The operators tag has no wiki summary.
30
votes
2answers
221 views
Physical interpretation of different selfadjoint extensions
Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. ...
12
votes
4answers
598 views
Energy is actually the momentum in the direction of time?
By comparatively examining the operators
a student concludes that `Energy is actually the momentum in the direction of time.' Is this student right? Could he be wrong?
12
votes
1answer
386 views
Intuitive meaning of Hilbert Space formalism
I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points:
The observables are given by self-adjoint operators on the ...
11
votes
1answer
85 views
Metric interpretation of self-adjoint extensions?
I am wondering if beyond physical interpretation, the one dimensional contact interactions (self-adjoint extensions of the the free Hamiltonian when defined everywhere except at the origin) have a ...
10
votes
3answers
363 views
How to tackle 'dot' product for spin matrices
I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as
$$
H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
10
votes
2answers
1k views
Meaning of the anti-commutator term in the uncertainty principle
What is the meaning, mathematical or physical, of the anti-commutator term?
$\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle [ A,B ] \rangle \vert^{2} ...
10
votes
2answers
903 views
Applications of the Spectral Theorem to Quantum Mechanics
I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
8
votes
2answers
503 views
Regularisation of infinite-dimensional determinants
Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM?
Edit: I failed to make myself clear. In finite ...
7
votes
1answer
75 views
Representation on Hilbert space of the product of two symmetry transformations
We know by Wigner's theorem that the representation of a symmetry transformation on the Hilbert space is either unitary and linear, or anti-unitary and anti-linear.
Let $T$ and $S$ be two symmetry ...
7
votes
1answer
140 views
String theory - OPE and primary operators
First, a disclaimer: I am new to Physics SE, and I am primarily a mathematician, not a physicist. I apologise in advance for the possibly poor quality of the question, any and thank you for your ...
7
votes
1answer
133 views
Discreteness of set of energy eigenvalues
Given some potential $V$, we have the eigenvalue problem
$$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$
with the boundary condition
$$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$
If we ...
6
votes
2answers
736 views
Difficulties with bra-ket notation
I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with ...
6
votes
2answers
351 views
What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?
I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following:
$|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
$|p\rangle$ is an eigenvector of ...
6
votes
1answer
179 views
Operator Ordering Ambiguities
I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity.
What does that mean?
I tried googling but to no avail.
6
votes
1answer
240 views
What is the value of a quantum field?
As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point ...
6
votes
1answer
190 views
“An operator is hermitian”. Implications?
Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as:
Every dynamical variable may be ...
6
votes
3answers
247 views
What is the physical meaning of weak expectation values?
In the two-state formalism of Yakir Aharanov, the weak expectation value of an operator $A$ is $\frac{\langle \chi | A | \psi \rangle}{\langle \chi | \psi \rangle}$. This can have bizarre properties. ...
5
votes
4answers
4k views
What is the Physical Meaning of Commutation of Two Operators?
I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable ...
5
votes
4answers
301 views
Is the momentum operator diagonal in position representation?
The matrix elements of the momentum operator in position representation are:
$$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}$$
Does this imply that $\langle x ...
5
votes
2answers
119 views
Quantum Mechanical Operators in the argument of an exponential
In Quantum Optics and Quantum Mechanics, the time evolution operator
$$U(t,t_i) = \exp\left[\frac{-i}{\hbar}H(t-t_i)\right]$$
is used quite a lot.
Suppose $t_i =0$ for simplicity, and say the ...
5
votes
1answer
139 views
Can one define an acceleration operator in quantum mechanics?
It seems most books about QM only talk about position and momentum operators. But isn't it also possible to define a acceleration operator?
I thought about doing it in the following way, starting ...
5
votes
3answers
210 views
Why is this identity an if, rather than if and only if?
A recent question (Product of exponential of operators) asked who to proved that the exponentials of operators multiply in same manner as those of scalars if and only if the commutator of the ...
5
votes
3answers
209 views
What does it mean to apply an operator to a state?
Let's say I have an operator $\hat{A}$ and a state $|\psi\rangle$. What exactly is the state $\hat{A}|\psi\rangle$? Is it just another different state that I am describing using my $\hat{A}$ and ...
5
votes
3answers
314 views
Some questions on observables in QM
1-In QM every observable is described mathematically by a linear Hermitian operator. Does that mean every Hermitian linear operator can represent an observable?
2-What are the criteria to say whether ...
5
votes
2answers
601 views
Basic Question - Green's Functions in Quantum Mechanics
I am trying to learn about Green's functions as part of my graduate studies and have a rather basic question about them:
In my maths textbooks and a lot of places online, the basic Greens function G ...
5
votes
1answer
346 views
Simultaneously commuting set
How does one determine the members of an simultaneously commuting set (of operators)? For example, I have read that for orbital angular momentum, the set is {$H,L^2,L_z$}. How does one know that these ...
5
votes
2answers
123 views
Non-associative operators in Physics
Are non-associative operators (or other kind of elements) used in Physics?
For example, in QM I'm looking for something like this: $A(BC)|\psi\rangle \ne (AB)C|\psi\rangle$
NOTE: I think that this ...
5
votes
2answers
198 views
Weyl Ordering Rule
While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
4
votes
4answers
396 views
Unitary Operator as a complex valued function
A book on Quantum Mechanics by Schwinger states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator."
Please give a hint on how to prove this assertion.
4
votes
4answers
583 views
How to calculate the quantum expectation of frequency of a particle?
I know how to calculate the expectation of < $\Psi$|A|$\Psi$ >
where the operator A is the eigenfunction of energy, momentum or position, but I'm not sure how to perform this for a pure frequency.
...
4
votes
4answers
143 views
How to prove that the symmetrisation Operator is hermitian?
Let $\mathcal{H}_N$ be the $N$ particle Hilbert space. So a quantum state $\left| \Psi \right>$ may be representated by
$$\left| \Psi \right> = \left| k_1 \right>^{(1)}\left| k_2 ...
4
votes
2answers
238 views
Eigenvalues of a quantum field?
Fields in classical mechanics are observables. For example, I can measure the value of the electric field at some (x,t).
In quantum field theory, the classical field is promoted to an operator-valued ...
4
votes
2answers
123 views
Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$
I know how to derive below equations found on wikipedia and have done it myselt too:
\begin{align}
\hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\
\hat{H} &= ...
4
votes
2answers
194 views
When you apply the spin operator, what exactly is does it tell you?
The example I'm trying to understand is:
$ \hat{S}_{x} \begin{pmatrix}
\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}
\end{pmatrix} = 1/2 \begin{pmatrix}
\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}
...
4
votes
3answers
1k views
Proof of Canonical Commutation Relation (CCR)
I am not sure how $QP-PQ =i\hbar$ where $P$ represent momentum and $Q$ represent position. $Q$ and $P$ are matrices. The question would be, how can $Q$ and $P$ be formulated as a matrix? Also, what is ...
4
votes
1answer
722 views
Evolution operator for time-dependent Hamiltonian
When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation
$$
...
4
votes
1answer
126 views
Eigenvalue of $L_z$
In section 4.3 of Griffths' "Introduction to Quantum Mechanics", just below Figure 4.6, the sentence begins
Let $\hbar \ell$ be the eigenvalue of $L_z$ at this top rung...
Why is this valid? ...
4
votes
2answers
89 views
Translator Operator
In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for ...
4
votes
1answer
236 views
How do we measure $i[\hat\phi(x),\hat\phi(y)]$ in QFT?
What operational procedure is required to measure $i[\hat\phi(x),\hat\phi(y)]$ in an interacting (or non-interacting) QFT? [assume smearing by test-functions, or give an answer in Fourier space, for ...
4
votes
1answer
929 views
Compatible Observables
My QM book says that when two observables are compatible, then the order in which we carry out measurements is irrelevant.
When you carry out a measurement corresponding to an operator $A$, the ...
4
votes
3answers
123 views
Associating a Unitary operator to proper Lorentz transformations?
If one reads eg page 32 of Srednicki where he says:
In quantum theory, symmetries are represented by unitary (or
antiunitary) operators. This means that we associate a unitary
operator U(Λ) ...
4
votes
1answer
204 views
How to evaluate spin operators in second quantization for spin symmetry-broken Slater determinants?
Suppose we have the following Slater determinant:
\begin{equation}
| \Psi \rangle = \prod \limits_{i,i'} a^+_{i\alpha} a^+_{i'\beta} | \rangle
\end{equation}
where $a^+_{i\alpha}$ creates an electron ...
4
votes
1answer
131 views
The difference between projection operators and field operators in QFT?
Is there a good reference for the distinction between projection operators in QFT, with an eigenvalue spectrum of $\{1,0\}$, representing yes/no measurements, the prototype of which is the Vacuum ...
4
votes
1answer
109 views
Constructing the space of quantum states
I want to learn how to construct spaces of quantum states of systems.
As an exercize, I tried to build the space of states and to find hamiltonian spectrum of the quantum system whose Hamiltonian is ...
4
votes
1answer
198 views
Existence of adjoint of an antilinear operator, time reversal
The time reversal operator $T$ is an antiunitary operator, and I saw $T^\dagger$ in many places (for example when some guy is doing a "time reversal" $THT^\dagger$), but I wonder if there is a ...
4
votes
1answer
203 views
Holstein-Primakoff and Dyson-Maleev representation
In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
4
votes
0answers
69 views
Shape of the state space under different tensor products
I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this).
Recall: In a ...
3
votes
6answers
429 views
Is H=H* sloppy notation or really just incorrect, for Hermitian operators?
I saw it in this pdf, where they state that
$P=P^\dagger$ and thus $P$ is hermitian.
I find this notation confusing, because an operator A is Hermitian if
$\langle \Psi | A \Psi \rangle=\langle A ...
3
votes
1answer
108 views
The issue on existence of inverse operations of $a$ and $a^{\dagger}$
I have asked a question at math.stackexchange that have a physical meaning.
My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
3
votes
3answers
517 views
Matrix elements of momentum operator in position representation
I have two related questions on the representation of the momentum operator in the position basis.
The action of the momentum operator on a wave function is to derive it:
$$\hat{p} ...
