In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

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The uncertainty in angular momentum

It is known that the different spatial components of the angular momentum don't commute with each other. $$ [L_x,L_y] \propto L_z \\ [L_y,L_z] \propto L_x \\ [L_z,L_x] \propto L_y $$ Also it is known ...
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Difference between operators used to represent quantum gates vs that to represent physical observables?

I have learnt that informations about a physical observable property is buried in the state vector of a quantum system. To get the possible value of a property all we need to do is multiply the state ...
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Gauge covariant derivative of a creation operator

Suppose we define the (gauge) covariant derivative or as $$\tilde{\nabla}=\nabla+ie\textbf{A},$$ where the vector potential $\textbf{A}$ has a matrix structure where only the diagonal has nonzero ...
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149 views

Definition of the “support” of the reduced density matrix

Some of the papers in condensed matter physics use the word "support" (space). For example, the following papers use the support especially for the reduced density matrix. http://journals.aps.org/prb/...
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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 $\langle{x}|[X,P]|...
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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: $$ \langle\psi|\hat{B}\psi\rangle=\int_{...
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How/Why did Feynman relate the element of Hamiltonian matrix $H_{12}$ to the amplitude to go from $|1\rangle$ to $| 2\rangle$?

$$ \newcommand{\bk}[2]{\left\langle #1 | #2 \right\rangle} \newcommand{\ket}[1]{\left| #1 \right\rangle} \newcommand{\bra}[1]{\left\langle #1 \right|} \newcommand{\biik}[3]{\left\langle #1 | #2| #3\...
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271 views

Wick Theorem, ordering & CFT

I'm having a little trouble with correlation functions wick theorem and ordering in the context of OPE and CFT, for string theory. (1) My first question, the propagator is: $$<X(z) X(w)> = \...
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301 views

Free Vacuum vs Interacting Vacuum and Wick's theorem

I'm studying perturbation theory in QFT and I stumbled on a conceptual problem. My understanding of the interplay between LSZ reduction formula and the Gell-Mann & Low perturbation series is that:...
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190 views

Transformation of operator matrix under change of basis [duplicate]

How does operator matrix transform under change of basis? If $\rvert \beta\rangle$ and $\rvert \alpha \rangle$ are two bases related by transformation $ \rvert\beta_m\rangle = \sum_n S_{mn} \rvert\...
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How to determine the trace and determinat of a differential operator?

How to determine the trace and determinant of the operator like $\Box$ or $\nabla^2$ etc. But first of all how to find the same for the simpler operator $\frac{d}{dx}$? I proceeded as follows. What ...
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58 views

Deriving the form of generators of transformations

I'm struggling to understand a bit of quantum mechanics relating to the transformation generators. This specific bit contains quite a few guesses and assumtions which probably do make sense in ...
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156 views

Book question positive square root on quantum operator

On p.86 Section 2.2.4 of the Quantum computation and quantum information book by Nielsen, $M_{o}$ is defined as the positive square root of the positive operator. Is the "positive square root" ...
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260 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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57 views

Operator algebra in integral form

In QM courses one can quite often see expressions like: $ \langle x| \hat{p} | \psi \rangle = \int dp \langle x| \hat{p} |p\rangle \langle p| \psi \rangle $ but I'm a bit confused as to how it ...
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Is this treatment of the momentum operator in the Dirac formalism allowed?

I have a problem understanding a specific bit of Dirac notation. Take, as an example this derivation: I'm dubious about the step from line 3 to 4. When momentum operator acts on the momentum ...
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Determining this vacuum expectation

I am trying to find the analytic expression for the result that follows from evaluating this vacuum expectation value: $\langle0\vert;\prod_{i=1}^M \prod_{j=1}^N \hat{a}(y_{ij}) \hat{a}^\dagger(y'_{...
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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 wrong)...
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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 ...
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323 views

Quantum and Classical Liouville operators

In the Heisenberg picture of Quantum Mechanics, for an observable $\hat{A}$, we have the famous Heisenberg equation giving the time evolution of the operator: ($\hat{H}$ is the Hamiltonian operator) $$...
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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 ...
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Is it possible to define the logarithm of creation/annihilation operators?

This is just a curiosity I had recently. I am going to let the reader interpret the context of the question, in their own way!
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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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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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Infinitesimally change a operator in QM

Reading Balian, "From Microphysics to Macrophysics", I've found the following identity: If we change the operator $\hat{{\mathbf{X}}}$ infinitesimally by $\hat{{\delta\mathbf{X}}}$, the trace of an ...
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What special significance does the eigendecomposition of a mixed density operator hold over other pure state decompositions?

It is known that in general, a mixed state can have multiple pure state decompositions. However, it has a unique eigendecomposition in the absence of degenerate eigenvalues. What is the special ...
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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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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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“This operator is odd under parity”

In problem 8.10 of Schaum's Quantum Mechanics they say: "We see that under the parity operator $r \rightarrow r$, $\theta \rightarrow \pi - \theta$ and $\phi \rightarrow \pi + \phi$ .. since $\frac{d}...
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319 views

How do you prove that the number operator commutes with a general Hamiltonian?

If you have a hamiltonian, in the case of a bosonic system $$ H=\sum_{ij}H_{ij}a_i^\dagger a_j, $$ and the number operator $$ N=\sum_{i}a_i^{\dagger}a_i. $$ How do you show that they commute? I have ...
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Prove that if $\hat H | a_n\rangle=a_n|a_n\rangle$, then $f(\hat H)| a_n\rangle=f(a_n)|a_n\rangle$

In Quantum Mechanics you have the eigenvalue equation: $$\hat H | a_n\rangle=a_n|a_n\rangle \tag{1}$$ where $\hat H$ is the Hamiltonian operator, $\{|a_n\rangle\}$ is a complete set of eigenstates ...
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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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160 views

Conceptual questions on the path integral formulation of QFT

I'm currently trying to teach myself the path integral formulation of QFT (having studied the canonical approach previously), but I'm having some conceptual difficulties that I hope I can clear up ...
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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 ...
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What quantum measurement formalism is easiest to implement physically?

As part of my studies and research, I have learned to work with three different measurement formalism which I define to avoid any ambiguity with the nomenclature: General measurements, which are ...
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Electric field operator in 2D geomatry

In the free field (3D), transverse electric field operator is given by the below expression; $$e^{\bot}(\textbf{R}) =i \sum_{\textbf{p},\lambda}\Big( \frac{\hbar cp}{2V\epsilon_{0}}\Big)^{1/2} \{e^{(\...
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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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285 views

Meaning of expectation value of product of non-commuting operators

Let $\hat{A}$ and $\hat{B}$ be Hermitian observables with spectra labeled by $a$ and $b$. Then we can write \begin{equation} \hat{A} = \sum_a a\, \hat{P}_a \end{equation} \begin{equation} \hat{B} = \...
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Lippmann-Schwinger equation and $T$ expansion

Lippmann-Schwinger equation, in operator form, is: $$ T=V+V\frac{1} {E-H_0+i \hbar \varepsilon} T=:V+V\Theta_0T, $$ where $H_{tot}=H_0+{V}$ is the hamiltonian ($H_0$ is the free particle hamiltonian ...
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78 views

How to deal with eigenvectors which are not square integrable?

In Quantum Mechanics there is one type of situation I'm still unsure on how to deal with. First of all, I want to make clear I'm trying to understand how to deal with this rigorously. What I'm talking ...
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561 views

Schroedinger equation for hydrogen atom

I have got a problem understanding the meaning of the Laplace operator in the Schrödinger equation for the hydrogen atom. $$\Big(-\frac{\hbar^2}{2m_e} \Delta_{r_e} - \frac{\hbar^2}{2M_P} \Delta_{r_p} ...
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122 views

What is time evolution operator?

Could you explain to me (level 1 years undergrade) what is a time evolution operator? I read on Wikipedia, and it confuses me.
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Boson ladder operator $n+1$ factor [closed]

Looking at Boson creation and annihilation operators, I come across that \begin{equation} b_a|n_\alpha\rangle=\sqrt{n_\alpha}|n_\alpha-1\rangle \end{equation} and \begin{equation} b_a^+|n_\alpha\...
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Adjoint Fokker-Plank operator

In Zwanzig's book "nonequilibrium statistical mechanics" he defines the Fokker-Plank equation for a probability distribution $f$ and with it an operator $D$: $${ \partial f(a,t) \over \partial t} = \...
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Definition of Hamilton operator

The Hamilton operator is by definition a self-adjoint operator $H\text{: }D\left(H\right)\to\mathcal{H}$ with $D\left(H\right)\subset\mathcal{H}$ a dense linear subspace of the Hilbert space $\mathcal{...
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One-electron reduced density matrix: Argument for positive semidefiniteness

I cannot follow an argument for positive-semidefiniteness of the one-electron density matrix given in "Molecular Electronic-Structure Theory" by Helgaker/Jorgensen/Olsen. First some definitions: $F(M)$...
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How to construct the operator and the physical experiment needed to perform an arbitrary 'measurement in a basis'?

I have taken an introductory level course in QM and have covered some advanced topics by myself and don't really understand what it means to 'measure in a particular basis'. A projective measurement ...
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140 views

Derivation of the low-energy effective Hamiltonian

In the quantum mechanics, the Hamiltonian $H$ satisfies the Schroedinger equation $$ H\psi = E\psi. $$ Suppose that $P$ is a projection operator, and $Q=1-P$. The low-energy effective Hamiltonian is $$...
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332 views

Time-ordering and Dyson series

In Dyson series we use a time-ordered exponential by arguing that a Hamiltonian at two different instants of time does not commute. Why is it that so? Can anyone explain with example why should the ...
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138 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.. " ...