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

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86 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 ...
3
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2answers
200 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 ...
4
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0answers
114 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 ...
3
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1answer
90 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 ...
2
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1answer
86 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 ...
4
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1answer
148 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 ...
5
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2answers
375 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
90 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
75 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
92 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 ...
2
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2answers
131 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
501 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
569 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
54 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
54 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
340 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
258 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, ...
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4answers
179 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 $A : U\subset \mathcal{H}\to \mathcal{H}$ ...
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2answers
256 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
59 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
132 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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651 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
139 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 ...
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2answers
147 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
88 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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121 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
101 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
133 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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129 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
93 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, ...
2
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1answer
218 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 ...
2
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1answer
156 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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106 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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309 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
279 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
44 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
224 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 ...
2
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1answer
242 views

How to visualize a Schrödinger cat state?

I recently read about Schrödinger cat states, which are basically a superposition of two coherent states $|\alpha\rangle$ with opposite phases, that is, $$ |\mathrm{cat}\rangle = |\alpha\rangle \pm ...
3
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1answer
116 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 ...
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52 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
92 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
145 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 ...
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1answer
46 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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1answer
41 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 ...
7
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1answer
117 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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83 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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2answers
61 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}} ...
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282 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 ...
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3answers
93 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
332 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 $\ket{-1/2}$ ...