The operators tag has no wiki summary.
3
votes
3answers
572 views
Index Manipulation and Angular Momentum Commutator Relations
I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed.
We have the relations
$$[L_i,p_j] ~=~ i\hbar\; \epsilon_{ijk}p_k,$$
...
3
votes
2answers
515 views
Commutators and Hermiticity - Exam question
I'm doing old exam questions, and here is one that on first glance seemed rather simple to me, but I just can't get it:
Given are two operators $A$ and $B$, and all we know about them is that
...
3
votes
2answers
184 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 ...
3
votes
2answers
1k views
QM: Expectation value of x*p <xp>, how interpret?
$xp$ is not a hermitian operator and hence doesn't represent an observable. Then, how can we interpret the expression
$$
\langle x p \rangle \text{,}
$$
the expectation value of position times ...
3
votes
1answer
46 views
Linearizing Quantum Operators [duplicate]
Possible Duplicate:
Linearizing Quantum Operators
I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator.
$$ ...
3
votes
1answer
707 views
Why/How is this Wick's theorem?
Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$.
$$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 $$
...
3
votes
2answers
388 views
Why is the Dirac operator so important - in both physics and mathematics?
Why is the Dirac operator considered so important - in both physics and (pure) mathematics?
3
votes
2answers
112 views
Quantum commutator
I'm given this commutator:
$$\left[PXP,P\right]$$
Being $P\psi=-i\hbar\partial_x\psi$, and $X\psi=x\psi$
I've solved it in two ways, the first one is just aplying the commutator to some function ...
3
votes
2answers
143 views
Sum of two density matrices: $\rho=p_1\rho_1+p_2\rho_2$
Suppose we have
$$\rho=p_1\rho_1+p_2\rho_2$$
Where $\rho_1$ and $\rho_2$ are density matrices with $p_1+p_2=1$
I'm trying to show this is also a density matrix
If we let
$$\rho_1=\sum_i^n ...
3
votes
1answer
240 views
The cleverest way to calculate $\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]$ with $\left[\hat{a},\hat{a}^{\dagger}\right]=1$
Who can provide me some elegant solution for
$$\left[\hat{a}^{M},\hat{a}^{\dagger N}\right]\qquad\text{with} \qquad\left[\hat{a},\hat{a}^{\dagger}\right]~=~1$$
other than brute force calculation?
...
3
votes
2answers
161 views
Derivative of a Position Eigenket
I was flicking through Zettili's book on quantum mechanics and came across a 'derivation' of the momentum operator in the position representation on page 126. The author derived that ...
3
votes
4answers
386 views
Is the momentum operator well-defined in the basis of standing waves?
Suppose I want to describe an arbitrary state of a quantum particle in a box of side $L$. The relevant eigenmodes are those of standing waves, namely
$$ \left<x|n\right>=\sqrt{\frac{2}{L}}\cdot ...
3
votes
1answer
157 views
Wick Order and Radial Ordering in CFT
I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case):
...
3
votes
1answer
61 views
Spectral properties of CFT
What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac ...
3
votes
2answers
154 views
Ordering Ambiguity in Quantum Hamiltonian
While dealing with General Sigma models (See e.g. Ref. 1)
$$\tag{10.67} S ~=~ \frac{1}{2}\int \! dt ~g_{ij}(X) \dot{X^i} \dot{X^j}, $$
where the Riemann metric can be expanded as,
$$\tag{10.68} ...
3
votes
1answer
141 views
commutation of operator product expansion
In CFT, when we have an OPE:
$$O_1(z)O_2(w)=\frac{O_2(w)}{(z-w)^2}+\frac{\partial O_2(w)}{(z-w)}+...$$
this holds inside a time-ordered correlation function, so $O_1(z)O_2(w)=O_2(w)O_1(z)$. Does it ...
3
votes
2answers
141 views
How to express continuous values as a matrix
Usually a quantity of a matrix is defined as the eigenvalues of the matrix. If so, how can anyone express continuous values, as in Schrodinger picture, into a matrix?
3
votes
1answer
299 views
Can we solve the particle in an infinite well in QM using creation and annihilation operators?
The particle in an infinite potential well in QM is usually solved by easily solving Schrodinger differential equation. On the other hand particle in the harmonic oscillator oscillator potential can ...
3
votes
1answer
171 views
Linearizing Quantum Operators
I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator.
$$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$
The right hand side ...
3
votes
0answers
46 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
0answers
41 views
Spectrum of a quantum relativistic “distance squared” operator
This question disusses the same concepts as that question (this time in quantum context). Consider a relativistic system in spacetime dimension $D$. Poincare symmetry yields the conserved charges $M$ ...
2
votes
2answers
368 views
Operator relation involving the logarithm of an operator?
Dirac gives the relation: $\exp(iaq)f(q,p) = f(q, p - a\hbar)\exp(iaq)$ where $\hbar$ is Planck's constant. Can anybody give me the corresponding relation when the $\exp$ function is a $\ln$?
2
votes
2answers
110 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
5answers
3k views
Mathematical background for Quantum Mechanics
What are some good sources to learn the mathematical background of Quantum Mechanics?
I am talking functional analysis, operator theory etc etc...
2
votes
1answer
105 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 ...
2
votes
1answer
90 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 ...
2
votes
1answer
111 views
Physical Significance of Operator Norm/Spectral Norm of a Quantum Operator
Is there any physical significance of operator norm/spectral norm of a hermitian operator?
2
votes
2answers
120 views
What is the nature of the correspondence between unitary operators and reversible change?
Why does the formalism of QM represent reversible changes (eg the time evolution operator, quantum gates, etc) with unitary operators?
To put it another way, can it be shown that unitary ...
2
votes
2answers
104 views
Matrix representing the quantity - why can some matrices not be physical quantity?
In Heisenberg picture, my textbook says that the following matrix
$A = \frac{5}{3}\Sigma_1 + i\frac{4}{3}\Sigma_2$ cannot represent physical quantity.
the book says this is because ...
2
votes
1answer
175 views
Expectation of a commutation relation
Is there any significance to: $\langle[H,\hat{O}]\rangle =0$ (which can easily be shown) where $H$ is the Hamiltonian, $\hat{O}$ is an arbitrary operator? Thanks.
2
votes
1answer
54 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 $A(a_i^\dagger,a_i)$:
$$\langle \phi|A(a_i^\dagger,a_i)|\phi'\rangle=A(\phi_i^*,\phi'_i)e^{\sum ...
2
votes
2answers
286 views
Constructing the exponential form of a unitary operator
I think I've got this figured out but wanted to make sure I'm doing this right.
Working with operators that satisfy bosonic commutation relations $[b,b^\dagger] = 1$, I define a very general unitary ...
2
votes
2answers
134 views
Why must quantum logic gates be linear operators?
Why must quantum logic gates be linear operators? I mean, is it just a consequence of quantum mechanics postulates?
2
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1answer
263 views
How do I solve these integrals of wave function and operator?
First integral
$$\int \Psi^*({\bf r},t)\hat {\bf p} \Psi({\bf r},t)\, d^3r,$$
where the $\Psi({\bf r},t)=e^{i({\bf k}\cdot{\bf r}-\omega t)}\,\,\,$ and $\hat {\bf p}=-i\hbar \nabla$.
Second one
...
2
votes
1answer
250 views
Hermitian operator ?? is this possible
we know that the operator
$ H= - \hbar ^{2} \frac{d^{2}}{dx^{2}}+ V(x) $ is hermitian isn't it ??
however what would happen if the potential were still real but it depends on the Wave function, for ...
2
votes
1answer
312 views
Question on ladder operators
Suppose we have a finite , discrete set of orthonormal states $|k\rangle $
We can construct raising and lowering operators intuitively, for example $$a_+ =\sum_{k=1}^nC_{k+1}|k+1\rangle \langle k|$$
...
2
votes
1answer
78 views
Physics Applications of Fredholm Theory:
I find Fredholm theory beautiful, especially the Liouville-Neumann series for solving Fredholm integral equations of the second kind. There seems to be a consensus that these equations are quite ...
2
votes
1answer
124 views
Coordinate representation of quantum ladder operator?
I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics.
I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
2
votes
1answer
56 views
A physical quantity that is a real combination and commutability
Suppose that a matrix
$$A ~=~ x_1 B + x_2 C$$
is a linear combination of two self-adjoint matrices $B$ and $C$.
I'm interested in when $A$ represents a physical quantity.
When the linear ...
2
votes
1answer
161 views
Degeneracy and the Hamiltonian
How many linearly independent eigenfunctions can be associated with one degenerate eigenvalue of the Hamiltonian operator? (Is there a limit since it contains a 2nd order differential operator?) ...
2
votes
1answer
58 views
Quantum mechanical analogue of conjugate momentum
In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ...
2
votes
1answer
170 views
Show that for QM operator A: $\int_{-\infty}^{\infty}\psi A^{\dagger}A\psi dx = \int_{-\infty}^{\infty}(A\psi)^*(A\psi)dx $
I need to show for $$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
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1answer
371 views
A Short way to show Conservation of Quantum Laplace–Runge–Lenz Vector?
I had been asked to prove the conservation of Quantum Laplace–Runge–Lenz Vector:
...
2
votes
1answer
110 views
What is the operator for the edge current of a fracional quantum Hall state?
The edge of a fractional quantum Hall state is a chiral conformal field theory. In the Laughlin case it corresponds to the chiral boson,
$$ S = \frac{1}{4\pi} \int dt dx ...
2
votes
1answer
203 views
How could $\textbf{S}^2$ not be a multiple of the identity?
I'm self-studying quantum mechanics with Sakurai's book (Modern Quantum Mechanics, 2nd edition) and came across the following in reference to the operator $\textbf{S}^2$:
As will be shown in ...
2
votes
1answer
136 views
A missing factor of 2 in the standard Hartree-Fock mean field?
Let's start from a very simple argument: If $A$ and $B$ are some operators, then I can write their product as
$$AB = (A-\langle A\rangle)(B - \langle B \rangle) + \langle A \rangle B + A \langle B ...
2
votes
1answer
40 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?
$\mathbb{E}[C^2] = a^2 + 2a\mathbb{E}[B] + \mathbb{E}[B^2]$
...
2
votes
0answers
94 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), ...
1
vote
6answers
407 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 ...
1
vote
2answers
283 views
Dirac notation question
I don't understand this equality
$$\int \!d^3p~\langle\textbf{x}|e^{-i(\hat{\textbf{p}}^2/2m)t}|\textbf{p}\rangle\langle\textbf{p} | \textbf{x}_0 \rangle ~=~\int\! ...
