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

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54 views

Distinction between state space and space of functions

In Quantum Mechanics a particle is described by its wave function $\Psi : \mathbb{R}^3\times \mathbb{R}\to \mathbb{C}$. In that sense, the state of a particle at time $t_0$ is characterized by a ...
8
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1answer
251 views

Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
1
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3answers
53 views

Is the perturbation Hamiltonian an observable?

In fine structure calculation we use the perturbation theory. The basic Hamiltonian $H_0$ is perturbed as: $H = H_0 + W$ First, the basic problem assume that $H_0$ is an observable. That allows to ...
0
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2answers
114 views

Function of observables in mathematical words

In mathematical words, an observable is an operator that a set of linearly independent eigenfunctions constitutes a complete basis of the wave-functions' space. Now, let's consider some observables: ...
1
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2answers
152 views

$\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute… or do they?

So $\hat{L}_{x}$ and $\hat{L}_{y}$ do not commute: $ [ \hat{L_{x}}, \hat{L_{y}}] = i\hbar \hat{L_{z}}$ But, what if we perform this operation on a state such that: $\hat{L_{z}} \phi_{l, m_{l}} = ...
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1answer
39 views

Unitarily Inequivalent Representations

The definition of unitarily equivalent representations I am using is the one given here: https://en.wikipedia.org/wiki/Haag%27s_theorem. Now in this text ...
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3answers
94 views

Both Eigenvalues and Operators are “Observables”? [duplicate]

I am having a bit of difficulty wading through the what seems to be multiple usages for Observables in Quantum Mechanics. " Mathematically observables are postulated to be Hermitian operators.. " ...
2
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3answers
205 views

Physical interpretation of applying a unitary operator to a state

When we apply one of the Pauli matrices $\sigma_y$ on one of its eigen-vectors $| \odot \rangle$, what does the eigen-value tell us about $| \odot \rangle$? Is this considered a measurement of $| ...
3
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1answer
60 views

Self-adjoint extensions with 'teletransporting' boundary conditions

When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which (i) the probabilities teleport* between points on the boundary and (ii) boundary conditions ...
22
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4answers
2k views

Why do we need infinite-dimensional Hilbert spaces in physics?

I am looking for a simple way to understand why do we need infinite-dimensional Hilbert spaces in physics, and when exactly do they become neccessary: in classical, quantum, or relativistic quantum ...
2
votes
1answer
53 views

Representation of U(1) on fock space

I am currently reading up on the use of group theory in physics using Peter Woit's book draft (available on his homepage). I do understand the mathematical concepts but have a bit of a problem making ...
0
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3answers
69 views

Confusion about Fock subspace

I'm currently reading Folland's book on quantum field theory and came along some definitions. On p.90 of his book, Folland defines the symmetric Fock space as ...
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147 views

Why is only one quantity of angular momentum i.e. $L_z$ quantized & not $L_x$ & $L_y$?

This is quoted from Arthur Beiser's Concepts of Modern Physics: Why is only one quantity of $\mathbf{L}$ quantized? The answer is related to the fact that $\mathbf{L}$ can never point in any ...
2
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1answer
66 views

Basis states for many-particle system

I'm reading these notes about second quantization. In section 1.4 the author introduces many-particle wavefunctions. But I can't understand how basis are defined here. I know that if $\{\chi_i | i=1, ...
4
votes
1answer
682 views

Existence of adjoint of an antilinear operator, time reversal

The time reversal operator $T$ is an antiunitary operator, and I saw $T^\dagger$ in many places (for example when some guy is doing a "time reversal" $THT^\dagger$), but I wonder if there is a ...
2
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1answer
112 views

How to interpret vector operators in quantum mechanics?

To the point: How should I think about the equation $$\hat{\mathbf{x}}\mid\mathbf{x'}\rangle = \mathbf{x'}\mid\mathbf{x'}\rangle~?$$ Is it a triple of equations $\hat{x}\mid x'\rangle = x'\mid ...
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1answer
61 views

Hermitian conjugate of an antiunitary transformation

In quantum mechanics, one often considers symmetry transformations which are defined in terms of operators which do not change the norm of states in the Hilbert space. For the Wigner's theorem, this ...
0
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2answers
71 views

Partial Measurement and the Math Behind it

$\newcommand{\ket}[1]{\left| #1 \right>}$ $\newcommand{bra}[1]{\left< #1 \right|}$ Talking about the partial measurement the professor defines the state $\ket \psi$ to be $$\ket{\psi} = ...
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1answer
56 views

The definition of the vacuum state of quantum field by path-integral

In the review Entanglement entropy of black holes by Sergey Solodukhin (arXiv:1104.3712, equation 13), I see a definition of vacuum state of quantum field by path integral over half of the total ...
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1answer
1k views

Rigged Hilbert space and QM

Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.
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1answer
59 views
5
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108 views

Ground state for interacting field thoeries

Are there references where the ground state of an interacting quantum field theory is explicitly written in terms of states of the underlying free theory? For example, let us suppose to have a self ...
3
votes
1answer
227 views

How to write a generic density matrix for multi qubit system

I was reading the paper device independent outlook on quantum mechanics. The author defines a generic two qubit density matrix as $$ \rho=\frac{1}{4}\left( I \otimes I + \vec{r_{\rho}} \cdot ...
3
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3answers
219 views

Constructing solutions to the time-dependent Schrödinger's equation

The following question is from David Griffiths' Introduction to Quantum Mechanics: Problem 2.13 A particle in the harmonic oscillator potential starts out in the state $$\Psi(x,0) = A[3 ...
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2answers
104 views

Integration by parts to derive $d\langle x \rangle / dt$

I am reading "Introduction to Quantum Mechanics" by David Griffiths and I am having trouble understanding part of a derivation of $\frac{d\langle x\rangle }{dt}$ in section 1.5 - Momentum - of the ...
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2answers
101 views

A problem with indistinguishable fermions and the order of applying operators

This question comes in consequence of another one. I want to stress a problem that none of the answers addressed it. For making my problem more understandable let me first remind a well-known state, ...
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1answer
100 views

How to visualize a Schrödinger cat state?

I recently read about Schrödinger cat states (SCS), which are basically a superposition of two coherent states $|\alpha\rangle$ with opposite phases, that is, $$ |cat\rangle = |\alpha\rangle \pm ...
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6answers
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How is quantum superposition different from mixed state?

According to Wikipedia, if a system has $50\%$ chance to be in state $\left|\psi_1\right>$ and $50\%$ to be in state $\left|\psi_2\right>$, then this is a mixed state. Now consider state ...
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2answers
112 views

Is every bra vector the complex conjugate of ket vector?

$\renewcommand{\ket}[1]{\lvert #1 \rangle}\renewcommand{\bra}[1]{\langle #1 \rvert}$Suppose we are taking the inner product of two vectors, say $a$ and $b$ as $$\bra{a}b\rangle$$ where $\bra{a}$ is a ...
3
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2answers
929 views

Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?

These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
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1answer
38 views

How is measurement on system in a Hilbert space seen?

I am a bit confused about different kinds of measurements on a system in state $W$ where $W$ is the density operator in Hilbert space $H$. A general measurement can be given by POVM's, let ...
2
votes
1answer
63 views

Separability of the Hilbert space: countable orthonormal basis vs. continuous spectrum

Hilbert spaces are mostly assumed to be separable. A Hilbert space is separable if and only if it admits a countable orthonormal basis. How does this fit together with the possible existence of the ...
0
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0answers
46 views

Quantized Banach Spaces

Reading the paper here, it mentions on the very first page that "The requirement of 'closed'-ness is imposed because we want to think of operator spaces as 'quantized (or non-commutative) Banach ...
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0answers
49 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 ...
2
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1answer
60 views

Wick's Theorem: Why is the vacuum expectation value of uncontracted operators zero?

I'm am right now reading Chapter 4.3 (Wick's Theorem) in Peskin & Schroeder. It is said that In the vacuum expectation value, any term in which there remain uncontracted operators gives zero ...
0
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1answer
33 views

What is the condition for local operations on bipartite entangled state?

I have an entangled state between Alice and Bob $|\psi\rangle_{AB}$ ( both Alice and Bob have states in Hiblert space of dimension $n$ ). Alice and Bob can only perform local meaurements. I assumed ...
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1answer
34 views

“Independent simultaneous eigenbras” in Dirac's book 'Principles of Quantum Mechanics'

I've been puzzling through this book off and on and can usually work out what is going on via other external references on the Intertubes. But, this paragraph from pages 55 and 56 has me a bit ...
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48 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 ...
5
votes
1answer
977 views

Representation of operators in quantum mechanics

For which systems we represent the Hamiltonian as a differential operator and for which system we represent it by a matrix? Can the momentum be represented by a matrix operator?
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78 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 = ...
4
votes
1answer
73 views

Constructing differential equation from arbitrary Hamiltonian

Suppose I begin with the time-independent Schrodinger equation $$ \left(-\frac{1}{2m}\partial_x^2 + V(x)\right)\psi_n(x) = E_n\psi_n(x), $$ ordinarily we specify the function $V$ and then solve for a ...
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2answers
84 views

Using rotation matrix for spin to write x oriented spin in z-spin basis

$\newcommand{\ket}[1]{\left| #1 \right>}$The problem is to write the ket vector for a particle with spin +1/2 along the x axis, in terms of the standard basis vectors $\ket{+1/2}$ and $\ket{-1/2}$ ...
0
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2answers
50 views

Dirac notation and column representation

$\renewcommand{ket}[1]{|#1\rangle}$ I am facing difficulty in understanding how the right hand side is coming in equation A below In $H$ of dimention 4, the vector $$ \sqrt{\frac{2}{3}} ...
0
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2answers
101 views

What is the meaning of integrating over the state space?

If $\lvert\psi\rangle$ denotes the state space corresponding to a qubit, then what is the meaning of the $$\int d\psi$$ where the integral is over whole state space of a qubit? How do I evaluate it? ...
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3answers
62 views

Do we get the same answer at any time if we measure a system's energy?

Schrödinger's equation says that the only allowed energy states of a system are the eigenvalues of the energy operator $H$. This means that if we measure the energy of the system at any time we ...
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2answers
117 views

Separability in quantum physics

I am under the impression that violations of Bell's inequality as shown in e.g. the Aspect experiment can be explained by the fact that the particles where not separable rather then the non-existence ...
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0answers
42 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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3answers
66 views

Probability density for wavefunction given as infinite superposition of eigenstates

How do we find the probability density as a function of (x,t), if the wavefunction is expressed as an infinite superposition of eigenstates? When the wavefunction is expressed as a superpostion of ...
2
votes
1answer
125 views

What does it mean “Hawking radiation is in a pure state”?

I'm trying to understand black hole paradox but I'm not sure if I understand what does it mean "Hawking radiation is in a pure state". Does it mean if Hawking radiation is in a mixed state then ...