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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Translation operators and positive-semidefinite condition

Good day. I have an operator $\exp{(-i\mu\hat{q})}\exp{(-i\nu\hat{p})}$, where $\hat{q}$, $\hat{p}$ are the position and momentum operators, respectively. The parameters $\mu,\nu$ are some real ...
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Clarification on bound states: do "locally bound" states exist?

In Griffiths, a state with energy $E$ is said to be "bound" if $$E < \min\left(\lim_{x\to\infty} V(x), \lim_{x\to-\infty} V(x)\right)$$ (i.e. $E$ is less than both of those quantities). ...
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How many way $N$ bosons can occupy $M$ states? [closed]

Suppose each of $N$ identical particles may occupy $M$ states. In how many ways can this be done if there are no restrictions on the number of particles in each state (Bose particles)?
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Generation of a non-accidental degenerate eigenspace carrying out the symmetry operations on one eigenstate

Let us consider a system described by an Hamiltonian $H$ over an Hilbert space $M$, and the finite group $G$ of symmetry operations w.r.t $H$, i.e. R_g : M \rightarrow M \qquad g\in G ...
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Can an isolated quantum system thermalize in the sense that the time dependent density matrix of the state approaches the micro-cannonical ensemble?

Usually people are interested in ETH, or if subsystems of the closed system thermalizes, or in a restricted set of observables or arguments of cannonical typicality or quantum chaotic systems which ...
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1 vote
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Exercise on self-adjointness of Hamiltonian [closed]

I am struggling with some exercise I have to solve for my quantum mechanics class. PROBLEM: Suppose $|\psi\rangle, |\phi\rangle$ are normalised and linearly independent (but not necessarily ...
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How to rotate an electron mathematically?

Im a mathematics student who just learned about the fact that if you rotate an electron by $2 \pi$ its spin state changes but if you turn it by $4 \pi$ it stays the same. I understand all the ...
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What is the effect of the operator $\hat{O} = |\phi \rangle \langle \psi|$ on any state $| \psi \rangle$? [closed]

I have seen this operator in some questions,including the book Liboff,Introductory Quantum Mechanics. I am not sure how it would act on a state.
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Doubts on Particle in a box model [closed]

I have following three doubts. For a particle in a box problem, a particle is moving within a box of length a. The normalization constant is $\sqrt{\frac{2}{a}}$. My question is if we take a negative ...
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393 views

When do two state functions represent the same quantum state?

According to the standard quantum mechanics, quantum states are one-dimensional subspaces of a separable Hilbert space. In practice, this Hilbert space is $L^2(M)$ where $M$ is the classical ...
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What is the difference between a vector and a representation of a vector in QM?

What does the phrase The wave function is a representation of the abstract quantum state. Or more generally, $A$ is a representation of a vector $\vec V$ mean? What is the difference between a ...
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Is the initial state the eigenstate of a Hamiltonian?

Solutions to the Schrödinger equation can take the form $\psi(r,t)=\psi(r)f(t)$, where $f(t) = e^{\frac{-iEt}{\hbar}}$, $$H \psi(r) = E \psi(r) ,$$ where $\psi(r)$ is the eigenstate of a ...
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1 vote
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Azimuthal coordinate operator: Hermition or not? Self-adjoint or not?

I am told that the azimuthal coordinate operator $\hat{\phi}$ is not self-adjoint. I am told this by people who I am sure know much more about this stuff than I do. To my unsophisticated mind, "...
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1 vote
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Difference between stationary states, collision states, scattering states, and bound states

A few weeks ago, I was presented one-dimensional systems in my QM class, and of course one-dimensional potentials too. Nonetheless, I'm still a bit unclear about the terminology my professor uses. ...
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Creating Schrödinger cat states with trapped ions

We will consider an ion is in an harmonic trap. The ion has two internal states \lvert g\rangle and \lvert s\rangle and it interacts with a laser that induces a state-dependent force. The quantum ...
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Time Evolution of Eigenkets in the Heisenberg picture

I'm reading Modern Quantum Mechanics by Jun John Sakurai and in section 2.2 he talks about Base Kets and Transition Amplitudes. He goes to show, that $|a',t\rangle=\mathcal{U}^\dagger|a'\rangle$, (...
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The Jacobi's theta functions $$\theta_1(0,\tau )=0$$ $$\theta_2(\tau) =\sum_{n\in \mathbb{Z}} q^{(n+\frac{1}{2})^2 /2 }$$ $$\theta_3(\tau) =\sum_{n\in\mathbb{Z}} q^{n^2/2}$$ $$\theta_4(\tau) =\sum_{n\... • 3,224 1 vote 1 answer 54 views Derivative of c(t) in Adiabatic Approximation In Sakurai's Modern Quantum Mechanics, second edition, 5.6.10 is$$\begin{aligned} \dot{c}_m(t)=-\sum_nc_n(t)e^{i[\theta_n(t)-\theta_m(t)]}\langle m;t|\left[\frac\partial{\partial t}|n;t\rangle\...
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From the Bloch sphere, it is mathematically clear that a $720°$ rotation is necessary to bring a spin $1/2$ particle back to its initial state, as a full rotation changes the sign of the state. ...