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 ...
Kim's user avatar
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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). ...
Trisztan's user avatar
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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)?
user1's user avatar
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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. \begin{equation} R_g : M \rightarrow M \qquad g\in G ...
Gippo's user avatar
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2 votes
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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 ...
Fibonacci M's user avatar
3 votes
2 answers
141 views

Question about the identity operator and the bosonic ladder operators

Consider a self-adjoint operator $B$, such that for each mode $a_1,...,a_n$ [of a quantum bosonic system with Hilbert space $\cal H$ given by the corresponding Fock space] we have $B a_i B^\dagger = ...
Noobgrammer's user avatar
-6 votes
0 answers
51 views

Mathematical tools in quantum mechanics [closed]

Prove that unitary transformation transforms one complete set of basis vectors into another.
Raju Raj's user avatar
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94 views

Functional analysis question about operator on quantum wave functions

If I have two time-independent wave functions $\psi_{t_{1}}$ and $\psi_{t_{2}}$ and define an operator $\hat{U}$ such that $$\psi_{t_{2}} = \hat{U}_{t_{1},t_{2}}(\psi_{t_{1}})$$ and $$\psi_{t_{2}}(x) =...
Adam Kabbeke's user avatar
1 vote
0 answers
25 views

Degeneracy in time-independent perturbation theory

I have a problem understanding the reasoning of my professor in Quantum Mechanics. The topic is time-independent perturbation theory. Let $H = H_0 + \lambda H_1$ be a perturbed Hamilton-Operator and ...
Octavius's user avatar
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Time evolution using non-Hermitian (not a PT symmetric) Hamiltonian

I am currently dealing with non-Hermitian hamiltonian and dynamics using it. In general the diagonalizable non-Hermitian matrix might have complex eigenvalues and the eigenvectors may not be ...
user101134's user avatar
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How can interacting field operators in $2D$ still satisfy the canonical commutation relation?

Free fields in any dimensions are well-known to be Gaussian, act on the Fock space and satisfy the canonical commutation relations. By definition, interacting field operators are NOT such cases, as ...
Keith's user avatar
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+50

Computing Fubini-Study expectation values over $\mathbb{C}P^n$

In finite-dimensional textbook quantum mechanics, we postulate that states of our system are rays in a Hilbert space $\mathcal{H}$ with dimension $\dim{\mathcal{H}} = n+1$ where $n \in \mathbb{N}$, ...
Silly Goose's user avatar
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7 votes
1 answer
298 views

Determining Bound States from Møller Operator

Hello I came across an interesting property of the Møller operator, which I summarize below: The Møller operator $\Omega^{(+)}$ maps in-states that belong to the continuum spectrum of the free ...
StackUser's user avatar
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2 votes
1 answer
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Why does the Pauli objection not disqualify the existence of the position operator?

According to the Pauli objection (see for example here or the answer to this question) there can be no time operator $\hat{T}$ canonically conjugate to the Hamiltonian $\hat{H}$ of a physical system ...
Martin Vaughan's user avatar
1 vote
1 answer
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Physical meaning of the CG coefficient $\langle j_1 m_1 j_2 m_2 |00 \rangle $

This particular CG coefficient has the value $$ \langle j_1 m_1 j_2 m_2 |00 \rangle = \delta_{j_1 j_2 }\delta_{m_1,-m_2} \frac{(-1)^{j_1 - m_1 }}{\sqrt{2j_1 + 1 }}. $$ The interesting point is that ...
poisson's user avatar
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How to prove if ${\hat A}^{\dagger}{\hat A}\le {\hat 1}$ then ${\hat A}{\hat A}^{\dagger}\le {\hat 1}$ for any square matrix ${\hat A}$? [migrated]

My note says that for any square matrix ${\hat A}$, if ${\hat A}^{\dagger}{\hat A}\le {\hat 1}$, then ${\hat A}{\hat A}^{\dagger}\le {\hat 1}$, but I don't know how to prove this. Can anyone help me ...
Ketty's user avatar
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1 answer
41 views

Pauli matrix exponentials [closed]

Just a short query to confirm my understanding. Given the Pauli-X operator $\hat{X}$ and it's eigenstates $|+\rangle:=\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|-\rangle:=\frac{1}{\sqrt{2}}(|0\...
John Doe's user avatar
  • 275
0 votes
1 answer
77 views

Correct mathematical notation of Kets, Bras, etc [closed]

Let's consider two discrete basis $|u_i\rangle$ and $|t_k\rangle$. For this the following is true: $S_{ik}=\langle u_i|t_k\rangle$ $(S^\dagger)_{ki}=(S_{ik})*=\langle t_k|u_i\rangle$ If we go from $|...
imbAF's user avatar
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1 vote
2 answers
62 views

What do people mean when they say orbital and spin angular momentum operators act on different spaces?

I was looking for an explanation of why the orbital and spin angular momentum operators commute, and I found many sources saying they act on different vector spaces. I am confused about the use of ...
toomanyfeet's user avatar
-2 votes
0 answers
52 views

Is Schrödingers "quantization" as a symptom of the Infinite Potential Well the same as in the linear algebra equivalent? [closed]

Is the quantization observed when "simulating" the Schrödinger equation in an Infinite Potential Well equivalent to the certain stationary eigenvalues you can obtain from the Hamilton ...
Peter Hackman's user avatar
-2 votes
1 answer
75 views

How to prove that all extensions of a pure state are product states? [closed]

I tried using the $\rho$ operator for this, and I got nothing. I approached it assuming I have two subsystems $A$ and $B$, both inside their own Hilbert space $H_A$ and $H_B$. And even defined $\rho = ...
mirikubuar's user avatar
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1 answer
100 views

Mathematical meaning of a position eigenbra $\langle x_0 |$

Let $|x_0\rangle$ be an position eigenket. The physical picture I have for $|x_0\rangle$ is a particle located at $x_0$. Thus it should be represented by a delta function $\delta(x-x_0)$. For $f\in L^...
CBBAM's user avatar
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1 answer
47 views

Can any meaning be given to a path integral with no fixed end point?

A path integral has the interpreted as the probability a particle goes from $A$ to $B$ in time $t$. Such a path integral is given by $$\langle x_B, t|x_A, 0\rangle = \frac{1}{Z} \int_{\textrm{paths } ...
CBBAM's user avatar
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1 vote
1 answer
42 views

Choice of spacetime foliation while quantising a conformal field theory

I was reading Rychkov's EPFL lectures on $D\geq 3$ CFT (along with these set of TASI lectures) and in chapter 3, he starts discussing radial quantisation and OPE (operator product expansion). I ...
QFTheorist's user avatar
1 vote
0 answers
30 views

Multimode squeezed operator

Given CCR (bosonic) algebra, with creation / annihilation operators $a_{i}^{\dagger}, a_i$ acting on a single particle Hilbert space $\mathbb{h}$, let's introduce the multimode squeezed operator for $...
MBlrd's user avatar
  • 159
1 vote
1 answer
78 views

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 ...
Octavius's user avatar
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17 votes
4 answers
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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 ...
Henry T.'s user avatar
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-4 votes
1 answer
60 views

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.
blackholesandrevelations's user avatar
0 votes
1 answer
68 views

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 ...
str's user avatar
  • 9
4 votes
1 answer
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 ...
mma's user avatar
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0 votes
1 answer
78 views

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 ...
GedankenExperimentalist's user avatar
0 votes
1 answer
69 views

How to find the expectation value of momentum operator? [closed]

I have struggled with the following steps in finding the expectation value of the momentum operator $\hat{p}$. $$\left \langle \hat{p} \right \rangle=\int_{-\infty}^{\infty}\psi^{*}(x)\hat{p}\psi{x}dx=...
SaaN's user avatar
  • 41
0 votes
0 answers
57 views

Wigner's formula for the kinetic energy density in QM

In the Schroedinger equation the kinetic energy is represented by the operator $T = -\frac {\hbar^2} {2m} \Delta$ which acts on a wavefunction $\Psi$. If we multiply this by the complex conjugate of ...
M. Wind's user avatar
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1 vote
0 answers
52 views

Normalizable solutions to the time-independent Schrödinger equation are real [duplicate]

In Griffiths Introduction to Quantum Mechanics (3ed.) problem 2.1, we are asked to prove that the normalizable solutions to the time-independent Schrödinger equation can always be chosen to be real, ...
jajaperson's user avatar
2 votes
0 answers
84 views

Introduction of symmetries in quantum mechanics

The (Italian) book that I am currently reading introduces the topic of symmetries in quantum mechanics in the following way: Let O and O' be two distinct observers and let $A$ and $B$ be two ...
davise's user avatar
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0 votes
1 answer
251 views

What does the identity operator mean in Quantum Mechanics? [duplicate]

I'm new to quantum mechanics, and I am beginning to study Dirac notation, but I do not understand the significance or meaning of the following equation: $$\sum_n\left|e_n\right\rangle\left\langle e_n\...
cookiecainsy's user avatar
0 votes
2 answers
105 views

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 ...
ZhuanXu's user avatar
  • 45
1 vote
1 answer
63 views

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, "...
bob.sacamento's user avatar
-2 votes
1 answer
51 views

What does doubly-entangled $W$-like state do with three-particle setup?

First, is entanglement of three particles in $W$-like state deliberately possible (and not by chance)? Second, is the following statement correct? In the doubly entangled $W$ state, represented as $$ |...
K Ivanov's user avatar
3 votes
0 answers
51 views

Unitary evolution of the displacement operator in terms of positions and momenta

Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\...
Phil K.'s user avatar
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1 vote
0 answers
30 views

Multimode unitary channel in terms of action on characteristic function

Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\...
Phil K.'s user avatar
  • 81
2 votes
1 answer
82 views

Global phase of wave function in quantum mechanics and Fubini-Study metric

I have a basic question about projective representations in quantum mechanics. In projective representation we identify the class of normalized states in Hilbert space as the same physical state as ...
Ervand's user avatar
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5 votes
1 answer
554 views

Some questions about derivation of uncertainty principle

In Introduction to Quantum Mechanics by Griffiths and Schroeter, they derive the Uncertainty principle in the following way: First, they define $$f=\left(\hat A-\langle A\rangle\right)|\Psi\rangle$$ $$...
GedankenExperimentalist's user avatar
1 vote
1 answer
56 views

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. ...
AlanFox86's user avatar
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1 vote
1 answer
33 views

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 ...
Mario's user avatar
  • 13
4 votes
2 answers
498 views

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 $|a',t\rangle=\mathcal{U}^\dagger|a'\rangle$, (...
Florpsiturtle's user avatar
4 votes
2 answers
65 views

Is there any meaning or statistical distribution associated with the Jacobi's $\theta$ functions?

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\...
ShoutOutAndCalculate's user avatar
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\...
liZ's user avatar
  • 19
0 votes
3 answers
146 views

Physical meaning behind the double rotation of spin 1/2 particles [duplicate]

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. ...
QuantumQuasar's user avatar
2 votes
2 answers
116 views

How to Choose which space to work in for Schrodingers equation?

I'm working out of Shankar's principles of quantum mechanics book. And overall, I think I get the gist of how to solve problems with Schrodinger's Equation. I recall in my Modern Physics course, we ...
Ben Ray's user avatar
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