# 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. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete or closed. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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### How to know which states are entangled from a state vector? [closed]

consider the following state vector of three qubits $$(1/2)|000⟩+(1/2)|011⟩+(1/2)|101⟩+(1/2)|110⟩.$$ how to know which qubits are entangled with respect to their basis states, in other words, how do ...
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### Applying an operator on both sides of an equation [migrated]

I am doing a quantum mechanics question involving the positivity of the norm. So I'm using the fact that the norm will be greater than zero but i want to apply an operator onto the ket on one side of ...
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### Time evolution of operators in the Heisenberg picture

In the Schrodinger picture, a state $|{\psi_{S}(t)}\rangle$ at a time $t$ is given by applying the time-evolution operator $\hat{U}(t)=e^{-\frac{i\hat{H}t}{\hbar}}$ to the state $|{\psi_{S}(0)}\rangle$...
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### Confusion about Deriving Momentum Operator and Hamiltonian Operator

In Sakurai's quantum mechanics, the derivation of momentum operator and Hamlitonian operator is based on spatial translation and time translation as below, for spatial translation and momentum ...
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### What are some examples of bounded momentum/Hamiltonian operators in infinite dimensional Hilbert spaces?

It is well known that one of the operators satisfying the Canonical Commutation Relation $[x,p]=i$ must be unbounded. In most cases I have seen, either both are unbounded or only $p$ is (e.g. Particle ...
1 vote
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### Why assume no degeneracy in Spin operator?

The Stern-Gerlach experiment measures a physical property of a quantum system. We hence associate an operator $\hat{S_z}$ with the observable, in accodance with the postulates of QM. The eigenvalues ...
1 vote
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### Does positive-definite Hamiltonian for a fermion make sense? [closed]

I have been told that positive-definite Hamiltonian for a fermion doesn't make sense. Can anyone explain why is that the case?
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### Question about spin-$½$ particles

Spin-½ particles needs to rotate 720º to return to its original state. If you rotate it 360º, its state will become opposite, for example $\left| ↑ \right>$ to $-\left| ↑ \right>$. This is my ...
1 vote
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### Convergence of series of elements in a quasi-local algebra

I am studying the quasi-local algebra on Bratteli and Robinson Operator Algebras and Quantum Statistical Mechanics, but there is one thing that is not clear to me at the moment. Let's say that the ...
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### Projectors of unbounded operators in *-algebra

Let's suppose we have a Hilbert space $\mathcal{H}$ and the C*-algebra of the set of bounded operators $\mathcal{B}(\mathcal{H})$. For what I have understood, unbounded operators as for example ...
1 vote
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### Is really hermiticity necessary to be a physical observable? What about larger class of operators like PT invariant operators or pseudo hermitian one?

It's really necessary for an observable represented by an operator acting in a Hilbert space to be hermitian? It's known that not only hermitian operators have real eigenvalues and that also normal ...
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### How to apply operator on spin wavefunction of pauli spin matrices? [closed]

There are singlet and triplet spin wavefunctions which are shown below: For singlet: |↑↓−↓↑> and For triplet: |↑↑>, |1/√2(↑↓+↓↑)>, |↓↓> If we want to apply an operator σz which represents ...
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### 2D Quantum Hamonic oscillator in magnetic field with a shiftted position

Background Consider a hole in a 2D parabolic potential in a magnetic field which is generate by the following gauge: $$\vec{A} = \left( - \frac{B_z y}{2}, \frac{B_z x}{2},0\right)$$ Our quantum ...
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### Can you apply non-unitary operators to a qubit?

I am wondering if it is possible to apply continuous, invertible transformations to a qubit which are not linear, i.e. not elements of $U(N)$ where $N=2^n$ where we have $n$ qubits. Consider $n=1$. ...
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### A limit of a particular Quantum Fidelity

I have the following problem. Let $\mathbf{\hat{\rho}}(t)$ and $\mathbf{\hat{\sigma}}(t)$ be two trace class positive operators acting on a Hilbert space of infinite dimension for all $t > 0$. More ...
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### Relation between Stone von Neumann Theorem and Bargmann's theorem

I am trying to understand a relation between the two theorems stated in the title. What I observed so far is that since $H^{2}(\mathbb{R},U(1))=\{e\}$, using Bargmann's theorem, we have that ...
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### How can we figure out what fraction of pure states in a Hilbert space are entangled? [duplicate]

The full Hilbert space of a quantum system will generally contain entangled states, and thus when entanglement is lost through decoherence, parts of Hilbert space become inaccessible. Is there a ...
1 vote
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### What exactly is the definition of the representation of an operator in position or momentum space?

I apologize for this kind of silly question, I haven't brushed up on QM for a while. I was looking at a problem today, essentially I'm given some operator $V = \lambda |{\xi}\rangle \langle{\xi}|$ ...
1 vote
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### Does the linear combination of basis functions, need to use eigenfunctions as basis?

Given a trial function like this one: $$\lvert\hat{\Psi}\rangle = \sum_i c_i\lvert\psi_i\rangle$$ where the trial function is expanded using exact solutions $\psi_i$ to the Time Independent ...
63 views

### What is meant by "bases" not belonging to state space and how are they possible?

Section A3 of Chapter II on mathematical basis of QM in Cohen-Tannoudji’s "Quantum Mechanics" book has me a bit confused. The section is named ...
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### Wave function square integrable [duplicate]

In quantum mechanics, when showing that the momentum operator is Hermitian operator, we use the fact that the wave function and its derivative go to zero at infinity from the assumption that the wave ...
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### What is the single-particle Hilbert space in the Fock space of QFT?

In Quantum field theory, the fields are operator-valued functions of spacetime. So for a scalar (spin $0$) field $$\psi: \mathbb{R}^{3,1} \rightarrow O(F),$$ where $O(F)$ is the space of operators on ...
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### Does the momentum operator applied to a position state vanish?

In quantum mechanics we have \begin{equation*} \langle x|p\rangle=C\exp\left(\frac{ipx}{\hbar}\right) \end{equation*} where $C$ is a normalization constant. It follows that \begin{equation*} -i\hbar\...
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### Reference for mathematics of quantum mechanics with infinite degrees of freedom?

I am looking for a book, or lecture notes or even courses available on YouTube where there is a good and detailed discussion on the mathematical aspects of Quantum Mechanics with infinite degrees of ...
1 vote
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### Is it valid to say that if two wavefunctions are not orthogonal, they must be one and the same?

I'm working on a homework problem that states the following: My first thought was that the negative sign would simply add an additional phase of pi to the wavefunction, and hence the two ...
1 vote
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### How is the Wigner little group representation of Poincaré group Unitary?

From Weinberg's QFT Vol.1, eq(2.5.11): $$U(\Lambda)\Psi_{p,\sigma}=({N(p)\over N(\Lambda p)})\sum_{\sigma'}D_{\sigma'\sigma}(W(\Lambda,p))\Psi_{\Lambda p ,\sigma '}.\tag{2.5.11}$$ However, this is not ...
1 vote
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### Link between the charge and the phase in a superconducting circuit

I have a question related to superconducting quantum circuits. Especially regarding the derivation of the transformation of $\cos(\phi)$ in the charge basis. In this question, a user states that ...
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### Does quantum mechanics need projective representations only due to the Born rule?

In quantum mechanics, physical states don't live in the Hilbert space, but rather on the equivalence class of rays on the Hilbert space. This is called a projective space. This is the reason why when ...
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### Overlap between eigenstates of angular momentum operators

Consider the states $\left|j,m_x\right>_x$ and $\left|j,m_z\right>_z$ with total angular momentum $j$ and the angular momentum operators $\hat{S}_x$ and $\hat{S}_y$. In particular, assume that ...
1 vote
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### Relative "volume" of entangled vs product states [duplicate]

A system containing $n$ qubits is described by a $2^n-$dimensional Hilbert space. Some of these states can be decomposed as product states, but not all of them. The remaining ones are called entangled ...
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### Kraus Operator for two-qubit basis

Let A and B each be a single qubit so that $\mathbb{H_{AB}}$ is a two-qubit system. In the basis {$|\uparrow\uparrow>,|\uparrow\downarrow>,|\downarrow\uparrow>,|\downarrow\downarrow>$, the ...
1 vote
I am currently referring Sakurai. He introduces spin states and operators from general arguments and experimental evidence but ad hoc introduces that the eigenvalues of the pure states as $\pm \frac{\... 0 votes 0 answers 38 views ### Calculation about fermions via quantum field theory I want to ask a specific question occurred in my current learning about neutrinos. What I want to calculate is an amplititude: \begin{equation} \langle\Omega|a_{\bf k m}a_{\bf pj}a_{\bf qi}^{\dagger}... 0 votes 1 answer 72 views ### Finding a complete eigenbasis for an "entangled" Hamiltonian? Suppose we have a tensor product Hilbert space$\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$and we have a Hamiltonian defined thereon which is given by$H = H_{1e} \otimes I+ H_1 \otimes H_2$. ... -1 votes 1 answer 43 views ### Closed form for number of dimension of angular momentum eigenspace I am trying to construct the totally uncorrelated state for a subsystem$i$given the information that a spin has total angular momentum eigenvalue$s_i$. I therefore write that the state operator ... 6 votes 0 answers 117 views ### The unitarity of the$\delta(x)\$ potential
One of the common potentials to solve in quantum mechanics is when $$H=\frac{p^{2}}{2m}+\delta(x).$$ Is this Hamiltonian considered to produce unitary evolution? In particular, I'm not sure what is ...