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

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Decompose a Hermitian Operator into Eigenvalues and Projectors

Quantum Computing - A Gentle Introduction by Eleanor Rieffell and Wolfgang Polak states on p57 : Any Hermitian operator $O$ with eigenvalues $\lambda_j$ can be written as $O = \sum_j \lambda_j ...
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1answer
60 views

What is the time evolution operator in quantum mechanics [duplicate]

I'm curious about what happen to a system when the configuration of the system changes. If we have a system in a state $|\psi_{\textrm{in}}\rangle$ and we change the configuration of the system, the ...
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4answers
201 views

Trying to understand inner product notation

I I'm taking a QM course and I'm trying to make sense of why observables are sometimes conjugated for no apparent reason in their inner products. Right now I'm watching Dr. Susskind's lecture on ...
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2answers
282 views

How to describe time evolution in relativistic QFT?

I must confess that I'm still confused about the question of time evolution in relativistic quantum field theory (RQFT). From symmetry arguments, from the representation of the Poincare group through ...
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2answers
146 views

Probability to find a particle in a particlar state $\psi_{n}$ [closed]

I have a problem to understand the probabilities in QM. In particular, if I have a particle in state $\psi_{n}$, then we change the system and we ask for the probability to find the particle in a ...
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1answer
93 views

Continuous basis of eigenvectors

In Quantum Mechanics, when using Dirac's formalism one of its features is the expansion of state vectors into continuous basis of eigenvectors of unbounded self-adjoint operators. Let $\mathcal{H}$ be ...
3
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1answer
50 views

When can a Hilbert space with a given Hamiltonian be decomposed into non-interacting tensor product factors?

Let's say I have a Hilbert space $\mathcal{H}$ (either finite-dimensional or with a countably infinite basis) with a specified Hamiltonian $\hat{H}$, representing some quantum system. Under what ...
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2answers
144 views

How to prove $\mathrm{Tr}\left[|\alpha\rangle\langle\alpha|\hat{A}\right]=\langle\alpha|\hat{A}|\alpha\rangle$

For a coherent state $|\alpha\rangle=e^{-\frac{|\alpha|^2}{2}}\sum\frac{\alpha^n}{\sqrt{n!}}|n\rangle$, please show me how to prove, $$ ...
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3answers
375 views

Hilbert space of a quantum system

The first postulate of Quantum Mechanics as I've learned can be stated as: The states of a quantum system can be described by vectors in a Hilbert space. I've seem also some people also ...
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2answers
107 views

Commutator of position and momentum

I'm reading Sakurai's Quantum Mechanics. One of the problem in the book asks to use the relation $$ \langle{x}|p\rangle=\frac{1}{\sqrt{2\pi\hbar}}e^{\frac{ipx}{\hbar}} $$ to evaluate ...
2
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1answer
179 views

Superposition of two wave functions of different Hilbert spaces

I am trying to think of this problem for quite some time. Let's say, we have two sets of wave functions $\lbrace|\psi\rangle\rbrace$ and $\lbrace|\phi \rangle\rbrace$ and they belong to two different ...
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2answers
189 views

Hermitian conjugate of differential operator

Help me find $\hat{B^\dagger}$, when we know that $$\hat{B}=i\frac{d}{dr}$$ with the condition that $\hat{B}$ is defined in spherical coordinates. My approach: $$ ...
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0answers
98 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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1answer
70 views

Are the position eigenkets $\lvert x \rangle$ really a basis for the space of states?

In my current understanding, matrix formulation and wave-function formulation of QM are basically the same because $\left|\psi\right>$ and $\psi(x)$ are really the same mathematical object: A ...
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2answers
140 views

How the position operator and the position basis are correctly defined?

In Quantum Mechanics, if one deals with wave functions, the Hilbert space in question is $L^2(\mathbb{R}^n)$ for a particle in $n$-dimensions, and the position operator corresponding to the $i$-th ...
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96 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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1answer
86 views

State time evolution of a quantum harmonic oscillator with a Dirac-Delta function as an initial state [closed]

I have a question just like this Phys.SE problem here with a difference that our system is a harmonic oscillator (rather than a free particle). A particle with mass $m$ is connected to a string with ...
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1answer
80 views

Physical interpretation of the creation operators in string theory?

Is there any way to describe phsycially which each creation operator $a^{(i)+}_{n}$ in string theory does to the ground state string? Here would be my guess (although it is likely to be totally ...
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1answer
116 views

BRST quantization and norm

States with definite ghost number have zero norm (since ghost number is anti-hermitian and has real eigenvalues). E.G. when quantizing relativistic point particle, physical spectrum turns out to ...
4
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2answers
281 views

Why don't non-Hermitian operators with all real-eigenvalues correspond to observables? [duplicate]

Suppose you could construct an operator that was non-Hermitian but had all real eigenvalues or could at least be restricted in a way to create only real eigenvalues, why would this operator not ...
2
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1answer
79 views

When do two operators act on the same Hilbert space?

Suppose I want to represent the quantum state of a spinless particle. To do so, I employ a Hilbert space $\mathcal{H}_X$, which is an infinite-dimensional Hilbert space equipped with a position ...
2
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1answer
74 views

Probability amplitude for motion from $x_i$ to $x_f$ in Heisenberg picture

In M. Nakahara's book Geometry, Topology and Physics on page 19, the probability amplitude for a particle to move from $x_i$ at time $t_i$ to $x_f$ at time $t_f$ is given as $$ \tag{1} \langle x_f, ...
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2answers
80 views

Is there an intuitive explanation to the fact that the solutions to the time-independent Schrödinger equation form a complete basis?

We were always told that the solutions to the time-independent Schrödinger equation form a complete eigenbasis for the space of all functions (all functions?) but I never understood why this is the ...
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2answers
115 views

Time evolution in Quantum Mechanics abstract state space

As I've learned the first postulate from Quantum Mechanics can be stated as follows: The states of a quantum system are described by vectors in a complex Hilbert space $\mathcal{H}$. The book ...
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3answers
446 views

Physical meaning of quantum operators

Let's say we have a wavefunction $\psi$ and a measurement operator $\hat A$. I understand how eigenvalues and eigenvectors of $\hat A$ describe the possible outcomes of the measurement. I also ...
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3answers
412 views

Eigenvalue for the creation operator for a coherent state [closed]

For a coherent state $$ |\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle $$ I can't solve the eigenvalue problem for $\hat{a}^{\dagger}|\alpha\rangle$ where ...
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0answers
47 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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2answers
48 views

Expectation value of $S_x$ [closed]

$$\chi=A\begin{bmatrix} 3i \\ 4 \end{bmatrix}$$ I am asked to evaluate the expectation value of $S_x$. I understand the equation $$\langle S_x \rangle=\frac{\hbar}{50} \begin{pmatrix} -3i & 4 ...
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2answers
240 views

Hermitian 2x2 matrix in terms of pauli matrices [closed]

In my studies, I found the following question: Show that any 2x2 hermitian matrix can be written as $$ M = \frac{1}{2}(a\mathbb{1}+\vec{p}\cdot \vec{\sigma}) $$ with $a=Tr(M)$, $p_i = ...
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2answers
147 views

Electron Spin state probability [closed]

Suppose there is a spin 1/2 particle in a state $\chi = {1 \over \sqrt{5}} \begin{bmatrix} 1\\ 2\\ \end{bmatrix} $. To determine the probability of finding the particle in a spin up($\hbar/2$) state, ...
4
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4answers
158 views

How to determine the observables rigorously?

In Quantum Mechanics, as I know, if a system is described by a Hilbert space $\mathcal{H}$, each physical quantity is associated with some hermitian operator $A : U\subset \mathcal{H}\to \mathcal{H}$ ...
3
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2answers
211 views

Wavefunctions in different Hilbert spaces

The state of a quantum system is represented by a wavefunction usually in some specific Hilbert space, .e.g of position, spin, momentum etc. But before deciding in which of these bases to decompose ...
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1answer
55 views

Possible values for $L_x$

I've a physical system with $l=1$ and I have to calculate the values I can obtain if I measure $L_x$ and their probability. I know that: the values I can obtain are $\ m=0, \pm 1$ $\displaystyle ...
2
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2answers
121 views

Superposition in Quantum Mechanics

First of all, let $V$ be a vector space over the field $\mathbb{F}$. It is possible then to show, by Zorn's Lemma that there is a basis for $V$. The main point is that although basis are quite ...
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2answers
448 views

State of a system in Quantum Mechanics and state vectors

I'm taking a course in Quantum Mechanics and there is something I'm not being able to fully understand. On more elementary courses on Quantum Mechanics I've been told that the idea of Quantum ...
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1answer
109 views

Completeness relations of eigenstates in the Heisenberg picture

I've been reading Srednicki's introduction to path integrals and I'm slightly unsure of the notation that he uses for the completeness relation of position eigenstates in the Heisenberg picture. In ...
3
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2answers
139 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 ...
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3answers
75 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 ...
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118 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: ...
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1answer
83 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
117 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.. " ...
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3answers
117 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 ...
2
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1answer
85 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, ...
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1answer
139 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 ...
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1answer
115 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
95 views

Eigenstates of sum of creation and annihilation operators

Does the operator $a+a^\dagger$ have eigenstates? If yes, what are they?
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285 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
217 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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1answer
43 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 ...
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2answers
206 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 ...