The operators tag has no wiki summary.
2
votes
2answers
146 views
Heisenberg picture of QM as a result of Hamilton formalism
Let's have formula of full time-derivative of physical value in Poisson's formalism:
$$\tag{1}
\frac{df}{dt} = -[H, f]_{P. br.} + \frac{\partial f}{\partial t},
$$
where $[A, B]_{P. br.}$ is Poisson's ...
3
votes
1answer
43 views
What conserved quantities are in 1D free quantum particle
From Laundau & Lifshitz "Quantum Mechanics":
If there are two conserved physical quantities $f$ and $g$ whose operators do not commute, then the energy levels of the system are in general ...
3
votes
1answer
65 views
The square of Pauli-Lubanski operator
Let's have Pauli-Lubanski operator:
$$
\hat {W}^{\alpha} = \frac{1}{2}\varepsilon^{\alpha \beta \gamma \delta}\hat {J}_{\beta \gamma}\hat {P}_{\delta} = \frac{1}{2}\varepsilon^{\alpha \beta \gamma ...
1
vote
3answers
127 views
The Momentum Operator in QM
I've seen the 'derivation' as to why momentum is an operator, but I still don't buy it. Momentum has always been just a product $m{\bf v}$. Why should it now be an operator. Why can't we just multiply ...
5
votes
2answers
146 views
Bounded and Unbounded (Scattering) States in Quantum Mechanics
I understand that bounded states in quantum mechanics imply that the total energy of the state, $E$, is less than the potential $V_0$ at + or - spatial infinity. Similarly, the scattering state ...
1
vote
1answer
132 views
Unitary transformations in mixed discrete-continuous representations
I am having trouble with the unitary transformation of a certain Hamiltonian in the paper
Zhai, H. Spin-orbit coupled quantum gases. Int. J. Mod. Phys. B 26 no. 1, 1230001 (2012). arXiv:1110.6798 ...
3
votes
2answers
99 views
Quantum Mechanics Basics: product space
Consider a coupled harmonic oscillator with their position given by $x_1$ and $x_2$. Say the normal coordinates $x_{\pm}={1\over\sqrt{2}} (x_1\pm x_2)$, in which the harmonic oscillators decouple, ...
0
votes
0answers
37 views
Operator of full angular momentum and spin operator
In relativistic mechanics classical angular momentum of one particle with center of energy vector creates angular momentum tensor. It's algebra is the same as Poincare group algebra. So in operator's ...
1
vote
1answer
70 views
Compute the central charge of $bc$ conformal field theory
I have a s****d question, how to calculate the central charge of $bc$ conformal-field theory in Polchinski's string theory, Eq. (2.5.12)?
For a $bc$ CFT given by
$$S=\frac{1}{2\pi } \int d^2 z \,\,b ...
0
votes
0answers
31 views
What physical value is described by following operator?
Let's have the system of point-like non-interacting particles and it's own angular momentum
$$
\mathbf L_{1} = \mathbf L - [\mathbf R_{E} \times \mathbf P],
$$
where $\mathbf R_{E}$ - center of energy ...
2
votes
1answer
87 views
Axiomatic structure behind Dirac's formulation of QM?
According to the paper Quantum Mechanics Beyond Hilbert Space by J.P. Antoine, several mathematical structures have been devised to make mathematical sense of Dirac's formulation of quantum mechanics ...
1
vote
2answers
62 views
Identity of Operator Product Expansion (OPE)
I have one more s****d question in Polchinski's string theory book, Eqs. (2.3.14a)
$$ j^{\mu}(z) :e^{ik \cdot X(0,0)}:~ \sim~ \frac{k^{\mu}}{2 z} :e^{ik \cdot X(0,0)}:,$$
where $j^{\mu}_a ...
3
votes
3answers
110 views
Identify the coefficients of Operator Product Expansion (OPE)
Sorry I have a stupid question in Polchinski's string theory book vol 1, p46.
For a holomorphic function $T(z)$ with a general operator $\mathcal{A}$, there is a Laurent expansion
$$T(z) A(0,0) \sim ...
1
vote
1answer
44 views
Number operator and Dirac field (with anticommutation relations)
Before using anticommutation relatives the energy, momentum, charge and number operators of the Dirac field have following expressions:
$$
\hat {H} = \int \epsilon_{\mathbf p}\left( \hat ...
2
votes
1answer
96 views
Reason behind canonical quantization in QFT?
Reason behind canonical quantization in QFT?
In the scalar field theory we simply promote the scalar field, $\phi(x)$ to a set of operators: $\hat{\phi}(x)$. What is the reason behind this?
4
votes
2answers
127 views
What is the most general expression for the coordinate representation of momentum operator?
I have a question about deriving the coordinate representation of momentum operator from the commutation relation, $[x,p]= i$.
One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th ...
4
votes
1answer
77 views
Operator norm of creation and annihilation operators
Are the creation and the annihilation operators $a(f)$ and $a^{\dagger}(f)$ for the bosonic Fock space bounded? What is their norm? So far I did not have found any note about this in the linked ...
4
votes
1answer
142 views
quantum mechanics current operators
How to derive the charge current and the energy current operators in second quantized form in Quantum mechanics ?
Also if you could comment in a similar way on the entropy current operator, that will ...
0
votes
2answers
74 views
Operator on Function of Momentum (QM)
I have exactly 0 clue on how to start this problem, but I would be forever grateful for a hint in the right direction.
Given the operators $\hat x=x$ and $\hat p=-i\hbar \frac{d}{dx}$, prove the ...
2
votes
2answers
108 views
Unitary spacetime translation operator
Srednicki writes: We can make this a little fancier by defining the unitary spacetime translation operator
$$ T(a) \equiv \exp(-iP^\mu a_\mu/ \hbar) $$
Then we have
$$ T(a)^{-1} \phi(x) T(a) = ...
4
votes
1answer
48 views
Time-ordered Derivative and Equal-time Commutator
In Green, Schwarz & Witten Superstring theory, Vol. I, page 141, I don't understand how pulling the derivative inside the Time-ordered product can give an Equal-time Commutator:
$$\tag{3.2.44} ...
4
votes
1answer
109 views
Does the Renormalization of QFT Contradict Canonical Quantization?
Does the renormalization of QFT contradict canonical quantization?
In canonical quantization, you take the classical fields and canonical momenta and turn them into operators, and you require that ...
2
votes
3answers
116 views
Differences between symmetric, Hermitian, self-adjoint, and essentially self-adjoint operators
I am a physicist. I always heard physicists used the terminology "symmetric", "Hermitian", "self-adjoint", and "essentially self-adjoint" operators interchangeably.
Actually what is the difference ...
3
votes
0answers
87 views
Proof for the completeness of eigenfunctions of a self-adjoint operator
I always heard the eigenfunctions of a self-adjoint operator form a complete basis. Where can I find a proof in infinite dimension space? Presumably readable for physicists.
4
votes
2answers
105 views
Schrödinger equation in position representation
$$
\DeclareMathOperator{\dif}{d \!}
\newcommand{\ramuno}{\mathrm{i}}
\newcommand{\exponent}{\mathrm{e}}
\newcommand{\ket}[1]{|{#1}\rangle}
\newcommand{\bra}[1]{\langle{#1}|}
...
5
votes
2answers
143 views
How to promote algebraic expressions to operators in quantum mechanics?
Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription
$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} ...
2
votes
1answer
89 views
Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?
These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
4
votes
3answers
92 views
Commutator with a square root
How to find the commutator $[a, \sqrt{a^\dagger a}]$? Here $a$ is a usual bosonic annihilation operator, and $[a, a^\dagger] = 1$.
The first thing I tried is
$$
[x,A] = [x, \sqrt{A}]\sqrt{A} + ...
2
votes
2answers
60 views
Are higher order mixed partial derivatives of wave function with different ordination equal?
For example, given two operators:
$$A = \frac{\partial}{\partial x}+\frac{\partial}{\partial y},$$
$$B =\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} + 1.$$
Deriving commutator ...
0
votes
1answer
89 views
Expectation value of position in infinite square well
I'm looking for some help to a question.
I'm working in the infinite square well, and I have the wavefunction:
$$\psi(x,t=0)=A\left( i\sqrt{2}\phi_{1}+\sqrt{3}\phi_{2} \right).$$
For every time t, ...
6
votes
2answers
188 views
Non-associative operators in Physics
Are non-associative operators (or other kind of elements) used in Physics?
For example, in QM I'm looking for something like this: $A(BC)|\psi\rangle \ne (AB)C|\psi\rangle$
NOTE: I think that this ...
2
votes
1answer
87 views
Replacing an operator with its expectation value
While dealing with a circling particle in an spherical symetric potential our professor said that we can replace an operator of $z$ component of angular momentum $\hat{L}_z$ with the expectation value ...
5
votes
1answer
163 views
Can one define an acceleration operator in quantum mechanics?
It seems most books about QM only talk about position and momentum operators. But isn't it also possible to define a acceleration operator?
I thought about doing it in the following way, starting ...
3
votes
0answers
56 views
Non-Hermiticity when Fourier transforming onto a finite lattice
I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where
$$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$
Where $i$ belongs to sublattice $A$, and $j$ to ...
1
vote
2answers
114 views
From position space to momentum space
Lets say I have a state vector $\left|\Psi(t)\right\rangle$ in a position space with an orthonormal position basis. If I now use an operator $\hat{p}$ on this basis I will get basis which corresponds ...
1
vote
1answer
58 views
Observables - what are they?
I often read in books that an observable is represented by an Hermitean operator. But it is deceiving as operator isn't the observable.
As far as I've read the observable is denoted like $\langle ...
12
votes
1answer
265 views
Discreteness of set of energy eigenvalues
Given some potential $V$, we have the eigenvalue problem
$$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$
with the boundary condition
$$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$
If we ...
0
votes
0answers
46 views
An application of Toeplitz operators
I want to find an application of the Toeplitz operators. All I need is a known problem (not an open problem) which solution use the theory of Toeplitz operators. I don't need all the details but I ...
0
votes
1answer
46 views
Eigenvalue $a_n$
Q1:
In Zetilli's book page 166 (ch. "Postulates of QM", eq. 3.1) i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another ...
3
votes
2answers
147 views
Why is this not a realisable operation on a quantum system?
Let $\rho = \begin{bmatrix}\ 1&0 \\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\ 1&1 \end{bmatrix}$ ...
5
votes
2answers
128 views
Quantum Mechanical Operators in the argument of an exponential
In Quantum Optics and Quantum Mechanics, the time evolution operator
$$U(t,t_i) = \exp\left[\frac{-i}{\hbar}H(t-t_i)\right]$$
is used quite a lot.
Suppose $t_i =0$ for simplicity, and say the ...
0
votes
0answers
29 views
Schrodinger equation in momentum space [duplicate]
I have a problem this is:
When I solve the Schrodinger equation in momentum space, I had done as this:
$\begin{array}{l}
i\hbar \frac{{\partial \Psi }}{{\partial t}} = - \frac{\hbar ...
-1
votes
2answers
75 views
Proof $\left[ {\hat H,{{\hat p}_i}} \right] = - \frac{\hbar }{i}\frac{{\partial \hat H}}{{\partial {{\hat q}_i}}}$ [closed]
I have a problem with the Hamiltonian, I don't think anything to solve it!!
So could you give me some hints!
Knowing that:
$$\left[ {{{\hat p}_i},{{\hat q}_k}} \right] = \frac{\hbar }{i}{\delta ...
1
vote
1answer
127 views
Some Dirac notation explanations
Equation for an expectation value $\langle x \rangle$ is known to me:
\begin{align}
\langle x \rangle = \int\limits_{-\infty}^{\infty} \overline{\psi}x\psi\, d x
\end{align}
By the definition we ...
2
votes
2answers
139 views
How do we know that $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$?
I am kind of new to this eigenvalue, eigenfunction and operator things, but I have come across this quote many times:
$\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue
$W$.
...
2
votes
1answer
58 views
Statistical sum of physical quantities in a quantum system
Let $C = A + B$ (statistical sum, so $\mathbb{E}[C] = \mathbb{E}[A] + \mathbb{E}[B]$), and let $p(A = a) = 1$. Are the following true?
$\mathbb{E}[C^2] = a^2 + 2a\mathbb{E}[B] + \mathbb{E}[B^2]$
...
1
vote
1answer
70 views
Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$
I just finished deriving the commutators:
\begin{align}
[\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\
[\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\
\end{align}
On the ...
3
votes
0answers
60 views
A particlar normal ordering problem
Say we have an expression of the form:
$$
\left<0\right|:\phi(x)^2: : \phi(y)^2:\left|0\right>,
$$
where $\phi$ is some scalar field. I have heard the claim several times, that in evaluating ...
3
votes
2answers
246 views
Coherent State, Unitary Operators, Harmonic Oscillator
Consider the operator:
$$O = e^{\theta(a^\dagger b - b^\dagger a)}$$
where $\theta$ is a constant.
$O$ is a unitary operator.
$a$, $a^\dagger$, $b$, and $b^\dagger$ are ladder operators for two ...
4
votes
2answers
152 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} &= ...
