Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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Motivating Irreducibility of Hilbert Space for Quantization Axioms

In the context of geometric quantization, we usually look for a map from the Poisson algebra of classical observables to the algebra of quantum observables (or rather, a sub-algebra of the classical ...
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76 views

Underlying C*Algebra operators in standard quantum mechanics?

Linearity in standard quantum mechanics (QM) is the key to making the math possible in this field, but the presence of nonlinear operators in QM is what is more generally dealt with. Working with the ...
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120 views

Pole in reflection/transmission coefficient and bound states

I was working on a scattering problem in a quantum mechanical system with Hamiltonian $$H_1=A^{\dagger}A=(-\partial_x+W(x))((\partial_x+W(x))).$$ One can show that a 'supersymmetric' partner to this ...
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243 views

Is the Hilbert space spanned by both bound and continuous hydrogen atom eigenfunctions?

As e.g. Griffiths says (p. 103, Introduction to Quantum Mechanics, 2nd ed.), if a spectrum of a linear operator is continuous, the eigenfunctions are not normalizable, therefore it has no ...
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39 views

How to build QM with projective spaces from the beginning?

In conventional treatment of QM, one assumes that (1) physical states are normalized vectors in (rigged) Hilbert spaces and (2) operators correspond to observables, with their eigenvectors denoting ...
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51 views

commutation relations in terms of eigenstates scalar product

This question has caught my attention because I was unaware of the fact that the position-momentum canonical commutation relations could be derived out of the only assumption for $\langle x | ...
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80 views

Picture-independence of quantum mechanics

I've been thinking about the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics in the following terms lately: a quantum system is a Hilbert space $\mathcal{H}$ equipped with ...
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284 views

Derivation of the Lippmann-Schwinger equation

I was trying to understand the derivation of the Lippmann-Schwinger equation in Sakurai's Modern Quantum Mechanics, Section 6.1. Our teacher presented a much simpler derivation, similar to that on ...
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147 views

Understanding the states in Quantum Field Theory

I am self-studying quantum field theory, and I've been struggling to understand the nature of the states that emerge in quantum field theories. After thinking about it, what I think one has in the ...
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586 views

LSZ reduction theorem derivation in Weinberg QFT

When deriving LSZ reduction theorem Weinberg in his QFT book have assumed n-point generalized Green functions, $$ G(q_{1},...,q_{n}) = \int d^{4}x_{1}...d^{4}x_{n}e^{-i\prod_{i =1}^{n}q_{j}x_{j}} ...
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41 views

What do different supercharges corrrespond to? Is creation mathematically close to fock space?

We may have $I =1,2,..N$ where it is said that this corresponds to $N$ supercharges, $Q^I$. By the supersymmetry algebra, $$[M_{\mu\nu},{\overline{Q}}^{I\dot{\alpha}}] = ...
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104 views

Commutation relations in quantum mechanics

As we know, simple harmonic oscillator can be solved only by commutation relations between creation and annihilation operators, and the Hamiltonian expression. The spin energy is either solved only ...
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100 views

How to understand the relationship between these two geometrical structures?

During my study of quantum information processing, I occasionally meet two different geometrical structures: (a) The geometry of the Hilbert space of quantum state, where the superposition and ...
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56 views

Ordering of eigenstates in the quantum adiabatic theorem

Suppose we have an 'initial' Hamiltonian $H_{i}$ whose eigenvalues are all non-degenerate, which we order as follows: $ E^{0}_{i} < E^{1}_{i} < \dots < E^{N-2}_{i} < E^{N-1}_{i}$ Suppose ...
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161 views

A problem with the Gamow state

Consider a form of potential $U(r)$ as follows $$ U(r)=\begin{cases}0 & 0<r\leq a \\ U_0 & a<r\leq b \\ 0 &r>b\end{cases} $$ In this problem $r$ is the distance from the origin, ...
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151 views

Density matrix formalism and group representation

The postulates of quantum theory can be given in the density matrix formalism. States correspond to positive trace class operators with trace 1 on a Hilbert space $\mathcal{H}$. Composition is defined ...
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89 views

Symmetry and Algebra

I'm trying to get a more concrete idea how symmetry is understood in quantum theories, as broad as possible. Consider a infinitesimal transformation of states in quantum physics of the form: $$ ...
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166 views

What are you studying when you study a Harmonic Oscillator in QM?

This probably is a naive question - so please forgive a self-studier. In the text I am studying, one builds a HO by placing a particle in a potential that increases quadratically from the origin. The ...
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232 views

Why do some terms vanish in first-order perturbation theory?

In first order perturbation theory, we usually express the first order perturbation in the eigenket of the perturbed Hamiltonian in the basis of the unperturbed Hamiltonian $H_{0}$: ...
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90 views

What is a continuous superselection sector?

I'm studying the terrible subject of continuous superselection rules and I faced with the following problem. Usually (continuous or discrete) superselection rules are defined involving a direct ...
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119 views

Can we “safely” assume that quantum computing systems will be finite-dimensional?

This is a common assumption in the study of quantum computation to assume that the quantum systems involved are finite-dimensional, since qubits lives in the two-dimensional Hilbert space. According ...
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35 views

How to represent the spherical wave by using Fock basis?

Suppose I have two particles with opposite momentum: $$ |\psi \rangle_{\mathbf k} = |\mathbf k; -\mathbf k\rangle ,\quad |\mathbf k| = M $$ I want to represent the spherical symmetric distribution of ...
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72 views

CFT: from States to Operators

I'm having trouble finding the general algorithm for moving from states to operators under the state-operator correspondence in a CFT. Does anyone have any hints as to how one might go about ...
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86 views

Physical significance of non-commutativity of ladder operators in Quantum Harmonic Oscillator

If we apply the raising (creation) operator to $Ψ_n(x)$ and the apply to it the lowering (annihilation) operator, we get $Ψ_n(x)$ times a constant. Does it physically say something? Can we get any ...
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51 views

What are some resources for Algebraic quantum mechanics for Physicists

I am interested in the GNS construction and other stuff. I am aware of Valter Moretti's book but I want something that is more inclined towards physicists.
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49 views

The Hilbert space that contains the first order correction to the state vector in Time-independent Perturbation Theory

When deriving the expression for the first order correction to the state vector of the new hamiltonian( H = H0 + H' ) we assume that $|\psi$n1> = $\sum_{m \neq n}$ C$_m$(n) $|\psi ^0 _m>$ ...
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40 views

QFT: Limits in Time Ordered Correlation Function Derivation

Background In part of the derivation for the time ordered correlation function I have the following equation (This equation I am fine with - it is what follows that I am not), $$ ...
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62 views

QFT: Ground State Momentum - Normalisation of States

In my notes I have, $$ \left\langle \mathbf{p} \left| \mathbf{q} \right.\right\rangle = \left\langle 0 \left| {a(\mathbf{p})}\ {a(\mathbf{q})}^{\dagger} \right| 0 \right\rangle $$ I am not sure how ...
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123 views

Is there a physical significance to non-normal states of the algebra of observables?

Quantum theory may be formalized in several different ways. Generally, the physical discussion of different states of a quantum system distinguishes pure and mixed states, and then subsumes both in a ...
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111 views

Field operator eigenstate vs. single-particle state

I would like to make sure I understand some basic QFT. My understanding so far is that field operators measure field intensity and their Fourier transform measure intensity of field oscillation. In ...
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95 views

Dynamics and kinematics of quantum field theory

What is the difference between dynamics and kinematics of quantum field theory? I read that in QFT there is no possibility to keep the two things distinct because of a problem with the separability of ...
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85 views

Creating an arbitrary state of the quantum simple harmonic oscillator

Suppose $\mathcal{B}=\{|0\rangle, |1\rangle, |2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} |\Psi\rangle = ...
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62 views

About the definition of super Hilbert Spaces

I have founded in the literature at leas two different definition of Hilbert spaces: Definition 1: A super Hilbert space is a complex super-vector space $\mathcal{H}=\mathcal{H}_0\oplus ...
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214 views

Ishibashi states and Cardy states in CFT

What are the Ishibashi states and Cardy states in CFTs? I am familiar with conformal field theory language. It would be great if someone can discuss about the basic idea of these states and their ...
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71 views

Are hilbert spaces invariant under gauge transformations?

I'm trying to work out if the physical hilbert space is invariant under any gauge transformation? I have found situations where under some transformations they don't change but I've now gotten very ...
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32 views

How to count total spin degeneracies for many spin one half particles?

Given the spin operator for particle $j$ \begin{align} \bar{S}_{j} = \left( \bigotimes_{k=1}^{j-1} I_{k} \right) \otimes \left(\tfrac{\hbar}{2}\bar{\sigma}\right)_{j} \otimes \left( ...
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64 views

I have a problem with the variational method approximation in quantum mechanics. Is my issue valid, or am I misunderstanding something?

The variational method for approximating the ground state of a Hamiltonian $H$ by providing a lower bound is simple enough. If we construct any test wave function $|\bar{0}\rangle$ then the claim is ...
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168 views

Simultaneous eigenket

J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator): Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with ...
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105 views

Query about the proof why non-orthogonal states cannot be reliably distinguished

I have a query regarding the proof given in the book by Nielsen and Chuang on why two non-orthogonal states cannot be perfectly distinguished. The proof is given on page 87 of the 10th edition; here ...
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110 views

Non Hermitian Quantum Mechanics

I was just reading about Non-Hermitian Quantum Mechanics dealing with Hamiltonians $H$ that are not Hermitian operators. Then it is unclear that we get orthonormal eigenstates. Now, I was reading a ...
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80 views

Is operator $\hat{O}_{\alpha}:|\phi,\psi\rangle\mapsto |e^{i~\alpha[\phi,\psi]}~\phi,e^{-i~\alpha[\phi,\psi]}~\psi\rangle$ unitary?

Is the operator $\hat O_{\alpha}$ which is defined in the following a unitary operator? Operator $\hat O_{\alpha}$ is supposed to act on composite states with two explicit components, such that ...
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52 views

How does independence of the basic bras affect the choice of numbers used to represent a ket?

On page 54 of Dirac's book, 'The Principles of Quantum Mechanics', he states: Take an orthogonal representation with basic bras $\langle\lambda_1\lambda_2...\lambda_u|$, labelled by parameters ...
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152 views

One more time about LSZ-theorem

This question is the continuation of this one. For simplicity, let's use $(1)$ from the linked question (it is called n-point Green function and in particle case coincides with internal diagram), $$ ...
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394 views

Transition Between Position and Momentum Basis

I'm having some trouble following pages 55-56 of Sakurai's Modern Quantum Mechanics. We're trying to transfer from position space into momentum space. Here's a quote: Let us now establish the ...
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156 views

Quantum Rigid Rotor Perturbation

As the title says, I have a rigid rotor with a perturbation given below $$H=\frac{L^2}{2I}-\alpha B L_z.$$ So I know that the eigenvalues of $H$ will be $\ell(\ell+1)/2I -\alpha B m$ where $m$ is our ...
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126 views

Momentum and position operators in Schrödinger representation

I was going through some intro notes on path integral (for QFT), and am stuck with this equation for position and momentum in Schrödinger (position) representation, $$ \hat{1} =\int ...
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62 views

How do I properly express adding perturbed states to unperturbed states?

I have a problem set due tomorrow, and the last problem is driving me nuts. Been combing through griffiths trying to find similar examples to no avail, so it'd be greatly appreciated if stackexchange ...
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262 views

A basic question about Heisenberg State Kets (in particular the simple harmonic oscillator)

I know base kets in the Heisenberg picture are $U^\dagger |{a}\rangle$ but if the base kets are the base of the hamiltonian, and the hamiltonian is independent of time, are all of the base kets ...
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257 views

Representing a polarization vector for light as a 'manifold of two state'

Explain me these projections please Context: I was reading a paper (Phys. Rev. A 68, 052307) which involved mesoscopic coherent states of light. There, in order to calculate the uncertainty of a ...
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28 views

Homework question on the Poynting vector

I need to show what the Poynting vector becomes when i insert the definition of A through creation and annihilation operators, what i get is the expected answer plus a term containing two creation ...