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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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Formal Properties of Coherence relations

Suppose that two particles a and b are in a coherent state. Is the coherence relation between a and b satisfy all of the following? (1) Irreflexive: a and b are distinct particles (although they ...
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Why does Lebesgue measure undergird wavefunctions?

My understanding (and correct me if I am wrong, because I am not a physicist) is that wavefunctions are all normalized members of the Hilbert space of square-integrable functions over some subset of $\...
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Why we need $SU(2)$ symmetry? When we use it? [on hold]

I am trying to learn Quantum mechanics and I am familiar with Pauli matrice but not with group theory. I want to understand SU2 symmetry in common language. When we talk about Pauli matrix x we simply ...
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Quantum superposition in density matrix formalism

I was thinking about quantum superposition and stumbled into something that made me quite uncomfortable. Consider a qubit with Hamiltonian eigenstates $|0\rangle$ and $|1\rangle$. To each of these ...
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Constructing orthonormal basis with given kets

$\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}$ If $\ket{a}$ and $\ket{b}$ are given as the normalized set (but not orthogonal) such that $\langle a\ket{b}= z$ with $z$ being a complex ...
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How to undo or reverse a Kronecker product (tensor product) in QuTiP? [duplicate]

Let's say I have a qubit. A qubit is described by the two basis state $$|0\rangle= \begin{pmatrix} 1\\0 \end{pmatrix} \quad\text{and}\quad \ |1\rangle= \begin{pmatrix} 0\\1 \end{pmatrix}. $$ So a ...
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Can a second-order Schrödinger equation preserve the norm?

Suppose we lived in a universe in which the Schrödinger equation contains second order time derivatives, $$i\hbar \partial_t^2|\varphi(t)\rangle = \mathbb{H} | \varphi(t)\rangle.$$ Would it be true ...
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Operator in Dirac notation

Lets say I have a operator $\textbf{A}$ = $ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $ in a canonical basis {|a⟩ ,|b⟩}. The operator can be re-written in Dirac notation as $\textbf{A}$=...
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Translation operator in QM [closed]

Given the usual translation operator in quantum mechanics: $$\hat{T}(y)=\exp\left(-\frac{i\hat{p}y}{\hbar}\right)$$ I want to show that $\hat{T}(y)|x\rangle=|x+y\rangle$, as one would hope. I've ...
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Are tensored qubits commutative?

I am given a solution to a problem, saying that $$ |\psi\rangle_{ABC} = \frac{1}{\sqrt{2}}(|0\rangle_A \otimes |1\rangle_C + |1\rangle_A \otimes |0\rangle_C) \otimes |+\rangle_B $$ $$ = \frac{1}{2}(...
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Quantum mechanics or creationism [closed]

I read this article recently: https://arxiv.org/abs/1802.00227, where author claimed that quantum mechanics is a sort of creationism! He argued in three different cases to show quantum mechanics does ...
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Is $e^{-2x}\sinh x$ an acceptable state wavefunction?

I have the following function in the range $(0, \infty)$: $$\psi(x)=e^{-2x}\sinh x$$ I would like to know if it is acceptable as a wavefunction. At $x = \infty$, we have $e^{-2x} = 0$ ...
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5answers
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Why do we use vectors in quantum mechanics?

I've been trying to make my understanding of quantum mechanics more mathematically rigorous, but I'm struggling a bit with the lack of intuition behind the fact that we represent quantum states with ...
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2answers
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From spin representation of state vector to other representation

I have learned that the state vector (ket vector) can have many representations, and the laws of quantum mechanics give one the ability to change between representation. But there is a subtle point ...
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How does the momentum operator act on a multi-particle state?

My issue is about the proper development of the action of the momentum operator $P^{\mu}$ - the generator of spacetime translations - on multi-particle states. I'm a little clueless on this, so I'm ...
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What is the inner product between a position ket and a two-state (or multi-state) ket?

I am recently studying the two-state system in quantum mechanics. As I learned, in the Hilbert space of a spinless particle, the relation between a scalar function and a ket state is satisfied as, $$...
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Does $\langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}|\psi\rangle $ for all $|\psi\rangle$ imply that $\hat{A} = \hat{B}$?

I've to solve this simple problem: Let $\hat{A}$ and $\hat{B}$ be two Hermitian operators such that: $$ \langle\psi|\hat{A}|\psi\rangle = \langle\psi|\hat{B}|\psi\rangle \qquad \forall |\psi\...
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In what sense is it 'more complicated to navigate through a Hilbert space to find more complex states'?

In the lecture titled Entanglement and Complexity: Gravity and Quantum Mechanics, Professor Leonard Susskind implies (at time 17:49) that the complexity of a quantum state impacts how 'complicated' it ...
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2answers
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Putting the ket $|l,m_x \rangle$ in terms of the ket $|l, m_z \rangle$

Could someone guide me in my thought process of this problem? I don’t know if I’m thinking about it the right way. The problem is the following: I have a system which possible states are generated ...
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3answers
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Why is it not possible to describe a mixed quantum state by a Hilbert space vector?

I read (for instance in Landau/Lifshitz III) that if I know the wave function of a quantum state, I have the maximal information of the state available, in different words, the description of the ...
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What is the physical significance/importance of anti-unitary operators?

Time-reversal symmetry is an anti-unitary operator. I understand the mathematical definition of this, but what are the implications? What should/would one expect from anti-unitary operators? Are ...
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If momentum eigenstates are not in $L^2$, how can they form a basis for it?

When your matrices are all finite, the eigenvectors of a self-adjoint matrix $\mathbf{A}$ form an orthogonal basis for some space. It's almost too trivial to mention that each basis vector is, indeed, ...
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2answers
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I have a question about momentum and energy of the infinite square well in quantum mechanics

In Griffiths quantum mechanics, There is a problem that "Find the momentum-space wave function $\varphi(p,t)$ for the $n$th stationary state of the infinite square well." The $n$th stationary ...
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Translating between classical treatment of non-autonomous systems and time evolution in quantum mechanics

When I read an introduction to (classical) dynamical systems, the system was considered in a phase space, and the state of the system evolving in phase space. For a non-autonomous system, an ...
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what is the meaning of projection of spin along z axis

Spin is an internal degree of freedom, which has no classical analogue. So what is the meaning of its projection along z-axis? if it is the internal degree of freedom and its Hilbert space is a 2-...
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1answer
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Find the eigenvalues of operator sum

Sorry to ask, but I'm struggling all week with the following question, so I was hoping if you could orient me. Suppose we have an orthonormal basis ${|n>}$, and we can define some operators which ...
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The Field $\mathbb{F}$ of A Hilbert Space [duplicate]

Is it always necessary for the field of some arbitrary Hilbert space I define to describe a system be a field of complex numbers only? Is it possible to have a field of naturals, or reals? Since the ...
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1answer
547 views

Is purification physicaly meaningful?

Consider a quantum system with Hilbert space $\mathscr{H}$ and suppose the quantum state is specified by a density operator $\rho$. Since it is hermitian, it has a spectral decomposition: $$\rho = \...
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3answers
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Use of ladder operators

I used ladder operaters to solve for the wavefunctions of a harmonic oscillator. I want to know, are ladder operaters universal? What i meant by universal is that, can they used to solve for the ...
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Energy level in harmonic oscillator [duplicate]

ground state is always non degenrate in harmonic oscillator in quantum state, why?
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What is the adjoint of a ket-bra?

Let $T$ be a linear operator, then we can consider the rank-one operator $$\vert Tx \rangle \langle y \vert.$$ I am wondering what is its adjoint operator, is it $$\vert y \rangle \langle T^*x \...
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1D Infinite Potential Well: getting different values with different methods for $\langle H \rangle$

We have $$\psi(x,0)=3u_1(x)+4u_2(x)$$ where $u_1(x)$ and $u_2(x)$ are the ground state and first excited states of a 1D infinite potential well. I got the normalization constant. I have to calculate ...
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How is the product $L\cdot S$ between orbital and spin angular momentum operators defined? Do they act on the same or different Hilbert spaces?

For an electron, are the Hilbert spaces for the spin angular momentum and the orbital angular momentum the same or are they different? If they're different, how do we justify the operator $L\cdot S$ ...
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1answer
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Tracing over a Fock space?

Suppose you have a bosonic Fock space with a vacuum $|0\rangle$. A particular state is labeled by the parameter $N \in \mathbb{Z}$. You can construct states like $$ | n_{N} \rangle = \frac{ \left( \...
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Does it matter whether or not $\delta(x)$ is a valid wave function for a particle on the real line?

We model the wave-functions of a particle on the line by vectors $\psi \in L^2(\mathbb{R})$, and the position operator $X:D(X) \rightarrow L^2(\mathbb{R})$ as the operator such that $X\psi(x) = x\psi(...
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What is meant by the multi-particle state $|n\rangle^{(+)}$ here?

I am reading Takagi's paper `Vacuum noise and stress induced by uniform accelerator'. I will attach a screenshot of something I am confused about (from page 31). I have a simple question: what is ...
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How does the normalization of an $n$-particle state $|n_{\mathbf{k}}\rangle$ work?

You can expand the free, real scalar field in the following manner $$ \phi(x) = \int \frac{d^{3}\mathbf{k}}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf{k}}}} \bigg[ e^{- i \omega_{\mathbf{k}}x^0+i \...
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Is there any “singlet state” for 3 or more spin 1/2 particles?

Every system with $N$ or more electrons lies in a Hilbert space $H=H_{\text{space}} \otimes H_{\text{spin}}$, with $H_{\text{space}}=H_{\text{space}}^{1}\otimes\cdots\otimes H_{\text{space}}^{N}$ and $...
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Moving from eigenstate basis of a Hamiltonian back to operator basis [duplicate]

This is maybe a stupid question but one I have been bugged by a long time. How to translate back from an eigenstate basis to another operator-basis. Let me give an explicit example to show what I am ...
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Is the space of gauge invariant states a Hilbert space?

In a theory with a gauge symmetry, as I understand it, the gauge symmetry is not a symmetry of a system, but rather its a redundancy in description. The procedure goes like this start with some ...
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If spontaneous symmetry breaking only occurs in infinite systems, why do we observe similar effects in finite systems?

Background No SSB in finite systems Consider a system interacting with a heat bath at inverse temperature $\beta$, with the resultant dynamics of the system described by a Liouvillian superoperator $...
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Full solution of the wavefunction for the particle in a box problem

The particle in a box problem is a common question that people are taught in order to get some practice using Schrödinger’s equation. For this kind of problem one usually solves the equation for ...
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Integral representation for inner product in the Gupta-Bleuler quantization

Following the Gupta-Bleuler method for the quantization of the electromagnetic field, we have that: $$[a_{k'\lambda'},\hat{a}^\dagger_{k\lambda}]=-g_{\lambda \lambda'}\delta(k-k')$$ And the vacuum $$\...
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What is the 'state space' of a quantum field theory called?

This is just a terminological question, not a question about reality or mathematics. I often want to talk about state spaces in quantum field theory. For example the space of [all possible vector ...
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Conventions for density matrix and projections

From this link : [http://math.ucr.edu/home/baez/lie/node12.html][1], it is said that, from starting a quantum state, $$\vec{v}=a|\text{up} \rangle+b|\text{down} \rangle,$$ we can define the matrix ...
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Pure state before and after measurement

Before the measurement of an observable, the quantum state is $$|\Phi\rangle = \sum_i c_i |\psi_i \rangle,$$ with $|\psi_i \rangle$ called "pure states". Once the measurement is done, the quantum ...
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1answer
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Doubts on Choi-Jamiolkowsky (CJ) matrix formulation

We can represent a CP map $\mathcal{M}$ by a positive semidefinite matrix via Choi-Jamiolkowsky (CJ) isomorphism. The CJ matrix $M$ ∈ $L(H_1 \otimes H_2)$ corresponding to a linear map $\mathcal{M}$ : ...
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3answers
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Why are there three $p$-orbitals?

This question is specifically about Schrödinger quantum mechanics, but if an answer in some other mode would illuminate it could be acceptable, as demonstrating a physical or mathematical reason for ...
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Infinite-dimensional Hilbert spaces in QM vs. finite-dimensional Hilbert spaces in quantum gravity?

It seems to me that there are fairly good reasons to assume that quantum theories need to rely in their formulation on infinite-dimensional spaces (cf. Why do we need infinite-dimensional Hilbert ...
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Is Hermitian product symmetric?

Following Hermitian product: $$ \langle f,g \rangle =\frac{1}{T} \int_{-\frac{T}{2}}^{+\frac{T}{2}} f^*(x) g(x) dx$$ is it commutative or not? i.e does one get: $$ \langle f,g \rangle = \langle g,f \...