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.
5,373
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Derivative of $c(t)$ in Adiabatic Approximation
In Sakurai's Modern Quantum Mechanics, second edition, $5.6.10$ is
$$\begin{aligned}
\dot{c}_m(t)=-\sum_nc_n(t)e^{i[\theta_n(t)-\theta_m(t)]}\langle m;t|\left[\frac\partial{\partial t}|n;t\rangle\...
- 19
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3
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Physical meaning behind the double rotation of spin 1/2 particles [duplicate]
From the Bloch sphere, it is mathematically clear that a $720°$ rotation is necessary to bring a spin $1/2$ particle back to its initial state, as a full rotation changes the sign of the state.
...
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2
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How to Choose which space to work in for Schrodingers equation?
I'm working out of Shankar's principles of quantum mechanics book. And overall, I think I get the gist of how to solve problems with Schrodinger's Equation. I recall in my Modern Physics course, we ...
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How to get the factor of $n^{-27/4}$ in number of open string states from the calculation in GSW's book?
In section 2.3.5 of Green, Schwarz, Witten's book on string theory (volume-1) pp. 116-118, the objective is to calculate an Asymptotic Formula for Level Densities $d_n$ for open bosonic string theory. ...
- 414
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1
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Measure of connectivity in Graph States
Given $G = (V,E)$, with the set of vertices $V$ and the set of edges $E$, the corresponding graph state is defined as
$$|G\rangle = \prod_{(a,b)\in E} U^{\{a,b\}} |+\rangle ^{\otimes V}$$
where the ...
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Why do the boundary terms vanish for a function in Hilbert Space?
Below I have attached a solution to a problem from a quantum mechanics textbook, and I'd simply like someone to explain why the boundary terms vanish in Hilbert Space for the functions $f(x)$ and $g(x)...
- 194
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Showing the Variance of an observable in a determinate state is always zero
I am working through Introduction to Quantum Mechanics by David J. Griffiths, and part 3.2.2 shows that the standard deviation of an obervable, $Q$, is always $0$ but I do not understand the steps ...
- 194
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Square Integrability of spherical symmetric wave
In my class we were discussing some wave equation for a spherical symmetric wave $u(t,r)$ and my professor investigated the behaviour of the solutions asymptotics $r \rightarrow \infty$. The solution ...
- 593
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Usage in the Clebsch-Gordan (CG) coefficients' recursive relation
In Sakurai's Modern Quantum Mechanics, its 3.8.39 is
\begin{aligned}\sqrt{(j\mp m)(j\pm m+1)}\langle j_1j_2;m_1m_2|j_1j_2;j,m\pm1 \rangle \\=\sqrt{(j_1\mp m_1+1)(j_1\pm m_1)}\langle j_1j_2;m_1\mp1,...
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If $H$ anniliates a state, must $Q$ and $Q^\dagger$ also annihilate the state?
Suppose we have a a Hamiltonian, $H$. And suppose also we have some operator $Q$ such that $\{Q, Q^{\dagger}\} = H$, and $Q^2 = 0$.
If we find a state $|\psi \rangle$ such that $Q|\psi \rangle = Q^{\...
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Do the states $|01\rangle$, $|10\rangle$ span the space of a single photon?
i nave included some simple information about Fock states for photons in a work.
I wanted to know if:
in the context of a beam splitter when we suppose to have only one photon, always entering from ...
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How can a QFT field act on particle states in Fock space?
Recently I asked a question that was considered a duplicate. However I felt that the related question didn't answer my doubts. After a bit of pondering I have realized the core of my discomfort with ...
- 4,474
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The limit of time evolution operator
Through reading Nenciu's rigorous proof on the Adiabatic Theorem and Gell-Mann-Low Theorem, I found:
Since the limit $t_0\to-\infty$ does not exist for $U(t,t_0,\epsilon)$, in order to make use of ...
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Is integral of energy-momentum tensor in QFT over a region $R$ self-adjoint?
Consider a quantum field theory in flat 1+1D spacetime for simplicity. Let $T_{\mu\nu}$ be the conserved symmetric stress tensor. One writes operators by integrating the tensor over the whole space, ...
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Why are the angular momentum raising and lowering operator coefficients real?
I had a homework problem where I had to find the coefficients for the angular momentum raising and lowering operators. I know the answer is supposed to be $\sqrt{l(l+1)-m(m\pm1)}$. I have figured out ...
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How to interpret QFT fields (in relation with QM)? [duplicate]
In QM we deal with the Schrödinger equation:1
$$i\frac{\partial}{\partial t}\psi = H \psi$$
the wave function $\psi(x)$ is the main object of interest: it can be interpreted as a scalar field, in the ...
- 4,474
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5
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What is the distinction between a ket and a state in quantum mechanics?
Sorry if this has been asked before in some manner, but I'm just a bit confused about the distinction between a state $\alpha$ and its ket $|\alpha\rangle$. I was recently told that a state $\alpha$ ...
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1
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Does the inner product of wavefunctions really have units? [closed]
Let $\psi(x)$ and $\phi(x)$ be wavefunctions. I usually see the inner product defined as
$$\int dx\, \overline{\psi(x)} \, \phi(x)$$
and interpreted, I think, as "the amplitude that state $\phi$ ...
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Parity operator action on quantized Dirac field
I am stuck on equation 3.124 on p.65 in Peskin and Schroeder quantum field theory book.
There they are claiming that:
$$P\psi(x)P=\displaystyle\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bf p}}}\...
- 413
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2
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Why are these unbounded operators (essentially) self-adjoint?
Can anyone provide exact mathematical reasoning as to why the following fundamental unbounded symmetric operators are essentially self-adjoint? I.e. on, their natural domains, they admit a unique ...
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2
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Are Hermitian operators Hermitian in any basis? [closed]
Given a Hilbert space and a Hermitian operator defined on it, will the operator exhibit Hermiticity in any basis used to span the space? My thought on this is that this must be the case, after all, if ...
- 4,088
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2
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192
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Difference between real operators and Hermitian operators in quantum mechanics
I'm reading some lecture notes on quantum mechanics, while describing the rigid rotor in bra-ket notation, the author mentions the parity operator $\hat{P}$ acting on kets as $\hat{P} \left \lvert m \...
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1
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162
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Hermiticity of a projection operator
I have managed to get myself quite confused about something. Consider a two level system consisting of states $|0\rangle$ and $|1\rangle$. the projection operator onto the state $|0\rangle$ is given ...
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2
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100
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How to determine parameters such that the state $|\psi\rangle=\frac1{\sqrt2}|+\rangle|+\rangle+a|+\rangle|x+\rangle+b|-\rangle|-\rangle$ is separable?
Suppose that two spin-1/2 are in the state:
$$ |\psi \rangle = \frac{1}{\sqrt{2}} |+\rangle|+\rangle + a|+\rangle|x+\rangle + b|-\rangle|-\rangle $$
and we want to find values for a & b such that ...
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Unitary Representation of $\text{SO}(3)$ in Position Representation
Let $R\in\text{SO}(3)$ be an arbitrary rotation, and let $U_R$ be the unitary representation of $R$ on some Hilbert space $\mathcal H$. To me, the defining property of $U_R$ is how it conjugates the ...
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On the derivation of path integral and completeness identity
I know that the unitary propagator between two points $x_{i}, x_{f}$ at instants $t_i, t_f$ is given by
$$K(t_i, t_f, x_i, x_f) = \langle x_i, t_i| x_f, t_f\rangle = \langle x_f|e^{i\frac{(t_f-t_i)}{\...
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What does the state $a_k a_l^\dagger|0\rangle$ represent?
Consider the action of the operator $a_k a_l^\dagger$ on the vacuum state $$|{\rm vac}\rangle\equiv |0,0,\ldots,0\rangle,$$ the action of $a_l^\dagger$ surely creates one particle in the $l$th state. ...
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Expansion of a wavefunction of a two-particle system in one dimension in an arbitrary basis without operations associated with the tensor product
In Shankar's Principle of Quantum Mechanics, Section 10.1, part The Direct Product Revisited (he calls tensor products direct products), he attempts to show that a two-particle state space is the ...
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Can the hybridization of edge states in the 1D SSH model be observed numerically?
So I was reading the lecture notes by Asboth on topological insulators . In the first chapter the SSH model is discussed :
$H_{SSH} = \sum_{i = 1}^N v|i,A\rangle \langle i,B | + h.c. + \sum_{i = 1}^{N-...
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2
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A question from S. Weinberg's book (Sec. 2.7)
S. Weinberg in his book "The quantum theory of fields" page 82 says: the elements $T,\bar{T}$, etc, of the symmetry group may be represented on the physical Hilbert space by unitary ...
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Wigner-Eckart theorem: Completeness relation
Consider the Wigner-Eckart theorem given by
$$\langle \alpha' j m'|A^q|\alpha j m\rangle = \frac{\langle \alpha' j m'|\mathbf{J}\cdot\mathbf{A}|\alpha j m\rangle}{j(j+1)}\langle j m'| J^q|j, m\rangle$$...
- 1,171
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Coherent creation operator: unitary or not?
In Quantum Mechanics, for coherent states $|z\rangle$ it can be prooved that if $|0\rangle$ is the vacuum state for an harmonic oscillator, therefore:
\begin{equation}
|z\rangle=e^{za^{\dagger}-z^*a}|...
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1
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63
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How does an operator in the denominator act on a state?
I am reading The Quantum Theory of light by Rodney Loudon.
In Chapter 7 the author defines a phase operator $\hat{\phi}$ using the destruction $\hat{a}$ and creation $\hat{a}^{\dagger}$ operators of ...
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Notation for spatially superposed particle using vacuum modes - Why the state of the particle and vacuum is entangled, not separable?
We have two places, call them Here and There where a particle could be. When the particle isn’t spatially superposed we write its state as $ |1\rangle_{Here}\otimes|0\rangle_{There}$ zero being the ...
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Schmidt decomposition of density operators
Consider a bipartite quantum system described by the density operator, $\hat{\rho}$, an operator acting on the Hilbert space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. This matrix will be a ...
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Hermiticity of a radial momentum operator $\hat{p}_r$ and the spectral theorem
In Nolting's QM book (Theoretical Physics 7), in the chapter on central potentials, a radial momentum operator $\hat{p}_r$ is defined as
\begin{equation}
\hat{p_r} = -i \hbar \Big( \frac{\partial}{\...
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How to actually implement a global phase gate?
I've been trying to model some single qubit gates via Rabi oscillations for a project. In this we have some Hamiltonian (in this case it's due to an oscillating electromagnetic field) that when ...
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380
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Outer product as an operator in an infinite dimensional Hilbert space
The outer product between a bra-ket $|a\rangle\langle a|$ where if $|a\rangle\in\mathcal{H}$ and $\langle a|\in\mathcal{H}_{dual}$ is a vector in the tensor vector space formed by the Hilbert space ...
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Angular momentum completeness relation
Can anyone tell me if the angular momentum completeness relation is given by
$$
\sum_{l=0}^{\infty} \sum_{m=-l}^{l} (2l + 1) |l,m\rangle\langle l,m| = I
$$
or
$$
\sum_{l=0}^{\infty} \sum_{m=-l}^{l} |l,...
- 1,171
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Continuum bases: why do we use dirac delta function? [duplicate]
In discrete bais, we can express a vector as
$$ |\psi\rangle=\sum_{i} c_i|e_i\rangle $$
with orthonormality
$$ \langle e_i|e_j\rangle=\delta_{ij}.$$
$\delta_{ij}$ is usual kronecker delta. If we ...
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In trouble with QFT field operators
I'm stuck on the contents of a side box in "QFT for the Gifted Amateur", chapter 4, dealing about field operators. The side box sets the scene about the use case the section will be ...
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Interpretation of the vector space in notation bra ket
I have read all of the question related to "bra-ket" but no one seems to take the same perspective as I am going to try to give. I know it might be a rather simple and short question, but I ...
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Unbound States of the 1D Finite Potential Well [closed]
Edit: After writing a Python code to numerically solve the constraint problem on the coefficients with Gauss-Jordan elimination, it seems that the biggest problem was that I was treating the ...
- 2,692
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Path Integrals for entangled states
Is there a way of characterizing entanglement between states in a path integral formalism? If so, does this shed some light on the apparently non-local effects of quantum mechanics?
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How do we know the creation and annihilation operators for angular momentum give rise to a complete basis?
In the case of angular momentum in quantum mechanics, a common exercise is to derive the spectrum for $J_z$ and $J^2$. I follow most of this argument, but I get a bit lost at the end.
To my ...
- 1,222
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Normalization of one particle state wave function in fock space - commutator
In deriving the 1/$\sqrt{N!}$ normalization factor the first step is looking at the one particle state (see image below). I am confused about how we got from the first line to the second? Maybe I am ...
