Questions tagged [operators]

In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as observables and as time evolution operators in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!

Filter by
Sorted by
Tagged with
0 votes
0 answers
17 views

Can I define $\hat{x}$ as $x\delta(x'-x)$?

Operators can be thought of as matrices. Since matrices have two indices and involve summing: $A^i_j \psi^j\equiv A^i_0\psi^0 + A^i_1\psi^1 + A^i_2\psi^2 \dots$ and a summation between 2 vectors turns ...
user avatar
  • 965
1 vote
2 answers
53 views

Thinking of a linear operator as a (1,1) tensor

I am reading that a linear operator $A$ can be thought of as a (1,1) tensor [where $(r,s)$ corresponds to $r$ vectors and $s$ dual vectors]. This can be done by saying $$A(v,f) \equiv f(Av)$$ where $v$...
user avatar
  • 459
0 votes
1 answer
25 views

Relation between diagonal and off-diagonal entries of Hermitian Operator

I am started doing a project in Quantum Chemistry and stumbled upon a problem which I can not seem to find the answer to. As the title suggests, I am looking for a relation between the diagonal and ...
user avatar
1 vote
1 answer
50 views

$(a - a^t)e^{\frac{1}{2} a^ta^t} | 0 \rangle = 0$ [closed]

When talking about the limits of squeezed states, we reach the conclusion that $e^{\frac{1}{2} a^ta^t} | 0 \rangle$ must be an eigenstate for momentum since the uncertainty in momentum becomes zero. ...
user avatar
3 votes
1 answer
58 views

Rigorously building a Fock space, creation/annihilation operators and inner products in a QFT

I would like to understand how one can install a set of states, starting with a vacuum, define creation/annihilation operators for the vacuum, solve for mode functions and define inner products in a ...
user avatar
  • 572
5 votes
2 answers
170 views

Condition for an operator on a quantum Hilbert space to behave like vector

In QM we work with $H=L_2(\mathbb{R}^3)$ as a Hilbert space of square-integrable complex-valued functions. Now we define a special set of three operators $L_x, L_y, L_z$ by $L_i = \varepsilon_{ijk} \...
user avatar
  • 61
0 votes
2 answers
85 views

Hermitian operators and their physical meaning

I read around about Hermitian operators recently. Even though I understood their "essential" or "mathematical" role, I didn't quite get what they physically mean, if they mean ...
user avatar
1 vote
0 answers
65 views

Primary fields in di Francesco's CFT

In the CFT book by Di Francesco et al. they use conventions such that part of the conformal algebra (see eq. 4.19) is $$ [D,P_\mu]=iP_\mu, \\ [D,K_\mu]=-iK_\mu, \tag{1} $$ where $P_\mu$, $D$ and $K_\...
user avatar
0 votes
1 answer
42 views

Prove that every component of angular momentum commutes with $f$

Let $f( \hat {\vec r} , \hat {\vec p} )$ be the any polynomial in variables $r^2, p^2 $ and $(\vec r \cdot \vec p)$. prove that every component of angular momentum commutes with $\hat f$: $[\hat L_k ,...
user avatar
  • 3
3 votes
2 answers
107 views

What does $\dot x$ mean as an operator in quantum mechanics?

I've been looking at a paper titled "Feynman's proof of the Maxwell Equations" by Freeman Dyson (American Journal of Physics 58, 209 (1990); https://doi.org/10.1119/1.16188) and I'm confused ...
user avatar
-1 votes
0 answers
45 views

How do you add two spins? [closed]

I have read texts about it, but I still haven't understood what the steps are. Every author seems to speak about it in a different way or order. What are the steps of adding two spins? Where I can ...
user avatar
3 votes
0 answers
69 views

Generalized momentum in terms of wavefunction: Is it always $-i\hbar \partial/\partial q$?

I saw this kind of derivation several times in different notes/review/educational articles. (For example https://arxiv.org/abs/1904.06560 or http://wcchew.ece.illinois.edu/chew/course/QMALL20121005....
user avatar
0 votes
1 answer
95 views

How can an operator be proportional to a scalar?

I am an undergraduate physics student reading through some parts of Griffiths's Quantum. I recently saw that $k$ is proportional to momentum $p$ via the De Broglie relation. But, to my understanding $$...
user avatar
0 votes
1 answer
79 views

Tong QFT Problem set 2, question 6: Normal ordering of angular momentum operator

I've been studying Tong's QFT notes and am trying to do problem sheet 2, question 6. here. We are asked to take the classical angular momentum of the field, $\begin{align} Q_i &= \epsilon_{ijk}\...
user avatar
  • 21
2 votes
0 answers
26 views

Is bosonic normal-ordering equal to fermionic normal ordering?

I'm trying to understand Jan von Delft's "Bosonization for Beginners — Refermionization for Experts", in which he uses boson normal ordering and fermion normal ordering interchangeably, and ...
user avatar
  • 1,688
4 votes
0 answers
55 views

Operator inequality between the Heisenberg Hamiltonian and the total spin

Consider a collection of $N$ spin-1/2 particles (qubits) with total spin $$\vec{S} = \frac{1}{2}\sum_{n=1}^N \vec{\sigma}_n$$ and a Heisenberg Hamiltonian $$H = -J \sum_{\langle n,m\rangle} \vec{\...
user avatar
  • 3,498
3 votes
1 answer
194 views

I'm getting imaginary eigenvalues of $X^2-P^2$

This is hermitian but I'm getting imaginary eigenvalues. Start with the commutator: $$[X^2-P^2,X+P]$$ $$=[X^2,P]-[P^2,X]$$ $$=X[X,P]+[X,P]X-(P[P,X]+[P,X]P)$$ $$=2i(X+P)$$ Now consider the ket $|E\...
user avatar
  • 2,929
6 votes
3 answers
942 views

Confusion regarding Heisenberg Uncertainty Principle

I've been studying some Quantum Mechanics recently. I am a mathematics student but I've always been interested in physics so I am currently learning this material from a more mathematical treatment of ...
user avatar
-1 votes
1 answer
52 views

What is the angular momentum operator? [closed]

On one hand, since the angular momentum is: $$L^{ij} = r^i p^j - r^j p^i$$ so it makes sense for the angular momentum operator to be: $$\hat{L}^{ij}= -i\hbar (r^i \partial^j - r^j \partial^i)$$ On the ...
user avatar
  • 965
0 votes
0 answers
22 views

State Vector and Density matrix [duplicate]

I have a slight confusion with the two. From what I have understood, state vector describes pure states, which means that with a probability $1$, our state is going to be in that state. However, if ...
user avatar
2 votes
0 answers
137 views

Is $\mathcal{L}^{*}_{L}(\rho) = 0$ $\forall$ $\rho$ $\in\{\rho: \langle\mathcal{L}_{L}(V)\rangle_{\rho}= 0\}$ true?

[Context: I have observed the fact, $\mathcal{L}^{*}_{L}(\rho) = 0$ $\forall$ $\rho$ $\in\{\rho: \langle\mathcal{L}_{L}(V)\rangle_{\rho}= 0\}$ (meaning of the symbols are below) true numerically, ...
user avatar
1 vote
0 answers
40 views

Commutation relation between squares of angular momentum [duplicate]

We usually come across the formula $$\vec{L}.\vec{S}=\frac{1}{2}\left[\vec{J}^2-\vec{L}^2-\vec{S}^2\right].$$ Do $\vec{J}^2$, $\vec{L}^2$ and $\vec{S}^2$ commute always, or do they commute under ...
user avatar
-2 votes
2 answers
81 views

How to calculate the commutator between Hamiltonian and momentum operator squared?

I want to calculate the commutator $[H,p^2]$, where H is the Hamilton operator in one dimension and $p$ the momentum operator in one dimension. I tried it the same way as for $[H,p]=i\hbar(\frac{d}{dx}...
user avatar
  • 157
1 vote
1 answer
41 views

Equal-time Canonical Commutation Relation for a scalar field

In chapter 2 of Quantum Field Theory and the Standard Model, Schwartz derives the equal-time commutation relations of the second-quantised field. Using $$ \phi(\vec{x}) = \int \frac{d^3p}{(2\pi)^3} \...
user avatar
1 vote
1 answer
48 views

Product notation for operators

If I have a Hamiltonian $$\mathcal{H} = \prod_j^N Z_j$$ where $j$'s are different sites on a lattice and $Z$'s are Pauli $Z$ operators does that mean that the Hamiltonian can also be written as $$\...
user avatar
  • 459
0 votes
0 answers
21 views

Normal ordering of an exponential [duplicate]

I would like to recalculate Eq.(2.4) in PRA, 31,4,(1985), which expresses the exponential of operators as a normal ordering form. This equation reads \begin{equation} D=e^{\alpha K_{+} - \alpha^{*} K_{...
user avatar
  • 354
-2 votes
1 answer
43 views

Expectation value of $\langle H^n \rangle_{t=0}$ [closed]

I am trying to find the expectation value of $\langle H^n \rangle_{t=0}$ I am given some normalized state $| \psi, 0 \rangle$ (at time equals zero). Since the hamiltonian is a hermitian operator, the ...
user avatar
1 vote
1 answer
27 views

What's the contraction for non-adjacent fields?

In section 8.2 of Coleman's QFT lectures, he introduces the definition of contraction of two fields, where $T$ denotes time ordering and the colons normal ordering. Then he proceeds to contraction in ...
user avatar
  • 144
1 vote
1 answer
31 views

Expression of Klein-Gordon field in Heisenberg picture

In Schrodinger picture, the scalar field is $$ \phi(\vec{x}) = \int \frac{d^3 p}{2E(\vec{p})} \left( a(\vec{p}) e^{i\vec{p}\cdot\vec{x}} + a(\vec{p})^{\dagger} e^{-i\vec{p}\cdot\vec{x}} \right). \...
user avatar
  • 43
6 votes
2 answers
256 views

Normal ordered exponential of one-body operators

Let $\{a_i\}_{i=1}^N$ be a set of annihilation operators (they are either all bosons, or all fermions) satisfying the canonical commutation or anti-commutation relation. In the book Quantum Theory of ...
user avatar
1 vote
1 answer
89 views

Decomposing a coherent state? [closed]

Can you decompose a coherent state $|\alpha\rangle$ into $p|p\rangle+q|q\rangle$, where $|p\rangle$ and $|q\rangle$ are eigenstates of the $P$ and $Q$ operators respectively with eigenvalues p and q? ...
user avatar
1 vote
2 answers
70 views

The composition property of the time-evolution operators

For time-dependent Hamiltonians in Quantum Mechanics and QFT, we define the time-evolution operator as a unitary operator $U (t_2, t_1)$ such that $$ \tag{1} |\psi (t_2) \rangle = U (t_2, t_1) |\psi (...
user avatar
0 votes
0 answers
39 views

How to show $SU(3)$ symmetry of the following hamiltonian?

I have a hamiltonian of the form: $H = \sum_i (\hat{U^+_i}\hat{U^-_{i+1}} + \hat{U^-_i}\hat{U^+_{i+1}}+\hat{V^+_i}\hat{V^-_{i+1}}+\hat{V^-_i}\hat{V^+_{i+1}}+\hat{T^+_i}\hat{T^-_{i+1}}+\hat{T^-_i}\hat{...
user avatar
  • 57
0 votes
1 answer
40 views

In what way are eigenfunctions of an observable operator complete? [duplicate]

I am a physics undergraduate reading through Griffiths's 2ed Quantum book. In section 3.4 (Generalized Statistical Interpretation), Griffiths states: The eigenfunctions of an observable operator are ...
user avatar
2 votes
2 answers
63 views

Srednicki's QFT: Why $\langle p|\phi(0)|0\rangle$ in the interacting theory is Lorentz invariant?

I am reading Srednicki's QFT and I have met a problem. In its section 5, (5.18) , after deducing the LSZ formula, in order to check whether his supposition "that the creation operators of free ...
user avatar
0 votes
1 answer
39 views

When does an operator in Quantum Mechanics have a discrete spectrum? [duplicate]

Say one has a classical Hamiltonian system with generalised coordinates $q$ and conjugate momenta $p$. After canonical quantization, promoting them to operators $\hat{q}, \hat{p}$, how can one ...
user avatar
2 votes
2 answers
122 views

Calculating $\langle p | [x,p] | \psi \rangle $ using Dirac notation

Calculating $\langle p | [x,p] | \psi \rangle $ using Dirac notation. I am aware of the relations $$\langle p|x| \psi \rangle = i \hbar \frac{d}{dp}\langle p| \psi \rangle, \langle x | p|\psi\rangle = ...
user avatar
0 votes
1 answer
62 views

Weinberg Chapter 10: Sign convention for momentum operator

Weinberg says that translational invariance produces a conserved momentum, i.e., $P^\mu$, such that (Eq. 10.1.1): \begin{align*} [P_\mu, O(x)] = +i\hbar \frac{\partial}{\partial x^\mu} O(x). \tag{...
user avatar
2 votes
0 answers
68 views

Why do we need a whole field of operators in QFT?

This is probably a very silly question, so I hope you'll excuse my ignorance of QFT. As far as I can tell, there are really two mathematical objects in the Hamiltonian formulation: The state vector, ...
user avatar
  • 1,632
1 vote
1 answer
64 views

Hamiltonian Open String

On page 38 of Becker Becker Schwarz, we're given equation 2.69 which is the Hamiltonian for a string given as $$H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2}).\tag{2.69}$$ Considering the open string ...
user avatar
0 votes
1 answer
80 views

Conservation of momentum in quantum mechanics

Let $P_x=P_{1x}+P_{2x}$ be the operator for the total linear momentum of an isolated system of two particles labeled 1 and 2. The questioned asked to show that $\langle P_x \rangle$ remains constant ...
user avatar
0 votes
1 answer
39 views

Commutator property proof

I am working through Griffiths, and about a chapter or so ago, I came across the following commutator identity: $$[AB,C] = A[B,C] + [A,C]B$$ I tried to prove this rule by calculating the commutator ...
user avatar
0 votes
0 answers
53 views

Is this proof that $a|0\rangle=0$ wrong (ladder oeprators and number operator)?

I understand everything except when he says "Then we get in particular $a|0\rangle=0$." It seems the author substituted the formula with $n=0$, But this seems unsound, since we get $|n-1\...
user avatar
5 votes
2 answers
435 views

Proper notation and definition of the Hamilton operator

The Hamilton operator is often defined as $$ \hat H = \frac{-\hbar^2 }{2m}\frac{d^2}{dx^2} + V(x) $$ but shouldn't it rather be $$\begin{aligned} \hat H &= \int\int dxdx' |x\rangle \langle x|\hat ...
user avatar
  • 1,776
3 votes
2 answers
59 views

Why can't $iℏ ∂/∂t$ be used when calculating commutators of $H$? [duplicate]

$[x, \hat{H}]$ or $[\hat{p}, \hat{H}]$ can be computed by substituting $\frac{\hat{p}^2}{2m} + V(x)$ for $\hat{H}$ and doing some simple calculations. e.g $$[x, \hat{H}] = [x, \frac{\hat{p}^2}{2m} + V(...
user avatar
  • 46
2 votes
1 answer
42 views

Worldsheet constraint Bosonic String

I am currently studying David Tong's notes on String theory and there’s a step taken in writing out the worldsheet constraint in lightcone coordinates $\sigma^{\pm}$ for the closed string that I’m not ...
user avatar
1 vote
1 answer
55 views

Why is there only one eigenket per eigenvalue of the number operator? [duplicate]

Let's "define" (I put quotes since it's not a definition, but just requiring a property) the operator $a$ such that: $$[a,a^\dagger]=1$$ then $$n=a^\dagger a$$ No other assumptions are made ...
user avatar
0 votes
1 answer
54 views

Why does the interaction hamiltonian not commute with itself at different times?

If you have a poincare invariant Hamiltonian $H$, then the Hamiltonian must commute with itself at different times and not explicitly depend on time. If the Hamiltonian $H$ can be written as $H$ = $H_{...
user avatar
  • 39
-2 votes
1 answer
38 views

Relation of two commuting operators to other operators

Consider two commuting quantum operators: $$\hat{A} \hat{B} = \hat{B} \hat{A} \quad $$ For any operator $\hat{C} $, how can we prove that: $$\hat{A} \hat{C} \hat{B} \hat{C} = \hat{B} \hat{C} \hat{A} \...
user avatar
  • 67
7 votes
3 answers
1k views

How to avoid paradoxes about time-ordering operation?

(Original title: is time-odering operator a linear operator?) I'm confused with two formulas, one of which is $$ \mathcal{T} \exp \left [-\frac{\mathrm{i}}{\hbar} \int_{t_0}^t \mathrm{d} t' \hat{H}_I(...
user avatar
  • 144

1
2 3 4 5
79