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 ...

learn more… | top users | synonyms

0
votes
1answer
80 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.
0
votes
2answers
54 views

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} ...
0
votes
1answer
24 views

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} = ...
0
votes
0answers
144 views

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 ...
0
votes
1answer
52 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: ...
0
votes
1answer
77 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 ...
0
votes
1answer
104 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 ...
0
votes
2answers
232 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 ...
0
votes
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.. " ...
2
votes
1answer
151 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 ...
3
votes
1answer
175 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}\ ...
1
vote
1answer
75 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 $$ a\psi_n = c_n \psi_{n-1}$$ $$ ...
0
votes
1answer
49 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 ...
1
vote
1answer
64 views

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 ...
0
votes
1answer
42 views

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) ...
1
vote
1answer
145 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.
3
votes
1answer
295 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 ...
5
votes
6answers
418 views

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 ...
2
votes
1answer
76 views

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: $$ ...
0
votes
2answers
96 views

$[A_1, H] =[A_2, H] = 0$ 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 ...
1
vote
3answers
224 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 ...
0
votes
1answer
92 views

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 ...
1
vote
0answers
40 views

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 ...
7
votes
1answer
80 views

Polchinski Exercise 2.2, can I show that a function is harmonic by applying $\partial\bar{\partial}$?

I'm working on the following exercise: Exercise 2.2: Work out explicitly the expression $$:X^{\mu_1}(z_1, \overline{z}_1) \dots X^{\mu_n}(z_n, \overline{z}_n): \qquad \qquad\qquad $$ $$ ...
2
votes
1answer
69 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 ...
0
votes
2answers
39 views

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 ...
3
votes
1answer
358 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 ...
2
votes
1answer
122 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 ...
3
votes
1answer
238 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 ...
2
votes
1answer
73 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 ...
1
vote
1answer
142 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. ...
0
votes
3answers
97 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 ...
-3
votes
1answer
95 views

Eigenstates of sum of creation and annihilation operators

Does the operator $a+a^\dagger$ have eigenstates? If yes, what are they?
4
votes
0answers
104 views

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 ...
0
votes
0answers
107 views

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 ...
1
vote
0answers
74 views

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 ...
0
votes
3answers
110 views

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 = ...
1
vote
1answer
104 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 ...
2
votes
1answer
120 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} ...
5
votes
1answer
107 views

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 ...
0
votes
1answer
102 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 ...
2
votes
1answer
91 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
votes
1answer
137 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 ...
2
votes
1answer
123 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 ...
1
vote
1answer
43 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 ...
0
votes
1answer
39 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 ...
1
vote
1answer
118 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: ...
7
votes
1answer
111 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 ...
1
vote
3answers
245 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 ...
3
votes
1answer
72 views

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 ...