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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Why Dirac delta function is considered as wave function?

I have read in my textbook that dirac delta function is regarded as wave function. But isn't it violating the condition required for a wave function ie . It becomes infinite $ δ(x-x_0)$ at $x=x_0$ ...
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Probability Measurement in Quantum Mechanics

In Sakurai's book it is given that, The state ket for an arbitrary physical state can be expanded in terms of $|x'\rangle$ In practice, the best the detector can do is to locate the ...
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Show $\langle \alpha| (\cos \mu)^{-a^{\dagger} a} | \alpha \rangle =e^{|\alpha|^2(\frac{1}{\cos \mu} -1)}$ for coherent states $|\alpha \rangle$

I am reading"Quantum continuous variables, A primer of Theoretical Methods" by A.Serafini, page 120. Let $ a $ be the annihilation operator for the Fock basis. I want to show $$\langle \alpha| (\...
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What is the value of the inner product $_x\langle+|-\rangle$?

I know that $\left|_x\langle+|-\rangle\right|^2 = 1/2$, so is it just as simple as taking the square root of $1/2$? Thanks.
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Do pure states actually exist?

Is it possible to ever actually measure something so precisely that it actually collapses to a pure state, or do we really just get arbitrarily close? If a wavefunction never actually collapses to a ...
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53 views

Difference between eigenstate and basis vector?

My understanding is that any wavefunction can be decomposed into a linear sum of basis vectors, which for momentum are something like sine waves and for position are delta functions. And then ...
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59 views

Can a density matrix have more than two dimensions?

It crossed my mind when reviewing density matrices that if you were looking at a composite system consisting of three subsystems, (indexed by three quantum numbers: $<i,j,k|\rho|i,j,k>$) then ...
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Are states (always) a fundamental notion in QM?

When dealing with quantum mechanics one usually postulates states* as fundamental notions. They form basis of the Hilbert space(H) and are used to compute expected values of observables which we "...
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Possible values of the total orbital angular momentum quantum number $L$

I'm reading Gerhard Herzberg's "Atomic spectra and atomic structure" (can be accessed on archive.org, the relevant part is from p.82-87) and I don't quite grasp how to determine the possible values of ...
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How to prove that a $d$-dimensional Hilbert space can only have $d^2$ equiangular vectors (i.e. that a SIC is a maximal collection of that kind)?

It is an open question if every $d$-dimensional Hilbert space contains a collection of $d^2$ states, such that every two have a scalar product of $\frac{1}{d+1}$, i.e. if a SIC-POVM exists for every ...
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Show for $r \in \mathbb R$, $e^{(a_1 ^\dagger a_2^\dagger - a_1a_2)r}|0,0 \rangle = \frac{1}{\cosh r} \sum_{j=0}^\infty (\tanh r)^j |j,j \rangle$

Let $ a_1, a_2 $ be annihilation operators for the first and second component in the product state $|m,n \rangle$ using Fock basis. Following "Quantum continuous variables, A primer of Theoretical ...
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Trying to understand spin in quantum mechanics

I'm trying to understand the concept of spin in Quantum Mechanics. I'm reading "Road to Reality" by Penrose, which despite not being a textbook, is reputed to give one a deep insight into physical ...
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Inequality for quantum probability

Let $H$ be a separable Hilbert space for a quantum mechanical system then $$w (x, y) = {{\langle y \mid x\rangle\langle x \mid y \rangle} \over \langle x \mid x \rangle\langle y \mid y \rangle}$$ is ...
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1answer
50 views

Proof that momentum operator acting on a wavefunction gives the expectation value

I'm following a book where the author tries to prove that $$\langle p \rangle = \langle \psi \vert \hat{p} \vert \psi \rangle,\tag{0}$$ so he just computes the integral $$ \langle \psi \vert \hat{p} \...
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What constitutes a quantum mechanical state? [duplicate]

In quantum mechanics, what exactly is the "state" of a quantum system and what constitutes the "state" of a quantum system? i will appreciate an intuitive and as well a mathematical answer.
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What's the momentum-space vacuum wave-functional of a fermion?

In the Schrödinger picture, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\phi(\mathbf x)|\phi\...
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Is every operator a power series of creation and annihilation operators (in a rigorous mathematical sense)?

Let $\mathscr{H}$ be a Hilbert space denoting the single-particle states and $c_k^*,c_k$ denote creation and annihilation operators of orthonormal basis $\phi_k\in \mathscr{H}$. Let $\mathscr{F}$ ...
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Selection rules on quadratic terms

Let's assume I have to find expectation value of $z^2$ in the $|l_im_i\rangle$ state. Can I use the selection rules in this way? $$\langle 10|z^2|10\rangle$$ $$=\langle 10|z \color{red}{|10\...
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Non-normalised Bargmann Coherent States

Currently, I am reading a paper which involves non-normalised Bargmann coherent states as my basis. I am interested in knowing how the creation and annihilation operator acts on these coherent states. ...
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1answer
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What is $\frac{d}{d\psi}\langle\psi| \hat{O} | \psi\rangle$?

I would like to know what is the derivative of an expectation value with respect to the molecular state $$\frac{d}{d\psi}\langle\psi| \hat{\mathbf{O}} | \psi\rangle$$ Note that here $|\psi\rangle$ ...
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What is the (triplet) eigenstate for two electrons? [duplicate]

Let's assume I have a harmonic oscillator which is one dimensional. What is my plan is to work the the two electron's spin states and my requirement is that they have to be in the triplet sates. Lets ...
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Why does the lowering operator applied to lowest state have to be 0? [closed]

When solving a problem in QM with raising and lowering operators ($\hat{a},\hat{a}^\dagger,L_{\pm},..$) it is often assumed that: $$ L_-|\Omega>=0 $$ Why is this assumed? Couldn't the result ...
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Mathematical Rigorousness of Taking the Thermodynamical limit of a finite size quantum model

Suppose I have nearest Neighbour Quantum Ising model with a transverse field. $$\hat{H} = \sum_{i}S^{x}_iS^{x}_{i+1} + h\sum_i S^{z}_i$$ Through Jordan-Wigner and Bogoliubov transformation, one finds ...
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Hamiltonian for a mode-shift operator

I have a discrete multi-level degree of freedom in my quantum system (for photons, for example this), which I write as $|l\rangle$. The degree of freedom is unbounded, i.e. $l$ can take ever positive ...
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Density overlap of orthogonal wavefunctions

Intuitively, I suspect that orthogonal wavefunctions don't have much overlap in their densities. For example, in separable approximations of many-body fermions like Hartree-Fock, the wavefunctions of ...
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Normalization constant of a planar wave

As we know for the plane waves ( $ae^{i k x}+b e^{-i k x}$), the normalization constant can be easily obtained from the integral $\int^{x_{2}}_{x_{1}}\psi^{*}\psi dx=1$ by the relation $|a|^{2}+|b|^{2}...
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Is there a good notation for basis states in quantum mechanics? [closed]

Given a complete set of orthogonal basis states $|e_i\rangle$ one can form any vector $|a\rangle = \sum_i a_i |e_i\rangle$. Is there a standard notation to separate out a basis state from a general ...
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Reasons for problematic behaviour of hydrogen atom states with azimuthal quantum number $\ell=0$

I have been reading "Introduction to Quantum Mechanics by David J. Griffiths" and in the chapter 6- Time-Independent Perturbation theory, when we are explaining the fine structure via relativistic ...
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60 views

Showing in the classical limit an operator $U = \langle \Omega | U |\Omega \rangle$

Let $\Omega$ be the ground state of the Hilbert space $\mathcal H$. How can I show that in the classical limit, an operator $U$ goes to $ \langle \Omega | U |\Omega \rangle$? Attempt. This is ...
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Is this operator related to the resolvent formalism (aka Green function)? [migrated]

I am working in the area of graph theory, where the matrix $L$ is the combinatorial matrix of a graph, a positive semidefinite matrix. The resolvent of a matrix $L$ (improperly called Green function) ...
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Is the evolution operator well-defined mathematically?

We know that in order to solve the time-dependent Schrodinger equation $i\partial_t \psi = H(t) \psi$, we need the evolution operator $$U(t) = T \exp{\left(-i\int_0^t H(t')dt'\right)}$$ where $T$ is ...
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Must an operator that preserves probability be unitary?

One property of the unitary operator is that it preserves the norm of the state-vectors: $$ \langle \Psi | U^\dagger U | \Psi \rangle = \langle \Psi | \Psi \rangle $$ If $U$ is unitary. Is the ...
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Why is the Bogoliubov transform unitary of $H\oplus H$

In Bach, V., Lieb, E.H. & Solovej, J.P. J Stat Phys (1994) 76: 3. https://doi-org.stanford.idm.oclc.org/10.1007/BF02188656, page 10, the Bogoliubov transform on the Fock space $\mathscr{F}$ is a ...
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1answer
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How Gupta-Bleuler condition implies $(a_p^3-a_p^0)| \phi \rangle=0$?

Gupta-Bleuler condition is $$\partial^\mu A_\mu^+ | \phi \rangle=0\tag{6.54}$$ where $$A_\mu^+= \int\frac{d^3\mathbf p}{(2\pi)^3 \sqrt{2|\mathbf p|}} \sum_{\lambda=0}^3 \epsilon^\lambda_\mu a_p^\...
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Contradiction between aymptotically free particles in QFT and unlocalization

When studying different interactions in any QFT, one always assumes that the IN and OUT states are asymptotically free particles with definite momenta. For example, one assumes that an electron and a ...
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$\alpha_n^{i}$ excitations in superstring vs $\psi_r^{i}$ excitations

Perhaps due to a gap in my learning I don't know why I don't see bosonic excitations $\alpha_n^{i}$ discussed in superstring theory, only fermionic excitations $\psi_r^i$. I know the NS sector ...
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Can one derive Schrodinger's Equation from quantum information theory?

I know that some people think that quantum information theory/science is fundamental physics. I also know that there are many definitions, theorems and rules in the field of quantum information. They ...
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How are anti-unitary transformations symmetric?

In the article on Wigner's theorem, unitary transformations ($U$) can be clearly seen as symmetric from: $T: \Psi =\{e^{ia} \Psi|a \in R \} \mapsto \Psi^{'} =\{e^{ib}U \Psi|b \in R \} $ and hence, ...
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Are the creation operators on the fermionic Fock space bounded linear operators (do they have finite operator norm)?

Let $H$ be a Hilbert space, denoting single-particle states, and $\mathfrak{F}$ be the fermionic Fock space. If $f\in H$, then is the creation operator $c^*(f)$ a bounded linear operator on $\mathfrak{...
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How to determine initial quantum state? [closed]

A particle in an infinite square well has its initial wave function an even mixture of the first two stationary states: $$\psi(x,0)=A(\psi_1(x)+\psi_2(x)) $$ As you may know, for $\psi(x,t)$ we ...
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3answers
86 views

Quantum Field Theory: Number Operator $\hat{N} = a^\dagger a$ and bra-ket notation

My textbook, Quantum Field Theory and the Standard Model by Schwartz, says the following: The easiest way to study a quantum harmonic oscillator is with creation and annihilation operators, $a^\...
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What is the spectrum of $\hat x \hat p + \hat p \hat x$?

In quantum mechanics we know that the canonical position $\hat x$ and momentum operator $\hat p$ satisfying \begin{align} [\hat x,\hat p] = i \quad (\hbar = 1) \end{align} have continuous spectrum. ...
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Bloch sphere hyperspatial multiqubit superpostion

What potentials can extend for a hyper-spatial diagram from a Bloch Sphere in terms of delineating multi-qubit superpositions extending from a quasi-periodic phase index upon reduction of the wave ...
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Could the definition of the experiment result occur before measurement?

Is there any fundamental demonstration that result of an experiment cannot have been determined before the measurement, yet according to the probabilistic rules of quantum mechanics? I understand ...
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The “different Hilbert spaces” of a photon?

What I understand of Fock states so far: They describe the quantum state of a bunch of photons. A single photon can be in several different energy states, and when these photons are tensored together -...
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2answers
73 views

QM probability density function without Born's rule, invariant to wave-function phase

The QM probability density as a function of the wave function is given by Born's rule as a postulate. This leads to the probability density being invariant to the phase of the wave function. Suppose ...
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Why do we call $x$ a constant while mentioning eigenfunctions of the position operator?

For a position eigenstate $\psi_x$, we say $$\hat{x}\psi_x=x\psi_x$$ Since it's an eigenfunction, we say that the result of using the operator is "a constant times the function itself". But $x$ isn't ...
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Differentiation of a ket vector with respect to a spatial dimension

Consider a state $|\psi\rangle$. While discussing the Schroedinger equation, we say $$\hat{H}|\Psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle$$ We also define the hamiltonian operator ...
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Solutions that are part of the Hilbert space

Why do we omit solutions that do not converge at $\pm\infty$ from the physical Hilbert space, what is the argument for us being allowed to do so?
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Euler-Lagrange Equations for Molecular Dynamics

In the Car-Parrinello (CP) method for molecular dynamics simulation, the Euler-Lagrange equations are given as $$ \begin{aligned} \frac { d } { d t } \frac { \partial \mathcal { L } _ { \mathrm { ...