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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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 ...
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63 views

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: ...
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108 views

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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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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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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135 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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204 views

Why isn't the time-derivative considered an operator in quantum mechanics? [duplicate]

Based on my understanding when doing quantum mechanics we deal with a small set of mathematical objects: namely scalars, kets, bras, and operators. But then in the Schrodinger equation we have this ...
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221 views

Can operators be argument of Dirac Delta function

In one part of Marc Bee's book on Quasielastic Neutron Scattering, he defines the pair correlation function $$ G(\textbf r,t) = \frac{1}{(2\pi)^3}\int I(\textbf Q,t)\text e^{-i\textbf Q.\textbf r}\ ...
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95 views

Eigenstates of a harmonic oscillator

Using ladder operators, I can find eigenstates $\psi_n$ with eigenenergies $$E_n=\hbar\omega\left(n+\frac{1}{2}\right). $$ In my textbook, ladder operators work like $$ a\psi_n = c_n \psi_{n-1}$$ $$ ...
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50 views

Find the expression of $j_+$ and $j_-$operators [closed]

I have just started to do exercises about quantum mechanics and I have some difficulties. I have a particle with spin $j=1$ and the Hamiltonian is $H=\gamma(j_xj_y+j_y j_x)$ (where $\gamma$ is a ...
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Can the inverse operator be expressed as a series?

I've seen the claim that a function of an operator can be defined as a series. For example, say $A: H_1 \mapsto H_2$ is an operator. Then $$ e^A \equiv \sum_{n=0}^\infty \frac{A^n}{n!}. \tag{1}$$ In ...
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Do I have some freedom when I define the quantum SHO ladder operators? [closed]

I tried to solve the quantum harmonic oscillator via the operator method. After doing it and looking up the solution I noticed that for some reason the ladder operators got an additional factor of (i) ...
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173 views

Continuous spectrum of hydrogen atom

I wonder if there is a nice treatment of the continuous spectrum of hydrogen atom in the physics literature--showing how the spectrum decomposition looks and how to derive it.
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401 views

A particle in a 1D box: what is the meaning of velocity?

In the box $x = 0$ to $x = L$, $V = 0$, and for $x < 0$ and $x > L$, $V = \infty$ (infinite potential well). The eigenvalues of the Hamiltonian are: $$E_n = \frac{n^2 h^2}{8L^2} \, .$$ Since ...
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Why does time evolution operator have the form $U(t) = e^{-itH}$?

Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through $$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$ and ...
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What is a single-phonon?

From what I understood from wikipedia, as well as some other resources, each phonon corresponds to a normal mode oscillation, and the creation operator to create a phonon of wavevector $k$ is: $$ ...
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$[A_1, H] =[A_2, H] = 0$ but $[A_1, A_2] \neq 0$?

I am having a difficult time understanding this problem. Suppose $[A_1, A_2] \ne 0,$ $[A_1, H] = 0,$ $[A_2, H] = 0.$ Show that the energy eigenstates of $H$ are in general ...
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267 views

Is Hamiltonian a differential operator in second quantization?

Normally, a free particle Hamiltonian is written $$ \hat{H} = - \frac{\hbar^2}{2m} \Delta $$ which is a differential operator because Laplacian $\Delta$ is. On the other hand, in second ...
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Commutation relations in second quantization

I know that for operators $a(\chi_1), a(\chi_2)$ of the same type (fermionic or bosonic) $$ [a(\chi_1), a(\chi_2)]_{-\xi} = [a^\dagger (\chi_1), a^\dagger (\chi_2)]_{-\xi} = 0 \tag{1}$$ where $$\xi ...
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How to write electron hole Hamiltonian into quasi-boson form?

V Chernyak, Wei Min Zhang, S Mukamel, J Chem Phys Vol. 109, 9587 (can be freely downloaded here http://mukamel.ps.uci.edu/publications/pdfs/347.pdf ) Eq.(2.2), Eq. (B1) Eq.(B4)-(B6). When I substitue ...
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70 views

Smoothness of the energy levels of a generic Hamiltonian

Let us take an Hamiltonian $H(\xi)$ which depends on a set of parameters $\xi$, and assume that the matrix elements $h_{ij}(\xi)$ of the Hamiltonian are smooth complex functions of the parameters ...
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How to express a convex function of a Hermitian operator in terms of its eigenvalues and eigenvectors?

The Hermitian operator $\hat O$ can be expressed as $$\hat{O}=\sum_i O_i|O_i\rangle\langle O_i|.$$ How to prove that a convex function $f(\hat O)$ can be expressed like $$f (\hat O)=\sum_i ...
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441 views

The dual role of (anti-)Hermitian operators in quantum mechanics

Hermitian (or anti-Hermitian) operators are of central importance in quantum mechanics in at least two different incarnations: Observables are represented by Hermitian operators on the quantum ...
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154 views

Prove: $A$ and $B$ commute, therefore functions $f(A)$ and $g(B)$ will always commute with one another [closed]

How do I / can I actually prove the relationship $[a,b]=0 \Rightarrow [f(a),g(b)]=0$ for all functions $f,g$. I'm asking because the following sentence in the solution to my quantum mechanics ...
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333 views

How is Lippmann-Schwinger equation derived?

I'd like to know the derivation of Lippmann-Schwinger equation (LSE) in operator formalism and on what assumptions it is based. I consulted the Ballentine book as advised in this Phys.SE post, but I ...
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1answer
88 views

Minus sign in the time ordering operator

The time ordering operator is usually defined as $$\mathcal{T} \left\{A(\tau) B(\tau')\right\} := \begin{cases} A(\tau) B(\tau') & \text{if } \tau > \tau', \\ \pm B(\tau')A(\tau) & \text{if ...
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1answer
172 views

Angular Momentum Operators - Commutation Relations

I was going over past PGRE exam questions, and came across this one. The components for the angular momentum operator $\mathbf{L}=(L_x,L_y,L_z)$ satisfy the following commutation relations. ...
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102 views

The $n$-th root of the NOT gate

I simply can not find material containing facts about the $n$-th root of the NOT gate and it's realization in Q.M. and also in C.M.. Does anyone have material? A comparison of the $n$-th root NOT ...
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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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Mode operators in the Virasoro algebra

This questions concerns Exercise 2.11 in Polchinski. We are asked to compute the commutator $$L_{m}(L_{-m}|0;0\rangle) - L_{-m}(L_{m} |0;0\rangle)$$ By plugging the mode expansions, we use the ...
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Calculation of OPE in Polchinski

Consider Exercise 2.8 in Polchinski's String Theory book. We are asked to compute the weight of $$f_{\mu \nu}:\partial X^{\mu} \bar{\partial}X^{\nu}e^{ik\cdot X}:$$ I have carried out the usual ...
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Probability flux

I was reading a text on Quantum Mechanics in which it said that $$\int{d^3 x \, j(x,t)} = \frac{\langle p\rangle}{m},$$ where $\langle p\rangle$ is the expectation value of the momentum operator at ...
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What is the algebraic form of the momentum eigenstate?

I'm asking this in the context of trying to verify the equation $a^{\dagger}_{p} \vert 0 \rangle = \frac{1}{\sqrt{2\omega_p}} \vert p \rangle$. So far I have calculated $\vert 0 \rangle = ...
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156 views

Time dependence of the displacement operator

I am following the derivation of the master equation (and application of this) in these lecture notes. Unfortunately I do not follow the step of eliminating the driving terms of the harmonic ...
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146 views

Operator product expansion energy momentum tensor

We have the following equation from Polchinski (2.4.6) $$ T(z)X^{\mu}(0) \sim \frac{1}{z}\partial X^{\mu}(0) , \tag{2.4.6} $$ where $T(z)$ is defined as $T(z) = -\frac{1}{\alpha'} :\partial X^{\mu} ...
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Quantum Mechanics - Lowering Operator [closed]

Let $a$ be a lowering operator. Show that $a$ is a derivative respects to raising operator, $a^\dagger$, $$a = \frac{\textrm{d}}{\textrm{d}a^\dagger}$$ Can someone please explain how to prove the ...
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154 views

Time reversal symmetry and real symmetric Hamiltonian matrix

In the literature (like those in quantum chaos), it seems that time-reversal symmetry implies that the Hamiltonian of the system is a real symmetric one, instead of just being complex Hermitian. Is ...
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120 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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165 views

Operators is a infinite dimensional matrix, how can it multiply by a wave function that is a n*1 (n is finite) matrix

My confusion started from thinking the quantum superposition principle. Several website say that the quantum superposition means all state can be represented as infinity superposition of orthogonal ...
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1answer
152 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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“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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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 ...
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126 views

Quantum mechanic particle

In non relativistic quantum mechanic, we are dealing with a problem involving a particle in one dimensional space, and it has been given the potential and it reads: ...
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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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299 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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Where can I find a detailed derivation of the form of two body operators in the second quantization?

I've been looking around online for a couple hours now and I can't find a very informative derivation of the form for two body operators in the second quantization. Is there a resource online ...
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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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129 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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Really how can an observable quantity be equal to an operator?

A wave-function can be written as $$\Psi = Ae^{-i(Et - px)/\hbar}$$ where $E$ & $p$ are the energy & momentum of the particle. Now, differentiating $\Psi$ w.r.t. $x$ and $t$ respectively, ...