\rangle, \tag{1}$$ ...
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3answers
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Real symmetric matrix in Wigner's theorem

A consequence of Wigner's theorem is that if a Hamiltonian matrix obeys time reversal symmetry then it is real-symmetric. It seems to me that for this to make sense then "real symmetric" should be a ...
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1answer
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Full Hilbert space of a particle

Given the Hilbert space $H$ of a single particle, we know we can write \begin{equation} H = H_{spatial}\otimes H_{other} \end{equation} where $H_{spatial}$ is spanned by the possible position states ...
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2answers
654 views

Rigorous justification for non-relativistic QM perturbation theory assumptions?

In perturbation theory for non-relativistic quantum mechanics, you begin with a Hamiltonian of the form $$H=H_0+\lambda H'$$ and assume that the perturbed eigenstates and eigenvalues can be written as ...
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1answer
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Verifying the “quantum teleportation identity” in $\mathbb{C}^2 \otimes \mathbb{C}^2$ (Bell basis)

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}$ Let $\{\psi_{i,j} : i, j = 0,1\}$ be the Bell basis of $\...
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1answer
133 views

Why should there be one-particle states in an interacting quantum field theory?

I'm a mathematician trying to learn quantum field theory. This question has two parts: first, I want to double check that I'm thinking about the surrounding issues correctly, after that I'll ask my ...
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2answers
95 views

Is a quantum harmonic oscillator always infinite dimensional?

Let us assume we have a quantum particle in a harmonic potential with the Hamiltonian $$H = \sum_n n \omega |n\rangle\langle n|$$ If I am not mistaken. Now when talking about harmonic oscillators ...
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Conceptual understanding of the Quantum Harmonic oscillator

First: When we consider a quantum particle in a harmonic (quadratic) potential we say that this particle is a harmonic oscillator, because it behaves like one. Is this correct? Now let us assume our ...
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2answers
64 views

Angular momentum coupling

I read about angular momentum coupling on wikipedia and there are a few things i dont understand. What does this mean "spin and orbital angular momentum of a single object belong to different Hilbert ...
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4answers
110 views

Why do wavefunctions for stationary states include $e^{-iEt/\hbar}$? [duplicate]

Stationary states are separable solutions with $\Psi(x, t)=\psi(x)e^{-iEt/\hbar}$. But why is that there? Griffiths (Section 2.1 Stationary states, equation 2.8) says that observables for these states ...
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What is the QFT state with two distinguishable fermions present?

I want to describe a system with two non-interacting and definitely different fermions, say an electron neutrino, $\nu_e$, and an electron, $e^-$. The state describing a single electron is given ...
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1answer
60 views

An intuitive explanation of the basis-independency of EPR pair

As usual, we write the EPR pair as $$ \frac{1}{\sqrt2}(\left|00\right> + \left|11\right>). $$ A property of the EPR pair is that this definition is basis-independent, which means $$ \frac{1}{\...
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Why is the orbital part of the ground state ket of two spin 1/2 particles a direct product?

In Shankar QM on page 406-407 he says that both electrons are in the lowest orbital state $|n=1, l=0,m=0>$ and have opposite spins so the orbital part of the ground-state ket is just the direct ...
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1answer
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Orthonormal basis written in Dirac Notation

$\left\{ e _ { i } \right\}$ is an orthonormal basis which has the orthonormal condition as following: $$e _ { i } ^ { T } \cdot e _ { j } = \delta _ { i j }$$ In Dirac Notation where $| i \rangle = | ...
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1answer
66 views

What is “Rotational Invariance” in the context of qubits

In this question the state, $\frac{1}{\sqrt{3}}\left|00\right\rangle +\frac{1}{\sqrt{3}}\left|01\right\rangle +\frac{1}{\sqrt{3}}\left|11\right\rangle$, has been said in the answers to not be able to ...
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0answers
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Need help understanding weird definition of pure states [duplicate]

So in many sources I have read that A pure state contains only one element, since the only entry on the density matrix will be 1. But what about superpositions?...
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2answers
94 views

How does a Hamiltonian 'generate' a unitary?

I know that the unitary (propagator) is given by $$U=e^{iHt}\tag{1}.$$ But I actually never saw how a Hamiltonian translates into a unitary. For example when I consider a two-level rotation in a ...
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2answers
75 views

Does something prevent superposition at our scale?

I often encounter the argument that quantum mechanics reduces to classical mechanics at sufficiently big scales, as soon as h becomes sufficiently small respect to the actions involved. I clearly ...
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2answers
70 views

Are superposition and uncertainty principles logically dependent?

If we assume superposition and define an Hilbert space with canonical commutation relations we can derive uncertainty relations. So it seems the uncertainty principle isn't required, or should be ...