The operators tag has no wiki summary.
3
votes
3answers
69 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
$$
[x,A] = [x, \sqrt{A}]\sqrt{A} + ...
2
votes
2answers
53 views
Are higher order mixed partial derivatives of wave function with different ordination equal?
For example, given two operators:
$$A = \frac{\partial}{\partial x}+\frac{\partial}{\partial y},$$
$$B =\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} + 1.$$
Deriving commutator ...
-3
votes
0answers
54 views
Raising/Lowering Operator [closed]
The question asks; For a harmonic oscillator,
$$
\langle2|a_{-}\,a_{+}^{3}|0\rangle
$$
which equals...
And there are numerous multiple choice answers. I calculated $$2\sqrt3$$ but the answer sheet ...
0
votes
1answer
54 views
Expectation value of position in infinite square well
I'm looking for some help to a question.
I'm working in the infinite square well, and I have the wavefunction:
$$\psi(x,t=0)=A\left( i\sqrt{2}\phi_{1}+\sqrt{3}\phi_{2} \right).$$
For every time t, ...
5
votes
2answers
123 views
Non-associative operators in Physics
Are non-associative operators (or other kind of elements) used in Physics?
For example, in QM I'm looking for something like this: $A(BC)|\psi\rangle \ne (AB)C|\psi\rangle$
NOTE: I think that this ...
2
votes
1answer
76 views
Replacing an operator with its expectation value
While dealing with a circling particle in an spherical symetric potential our professor said that we can replace an operator of $z$ component of angular momentum $\hat{L}_z$ with the expectation value ...
5
votes
1answer
139 views
Can one define an acceleration operator in quantum mechanics?
It seems most books about QM only talk about position and momentum operators. But isn't it also possible to define a acceleration operator?
I thought about doing it in the following way, starting ...
3
votes
0answers
48 views
Non-Hermiticity when Fourier transforming onto a finite lattice
I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where
$$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$
Where $i$ belongs to sublattice $A$, and $j$ to ...
1
vote
2answers
85 views
From position space to momentum space
Lets say I have a state vector $\left|\Psi(t)\right\rangle$ in a position space with an orthonormal position basis. If I now use an operator $\hat{p}$ on this basis I will get basis which corresponds ...
1
vote
1answer
55 views
Observables - what are they?
I often read in books that an observable is represented by an Hermitean operator. But it is deceiving as operator isn't the observable.
As far as I've read the observable is denoted like $\langle ...
7
votes
1answer
133 views
Discreteness of set of energy eigenvalues
Given some potential $V$, we have the eigenvalue problem
$$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$
with the boundary condition
$$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$
If we ...
0
votes
0answers
44 views
An application of Toeplitz operators
I want to find an application of the Toeplitz operators. All I need is a known problem (not an open problem) which solution use the theory of Toeplitz operators. I don't need all the details but I ...
0
votes
1answer
43 views
Eigenvalue $a_n$
Q1:
In Zetilli's book page 166 (ch. "Postulates of QM", eq. 3.1) i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another ...
3
votes
2answers
144 views
Why is this not a realisable operation on a quantum system?
Let $\rho = \begin{bmatrix}\ 1&0 \\ 0&0 \end{bmatrix}$, $\rho' = \begin{bmatrix}\ 0&0 \\ 0&1 \end{bmatrix}$, $\rho'' = \dfrac{1}{2}\begin{bmatrix}\ 1&1 \\ 1&1 \end{bmatrix}$ ...
5
votes
2answers
120 views
Quantum Mechanical Operators in the argument of an exponential
In Quantum Optics and Quantum Mechanics, the time evolution operator
$$U(t,t_i) = \exp\left[\frac{-i}{\hbar}H(t-t_i)\right]$$
is used quite a lot.
Suppose $t_i =0$ for simplicity, and say the ...
0
votes
0answers
29 views
Schrodinger equation in momentum space [duplicate]
I have a problem this is:
When I solve the Schrodinger equation in momentum space, I had done as this:
$\begin{array}{l}
i\hbar \frac{{\partial \Psi }}{{\partial t}} = - \frac{\hbar ...
0
votes
0answers
91 views
Prove that the position operator is $\hat{x} = i\hbar \frac{d}{{dp}}$ in the momentum representation [closed]
Proof that: $x = i\hbar \frac{d}{{dp}}$
I did this, could you tell me if I am false or true
$\begin{array}{l}
x{e^{\frac{{ipx}}{\hbar }}} = - i\hbar \frac{{d{e^{\frac{{ipx}}{\hbar }}}}}{{dp}} = ...
-1
votes
2answers
70 views
Proof $\left[ {\hat H,{{\hat p}_i}} \right] = - \frac{\hbar }{i}\frac{{\partial \hat H}}{{\partial {{\hat q}_i}}}$ [closed]
I have a problem with the Hamiltonian, I don't think anything to solve it!!
So could you give me some hints!
Knowing that:
$$\left[ {{{\hat p}_i},{{\hat q}_k}} \right] = \frac{\hbar }{i}{\delta ...
1
vote
1answer
100 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
124 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
49 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
58 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
56 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
220 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
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} &= ...
2
votes
1answer
56 views
The matrix element of a normal-ordered operator
Eq (1.137) in Negele and Orland gives the following identity for a normal-ordered operator $A(a_i^\dagger,a_i)$:
$$\langle \phi|A(a_i^\dagger,a_i)|\phi'\rangle=A(\phi_i^*,\phi'_i)e^{\sum ...
4
votes
2answers
89 views
Translator Operator
In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for ...
2
votes
1answer
108 views
Phys.org Spectral geometry to unite relativity and quantum mechanics, restate in laymens terms?
Lingua Franca links relativity and quantum theories with spectral geometry
Could someone give me a short synopsis of this article in laymens terms? What implications does this have in the physics ...
3
votes
1answer
119 views
How to define the mirror symmetry operator for Kane-Mele model?
Let us take the famous Kane-Mele(KM) model(http://prl.aps.org/abstract/PRL/v95/i22/e226801 and http://prl.aps.org/abstract/PRL/v95/i14/e146802) as our starting point.
Due to the time-reversal(TR), ...
0
votes
1answer
103 views
Matrix representation for fermionic annihilation operator
My guess it should look something like this:
$ c_\sigma = ...
0
votes
0answers
40 views
Time ordering and Fermions
Having time ordering operator for fermions, should it reverse sign if it swaps operators with opposite spin variable? In other words should
$T[c_{t_1,\uparrow}c_{t_2,\downarrow}^\dagger]$
return ...
1
vote
0answers
42 views
QFT basics for Klein-Gordon fields
I am teaching myself QFT from Peskin for next years maths course and I have two questions:
What is a c-number? Is it a complex number, and if so why does it mean, ...
3
votes
2answers
133 views
Quantum commutator
I'm given this commutator:
$$\left[PXP,P\right]$$
Being $P\psi=-i\hbar\partial_x\psi$, and $X\psi=x\psi$
I've solved it in two ways, the first one is just aplying the commutator to some function ...
1
vote
1answer
112 views
Hermitian Adjoint of differential operator
I came across this equation (identity) (Eq. 4 in this paper):
$\int(-i d\psi/dx)^*\psi dx = \int \psi^*(-i d\psi/dx) dx + id(\psi^*\psi)/dx\mid_{-\infty}^{+\infty}$
I have trouble proving it. I ...
4
votes
3answers
124 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) ...
2
votes
1answer
72 views
Quantum mechanical analogue of conjugate momentum
In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ...
-1
votes
1answer
87 views
Operators in quantum mechanics
According to the Quantum Mechanics, can we write $\langle q|p\rangle = e^{ipq}$?
If so then how?
And if we transfer to integrate formulation then how it will look like?
2
votes
1answer
83 views
Physics Applications of Fredholm Theory:
I find Fredholm theory beautiful, especially the Liouville-Neumann series for solving Fredholm integral equations of the second kind. There seems to be a consensus that these equations are quite ...
1
vote
0answers
29 views
Quantum graph theory: complex spectra
In quantum graph theory, what are the properties of a given graph to own complex conjugated complex eigenvalues, either finite or infinite? Spectral graph theory is as far as I know a not completely ...
3
votes
1answer
67 views
Spectral properties of CFT
What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac ...
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 ...
10
votes
3answers
365 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] + ...
3
votes
5answers
240 views
Math of eigenvalue problem in quantum mechanics
I learned the eigenvalue problem in linear algebra before and I just find that the quantum mechanics happen to associate the Schrodinger equation with the eigenvalue problem. In linear algebra, we ...
1
vote
1answer
71 views
Notational techniques for dealing with creation operators on Fock space
This question is trying to see if anyone has some simple notation (or tricks) for dealing with operators acting on coherent states in a Fock space. I use bosons for concreteness; what I'm interested ...
2
votes
1answer
175 views
Show that for QM operator A: $\int_{-\infty}^{\infty}\psi A^{\dagger}A\psi dx = \int_{-\infty}^{\infty}(A\psi)^*(A\psi)dx $
I need to show for $$A = \frac{d}{dx} + \tanh x, \qquad A^{\dagger} = - \frac{d}{dx} + \tanh x,$$ that
$$\int_{-\infty}^{\infty}\psi^* A^{\dagger}A\psi dx = ...
2
votes
1answer
128 views
Coordinate representation of quantum ladder operator?
I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics.
I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
7
votes
1answer
140 views
String theory - OPE and primary operators
First, a disclaimer: I am new to Physics SE, and I am primarily a mathematician, not a physicist. I apologise in advance for the possibly poor quality of the question, any and thank you for your ...
5
votes
3answers
210 views
Why is this identity an if, rather than if and only if?
A recent question (Product of exponential of operators) asked who to proved that the exponentials of operators multiply in same manner as those of scalars if and only if the commutator of the ...
1
vote
0answers
79 views
Explicit evaluation of a radially ordered product
I am trying to understand the application of the operator product expansion to calculate the radially ordered product in the complex plain of $T_{zz}(z)\partial_w X^{\rho}(w)$ which should result in
...
1
vote
2answers
300 views
Deriving a QM expectation value for a square of momentum $\langle p^2 \rangle$
I alredy derived a QM expectation value for ordinary momentum which is:
$$
\langle p \rangle= \int\limits_{-\infty}^{\infty} \overline{\Psi} \left(- i\hbar\frac{d}{dx}\right) \Psi \, d x
$$
And i ...
