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 $\times$, $+$!
3,528
questions
0
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16 views
Does the abstract wavefunction change in this following example?
Suppose, we have a basis $|u\rangle$, described by the function $u=g(x)$. We can normalize this basis, using our standard $|x\rangle$ basis using the following :
$$\hat{I}=\int |x\rangle\langle x|dx=\...
3
votes
0answers
20 views
Form of the interaction representation time evolution operator
I am a bit confused about the interaction representation picture. Consider the time independent Hamiltonian $H = H_0 + V$. My question concerns the interaction representation time evolution operator:
$...
0
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0answers
34 views
Are the Orbital Angular Momentum Operators Linear?
I'm a bit confused whether or not I can distribute out the operators in the definition of the raising and lowering operators. For example, given:
$$L_{+}|l \space m\rangle = a |l\space m+1\rangle$$
$$...
3
votes
2answers
78 views
Translation operator and position operator
I am a little bit confused about the translation and position operator and hope for some clarification.
Let $\hat{x}$ be the position operator, which satisfies $\hat{x} \vert x \rangle = x \vert x \...
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0answers
30 views
Two-particle operators in QFT and the factor 1/2
I am learning about QFT through the book Quantum Field Theory for the Gifted Amateur and I am having trouble understanding the factor 1/2 in the definition of two particle field operators. In the book ...
-1
votes
0answers
34 views
Wick contraction for fermions of different types
When we have an interaction hamiltonian with fermion field of two different types ($\Psi_a$ $\overline{\Psi_b}$) and a scalar field $\Phi_c$. How does one do the wick contraction? eg. Leptoquark decay ...
2
votes
1answer
59 views
Inserting a position operator in the path integral in QFT
With the usual path integral description, we have the formula
$$\langle q''t''|q't'\rangle =\int\mathcal{D}q \exp{(iS)}$$ where $S=\int_{t'}^{t''}L(q,\dot{q})$ is the action evaluated for $t\in (t',t''...
1
vote
2answers
77 views
Prove that $i\hbar\dot{\hat{\Omega}}=[\hat{\Omega},\hat{H}]$ , where $\hat{\Omega}(\hat{x},\hat{p})$ is Taylor expandable
How do I show that $i\hbar\dot{\hat{\Omega}}=[\hat{\Omega},\hat{H}]$ , where $\hat{\Omega}(\hat{x},\hat{p})$ is Taylor expandable, $\hat{H}(\hat{x},\hat{p})$ is the Hamiltonian of the system and $\hat{...
5
votes
1answer
69 views
Is there a (semiclassical) electric field operator?
So I come from a chemistry background, where the electronic structure of atoms and molecules is central. For practical purposes, we usually work with a charge density operator
$$ \hat{\rho}(r) = q \...
-3
votes
0answers
46 views
I need help with operators [closed]
So, my main question is just simply, what are physics operators and what are the most common ones?
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1answer
26 views
Ehrenfest theorem: On which classical circle can we find the electrons in an homogenous magnetic field?
In the French wiki article about the Ehrenfest theorem I found these formulas.
$${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\langle {\hat {x}}\rangle ={\frac {1}{m}}\langle {\hat {p}}\rangle }...
1
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0answers
25 views
Translation matrix for multiple spins
I need to do a translation, but not a translation in the classical definition, I need to, in a system with a operator, for example, $$ \sigma^z_j=1\otimes \cdots \otimes 1\otimes \sigma^z\otimes 1 \...
0
votes
1answer
33 views
Ladder Operators and Angular momentum
The angular momentum operator $L_z$ can be expressed in terms of $L_z = xp_y - yp_x$ where $ p = \hat{p} = -i\sqrt{hwm/2}(a^{\dagger} - a)$ and $x = \hat{x}\sqrt{\hbar/2mw}(a^{\dagger} + a)$.
We want ...
2
votes
0answers
73 views
Expectation values in path integral formalism
In quantum field theory, it is often assumed that the expectation value $\langle A\rangle$ of an operator $A$ can be written in the path integral formalism in the following way:
$$
\langle A\rangle = \...
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0answers
15 views
Many-body Bose-Hubbard Interaction Energy
I am trying to understand the derivation in this link.
They start with a slightly modified Hamiltonian (with a gauge field) (I call $H_j$ the tunneling, since my question is regarding the interaction ...
-2
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0answers
33 views
How can you calculate a reduced matrix element $\langle 0|\varepsilon|0\rangle $? [closed]
How can you calculate a reduced matrix element $\langle 0|\varepsilon|0\rangle $ ?
$\varepsilon$ is a polarization vector.
I also want to know the lists on the webpage that says some reduced matrix ...
23
votes
6answers
2k views
What is the actual use of Hilbert spaces in quantum mechanics?
I'm slowly learning the quirks of quantum mechanics. One thing tripping me up is... while (I think) I grasp the concept, most texts and sources speak of how Hilbert spaces/linear algebra are so useful ...
4
votes
1answer
79 views
Is there a well-defined association between abstract linear operators in Fock space and normal ordered polynomials of fermionic operators?
Suppose I have a fermionic Fock space $H$ of dimension $2^n$. If I fix an operator $O$ acting on $H$ that commutes with the number operator $N$, I typically make an assumption internally that such an ...
0
votes
1answer
42 views
An identity about exponential of operators
Suppose we have a finite set $\mathcal{S}$ and operators (actually, matrices):
$$b_{x-\frac{1}{2},x} = a_{x} + a_{x}^{\dagger} \quad \mbox{and} \quad b_{x,x+\frac{1}{2}} = i(a_{x}-a^{\dagger}_{x})$$
...
0
votes
0answers
45 views
Expectation value of position operator for driven quantum harmonic oscillator
I'm having quantum harmonic oscillator which is in ground state at $t=0$. Now it is subjected to arbitrary force $F(t)$. How can I calculate $\langle \hat{x} \rangle$ and $\langle \hat{p} \rangle$?
I ...
6
votes
2answers
512 views
Rotation of spin operator itself
Consider for example the rotation of the $x$ component of spin by $\pi$ about the $z$ axis. This flips the spin giving $$e^{i\pi S_z}S_xe^{-i\pi S_z}=-S_x.$$
However, can this be proved directly using ...
0
votes
1answer
44 views
Is the momentum operator in position basis a first-order approximation?
In my course we have just derived the momentum operator in the position basis to be the first-order derivative of the position. This was achieved by using a first-order taylor expansion:
$$
\psi(\...
1
vote
1answer
55 views
Mathematical expression of the coordinate exchange operator
So I know that the total exchange operator is the product of the coordinate, spin and isospin exchange operators, as,
\begin{equation}
P_{12} = P_{12}^r P_{12}^\sigma P_{12}^\tau
\end{equation}
The ...
1
vote
1answer
32 views
Parity transformation on the adjoint spinor
In eq. (3.128) of page 66 of "An Introduction to QFT", by Peskin & Schroeder, a step involves:
$$P\,\overline{\psi}\left(t,\textbf{x}\right)P^{-1}=P\,\psi^{\dagger}\left(t,\textbf{x}\...
3
votes
1answer
66 views
When is the Liouvillian superoperator not diagonalisable?
An $n \times n$ matrix, $L$, is diagonalisable if it has $n$ linearly independent eigenvectors.
I've recently been working with open quantum systems and come across the non-hermitian, Liouvillian ...
0
votes
1answer
64 views
Definition of invariance of a QM operator under a transformation
In the Sakurai book "Modern quantum mechanics" (pg. 263) an operator $S$ is said to be invariant under a unitary transformation $T$ if:
$$T^\dagger S T = S.$$
Where that definition come from?...
0
votes
0answers
30 views
Quadratic Expansion With Operators [duplicate]
I was looking at the hamiltonian of a particle confined to the $x$-$y$ plane when it has mass $m$ and charge $q$ coupled to the electromagnetic field. My question is actually a very simple one. During ...
1
vote
0answers
70 views
Lorentz transformation of operator
I am reading the Zee book on group and at pg.442 when explaining the Poincare' algebra an the infinity dimensional representation, it states that since $P_{\mu}P^{\mu}$ is a Lorentz invariant then it ...
2
votes
0answers
41 views
Find operator that produce a given amplitude
While I was studying spinor helicity formalism, in many articles, it gives the operators that produce a given amplitude. Is there a generic method of finding operators that produce a given amplitude?
...
0
votes
1answer
56 views
Can the Hermitian operator be related to state space to describe physical phenomena?
Can space-time, in which phenomena occur, and the space of states in which phenomena are described by means of the Hermitian operator be related? I suspect it is because the hermetic operator is built ...
0
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0answers
35 views
Sign of the momentum operator in QM [duplicate]
What is the physical reason or concept underlying the minus sign in the definition of the momentum operator in QM:
$$\hat{p}_i = -i \hbar\frac{\partial}{\partial x_i}~? $$
-1
votes
2answers
98 views
Why $⟨a│H'│a⟩=0$, when $H'$ has zero diagonal components?
On 'David J.Griffiths, Introduction to Quantum Mechanics' Chapter 11.1.1 equations (11.16)
When $H'$ has zero diagonal component, $⟨a│H'│a⟩=0$
However, I think that there is needed conditon '...
1
vote
0answers
31 views
Proof of the Glauber relation $e^{A}e^{B} = e^{A+B}e^{(1/2)[A,B]}$ [duplicate]
If $A$ and $B$ are two operators that commute with their commutators,
\begin{align}
e^{A}e^{B} &= e^{A+B}e^{(1/2)[A,B]}
\end{align}
That well-known statement is known as Glauber's Formula.
I am ...
1
vote
1answer
46 views
Heisenberg Equation for Raising and Lowering Operators of the Harmonic Oscillator
For a Schrödinger operator $O$ that is independent of time, the Heisenberg equation is
$$\dot O_H(t) = i[H, O_H(t)], \tag{1}$$
with solution given by
$$O_H(t) = e^{iHt}O_S e^{-iHt}. \tag{2}$$
I want ...
0
votes
0answers
76 views
Commutation between hermitian operator $H$ and the operator $U(m,n) = |\varphi_m\rangle \langle \varphi_n|$
I was trying to do this exercise from Cohen-Tannoudji Quantum Mechanics book:
$|\varphi_n\rangle$ are the eigenstates of a Hermitian operator $H$ ( $H$ is, for example, the Hamiltonian of some ...
2
votes
0answers
61 views
What does cohomology of $Q_B$ mean in BRST quantization in Polchinski?
While proving no-ghost theorem ($4.4$ Polchinski) the term cohomology of $Q_B$ is used quite a lot of time. From what I understand this has to be a set since "cohomology of $Q_B$" is ...
1
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0answers
76 views
Do fermionic creation/annihilation anticommutation relations fix the creation and annihilation operators?
If you define operators $a, a^\dagger$ which satisfy (e.g.) the relations $\{a,a\}=\{a^\dagger,a^\dagger\}=0$ and $\{a,a^\dagger\}=1$.
Will this uniquely define the operators such that $a |0\rangle \...
0
votes
1answer
53 views
Intuition for Spin operator in arbitrary direction
I understand why the Spin operators in $x$, $y$ and $z$ direction are given by : $\begin{align*}
S_x = \begin{pmatrix}
0 &\hbar/2\\
\hbar/2 & 0
\end{pmatrix}
S_y = \begin{pmatrix}
0 & -i\...
3
votes
1answer
58 views
Derivation of $T(z)(TT)(w)$ in CFT
I am trying to derive eq. (6.213) in Di Francesco's CFT book,
$$T(z)(TT)(w) \sim \frac{3c}{(z-w)^6}+\frac{(8+c)T(w)}{(z-w)^4}+\frac{3 \partial T(w)}{(z-w)^3}+\frac{4 (TT)(w)}{(z-w)^2}+\frac{\partial(...
1
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1answer
64 views
Confusion about interpretation of expectation values in quantum mechanics
Given a state $|\psi \rangle$ one can form the expectation value of an observable $O$ as:
$$
\langle \psi|O|\psi \rangle.
$$
For the case $O = H$, where $H$ is the Hamiltonian of the quantum system, ...
0
votes
0answers
26 views
Compatible observables for a quantum harmonic oscillator 3D
In quantum mechanics, from what I understand from the theory, if any operators commute then they have a complete set of simultaneous eigenvectors. This means that any vector of space can be expressed ...
6
votes
1answer
158 views
Confusion about in and out states, interacting Hilbert space etc, referring to Weinberg QFT
There are many posts related to this issue on this site, but I have found none that answer my specific questions about this matter.
I review my understanding of Weinbergs approach. There are probably ...
0
votes
0answers
31 views
First quantization Entropy of coherent state
I'm trying to follow the computations of example 5.1 in this paper. To begin with they have a symplectic Hilber space $(\mathcal{K},\tau,\sigma)$, where $(\mathcal{K},\tau)$ is a separable Hilbert ...
1
vote
2answers
83 views
Kinetic Energy of wavefunction
Suppose we have been provided the form of some wavefunction on a graph, but not the exact mathematical expression of the wavefunction $\langle x|\psi\rangle $. Now I'm asked to find the average ...
1
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0answers
17 views
How does commutator of $Q_B$ with change in $H$ results in moving around in gauge space
In ch-$4$ Polchinksi states following:
In order to move around in space of gauge choices, the BRST charge must remain conserved. Thus it must commute with change in the Hamiltonian.
Commutation with ...
2
votes
1answer
38 views
Counting sign definiteness in BRST cohomology of string
In Polchinksi ch-$4$ following manipulation is done:
$$|\psi_1\rangle=(e\cdot\alpha_{-1}+\beta b_{-1}+\gamma c_{-1})|0,\textbf{k}\rangle .\tag{4.3.25}$$
$$\langle\psi_1|\psi_1\rangle=\Big(e^*\cdot e+(\...
2
votes
1answer
68 views
What's the (intuitive) difference between a primary and a quasi-primary operator in a CFT?
I've come across many different definitions of primaries and quasi-primaries. Some references define them based on their transformation law
\begin{align}\hat\phi'(x')=\left|\frac{\partial x'}{\partial ...
1
vote
1answer
82 views
Is this identity true for operators?
We have the classical identiy:
$$ \boldsymbol{L}\cdot\left(\boldsymbol{p}\times\boldsymbol{L}\right) = 0$$
But I was wondering if that is also right in QM, considering $L$ and $P$ do not generally ...
0
votes
4answers
123 views
Shared eigenstates of angular momentum
The angular momentum operators $L_x$, $L_y$ and $L_z$ don't commute with each other but they do all commute with the operator $L^2 = L_x^2+L_y^2+L_z^2$. I know that if two matrices $A$ and $B$ commute ...
1
vote
1answer
74 views
Commutators in Poincare algebra
Consider the method of induced representations for the Poincare algebra, i.e. given a field $\phi$ (which need not be a scalar field despite its notation), we have the commutator $$[J^{\mu\nu},\phi(0)]...
