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 $\times$, $+$!
3,945
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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 ...
- 965
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2
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53
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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 $v$...
- 459
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1
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25
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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 ...
1
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1
answer
50
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$(a - a^t)e^{\frac{1}{2} a^ta^t} | 0 \rangle = 0$ [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.
...
3
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1
answer
58
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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 ...
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5
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2
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170
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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} \...
- 61
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2
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85
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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 ...
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0
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65
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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_\...
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1
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42
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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 ,...
- 3
3
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2
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107
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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 ...
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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 ...
3
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69
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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....
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95
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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
$$...
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1
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79
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Tong QFT Problem set 2, question 6: Normal ordering of angular momentum operator
I've been studying Tong's QFT notes and am trying to do problem sheet 2, question 6. here.
We are asked to take the classical angular momentum of the field,
$\begin{align}
Q_i &= \epsilon_{ijk}\...
- 21
2
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0
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26
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Is bosonic normal-ordering equal to fermionic normal ordering?
I'm trying to understand Jan von Delft's "Bosonization for Beginners — Refermionization for Experts", in which he uses boson normal ordering and fermion normal ordering interchangeably, and ...
- 1,688
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55
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Operator inequality between the Heisenberg Hamiltonian and the total spin
Consider a collection of $N$ spin-1/2 particles (qubits) with total spin
$$\vec{S} = \frac{1}{2}\sum_{n=1}^N \vec{\sigma}_n$$
and a Heisenberg Hamiltonian
$$H = -J \sum_{\langle n,m\rangle} \vec{\...
- 3,498
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votes
1
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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\...
- 2,929
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3
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942
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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 ...
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1
answer
52
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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 ...
- 965
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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 ...
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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, ...
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1
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0
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40
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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 ...
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2
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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}...
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1
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1
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41
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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
$$
\phi(\vec{x}) = \int \frac{d^3p}{(2\pi)^3} \...
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1
answer
48
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Product notation for operators
If I have a Hamiltonian
$$\mathcal{H} = \prod_j^N Z_j$$
where $j$'s are different sites on a lattice and $Z$'s are Pauli $Z$ operators does that mean that the Hamiltonian can also be written as
$$\...
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21
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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_{...
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1
answer
43
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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 ...
1
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1
answer
27
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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 ...
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1
answer
31
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Expression of Klein-Gordon field in Heisenberg picture
In Schrodinger picture, the scalar field is
$$
\phi(\vec{x}) = \int \frac{d^3 p}{2E(\vec{p})} \left( a(\vec{p}) e^{i\vec{p}\cdot\vec{x}} + a(\vec{p})^{\dagger} e^{-i\vec{p}\cdot\vec{x}} \right). \...
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2
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256
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Normal ordered exponential of one-body operators
Let $\{a_i\}_{i=1}^N$ be a set of annihilation operators (they are either all bosons, or all fermions) satisfying the canonical commutation or anti-commutation relation. In the book Quantum Theory of ...
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Decomposing a coherent state? [closed]
Can you decompose a coherent state $|\alpha\rangle$ into $p|p\rangle+q|q\rangle$, where $|p\rangle$ and $|q\rangle$ are eigenstates of the $P$ and $Q$ operators respectively with eigenvalues p and q?
...
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1
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2
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The composition property of the time-evolution operators
For time-dependent Hamiltonians in Quantum Mechanics and QFT, we define the time-evolution operator as a unitary operator $U (t_2, t_1)$ such that
$$ \tag{1}
|\psi (t_2) \rangle = U (t_2, t_1) |\psi (...
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0
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39
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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{...
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40
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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 ...
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2
votes
2
answers
63
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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 ...
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1
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39
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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 ...
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votes
2
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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
$$\langle p|x| \psi \rangle = i \hbar \frac{d}{dp}\langle p| \psi \rangle, \langle x | p|\psi\rangle = ...
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1
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62
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Weinberg Chapter 10: Sign convention for momentum operator
Weinberg says that translational invariance produces a conserved momentum, i.e., $P^\mu$, such that (Eq. 10.1.1):
\begin{align*}
[P_\mu, O(x)] = +i\hbar \frac{\partial}{\partial x^\mu} O(x). \tag{...
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0
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Why do we need a whole field of operators in QFT?
This is probably a very silly question, so I hope you'll excuse my ignorance of QFT. As far as I can tell, there are really two mathematical objects in the Hamiltonian formulation:
The state vector, ...
- 1,632
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1
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Hamiltonian Open String
On page 38 of Becker Becker Schwarz, we're given equation 2.69 which is the Hamiltonian for a string given as $$H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2}).\tag{2.69}$$ Considering the open string ...
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1
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80
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Conservation of momentum in quantum mechanics
Let $P_x=P_{1x}+P_{2x}$ be the operator for the total linear momentum of an isolated system of two particles labeled 1 and 2. The questioned asked to show that $\langle P_x \rangle$ remains constant ...
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39
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Commutator property proof
I am working through Griffiths, and about a chapter or so ago, I came across the following commutator identity: $$[AB,C] = A[B,C] + [A,C]B$$ I tried to prove this rule by calculating the commutator ...
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0
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53
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Is this proof that $a|0\rangle=0$ wrong (ladder oeprators and number operator)?
I understand everything except when he says
"Then we get in particular $a|0\rangle=0$."
It seems the author substituted the formula with $n=0$, But this seems unsound, since we get $|n-1\...
- 23
5
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2
answers
435
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Proper notation and definition of the Hamilton operator
The Hamilton operator is often defined as
$$
\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 ...
- 1,776
3
votes
2
answers
59
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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
$$[x, \hat{H}] = [x, \frac{\hat{p}^2}{2m} + V(...
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2
votes
1
answer
42
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Worldsheet constraint Bosonic String
I am currently studying David Tong's notes on String theory and there’s a step taken in writing out the worldsheet constraint in lightcone coordinates $\sigma^{\pm}$ for the closed string that I’m not ...
1
vote
1
answer
55
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Why is there only one eigenket per eigenvalue of the number operator? [duplicate]
Let's "define" (I put quotes since it's not a definition, but just requiring a property) the operator $a$ such that:
$$[a,a^\dagger]=1$$
then
$$n=a^\dagger a$$
No other assumptions are made ...
- 23
0
votes
1
answer
54
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Why does the interaction hamiltonian not commute with itself at different times?
If you have a poincare invariant Hamiltonian $H$, then the Hamiltonian must commute with itself at different times and not explicitly depend on time. If the Hamiltonian $H$ can be written as $H$ = $H_{...
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votes
1
answer
38
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Relation of two commuting operators to other operators
Consider two commuting quantum operators:
$$\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} \...
- 67
7
votes
3
answers
1k
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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(...
- 144