The operators tag has no wiki summary.
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1answer
73 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
107 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 ...
10
votes
3answers
321 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
$$
H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
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0answers
28 views
Schrodinger equation in momentum space [duplicate]
I have a problem this is:
When I solve the Schrodinger equation in momentum space, I had done as this:
$\begin{array}{l}
i\hbar \frac{{\partial \Psi }}{{\partial t}} = - \frac{\hbar ...
0
votes
0answers
71 views
Prove that the position operator is $\hat{x} = i\hbar \frac{d}{{dp}}$ in the momentum representation [closed]
Proof that: $x = i\hbar \frac{d}{{dp}}$
I did this, could you tell me if I am false or true
$\begin{array}{l}
x{e^{\frac{{ipx}}{\hbar }}} = - i\hbar \frac{{d{e^{\frac{{ipx}}{\hbar }}}}}{{dp}} = ...
-1
votes
1answer
65 views
Proof $\left[ {\hat H,{{\hat p}_i}} \right] = - \frac{\hbar }{i}\frac{{\partial \hat H}}{{\partial {{\hat q}_i}}}$ [closed]
I have a problem with the Hamiltonian, I don't think anything to solve it!!
So could you give me some hints!
Knowing that:
$$\left[ {{{\hat p}_i},{{\hat q}_k}} \right] = \frac{\hbar }{i}{\delta ...
1
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1answer
87 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
109 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
$W$.
...
3
votes
2answers
183 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 ...
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?
$\mathbb{E}[C^2] = a^2 + 2a\mathbb{E}[B] + \mathbb{E}[B^2]$
...
1
vote
1answer
44 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
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 ...
1
vote
2answers
238 views
Derivatives of operators
How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this?
For example: say you had the ...
4
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2answers
91 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
53 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 $A(a_i^\dagger,a_i)$:
$$\langle \phi|A(a_i^\dagger,a_i)|\phi'\rangle=A(\phi_i^*,\phi'_i)e^{\sum ...
3
votes
1answer
61 views
Spectral properties of CFT
What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac ...
0
votes
1answer
83 views
Matrix representation for fermionic annihilation operator
My guess it should look something like this:
$ c_\sigma = ...
4
votes
2answers
73 views
Translator Operator
In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for ...
2
votes
1answer
105 views
Phys.org Spectral geometry to unite relativity and quantum mechanics, restate in laymens terms?
Lingua Franca links relativity and quantum theories with spectral geometry
Could someone give me a short synopsis of this article in laymens terms? What implications does this have in the physics ...
2
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0answers
93 views
How to define the mirror symmetry operator for Kane-Mele model?
Let us take the famous Kane-Mele(KM) model(http://prl.aps.org/abstract/PRL/v95/i22/e226801 and http://prl.aps.org/abstract/PRL/v95/i14/e146802) as our starting point.
Due to the time-reversal(TR), ...
0
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0answers
38 views
Time ordering and Fermions
Having time ordering operator for fermions, should it reverse sign if it swaps operators with opposite spin variable? In other words should
$T[c_{t_1,\uparrow}c_{t_2,\downarrow}^\dagger]$
return ...
1
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0answers
36 views
QFT basics for Klein-Gordon fields
I am teaching myself QFT from Peskin for next years maths course and I have two questions:
What is a c-number? Is it a complex number, and if so why does it mean, ...
3
votes
2answers
112 views
Quantum commutator
I'm given this commutator:
$$\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 ...
2
votes
1answer
87 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
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3answers
104 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
57 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
82 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
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 ...
1
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0answers
26 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
104 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 ...
5
votes
1answer
343 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 ...
3
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5answers
226 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
66 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
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 = ...
1
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0answers
73 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
...
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 ...
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,
$$\tag{10.68} ...
7
votes
1answer
133 views
String theory - OPE and primary operators
First, a disclaimer: I am new to Physics SE, and I am primarily a mathematician, not a physicist. I apologise in advance for the possibly poor quality of the question, any and thank you for your ...
8
votes
2answers
499 views
Regularisation of infinite-dimensional determinants
Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM?
Edit: I failed to make myself clear. In finite ...
5
votes
3answers
197 views
Why is this identity an if, rather than if and only if?
A recent question (Product of exponential of operators) asked who to proved that the exponentials of operators multiply in same manner as those of scalars if and only if the commutator of the ...
1
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2answers
236 views
Deriving a QM expectation value for a square of momentum $\langle p^2 \rangle$
I alredy derived a QM expectation value for ordinary momentum which is:
$$
\langle p \rangle= \int\limits_{-\infty}^{\infty} \overline{\Psi} \left(- i\hbar\frac{d}{dx}\right) \Psi \, d x
$$
And i ...
0
votes
4answers
379 views
Product of exponential of operators
in the context of non-relativistic quantum mechanics I want to show that, for any $A$ and $B$ operators
$$e^{A}e^{B}=e^{A+B} $$
if and only if
$$[A,B]=0$$
I remember my professor told use about ...
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votes
4answers
132 views
How to prove that the symmetrisation Operator is hermitian?
Let $\mathcal{H}_N$ be the $N$ particle Hilbert space. So a quantum state $\left| \Psi \right>$ may be representated by
$$\left| \Psi \right> = \left| k_1 \right>^{(1)}\left| k_2 ...
1
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3answers
144 views
Operators explaination and momentum operator in QM
I know and understand why equation below holds. But i am new to operator thing in QM and would need some explaination on this.
$$\langle x \rangle = \int\limits_{-\infty}^\infty |\Psi|^2 x \, ...
5
votes
2answers
164 views
Weyl Ordering Rule
While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
5
votes
4answers
268 views
Is the momentum operator diagonal in position representation?
The matrix elements of the momentum operator in position representation are:
$$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}$$
Does this imply that $\langle x ...
1
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1answer
49 views
Question about the linearity of wave functions
For piece-wise constant potential, the potential energy is constant so the time dependent wave function can take the form $\psi(x,t)=C_1e^{i(kx- \omega t)}+C_2e^{i(-kx-\omega t)}$ where ...
1
vote
1answer
96 views
Klein-Gordon Canonical Commutation Relation (CCR)
In the complex Klein-Gordon field we regard as dynamical variables the field $\phi$, the complex conjugate of the field $\phi^*$, and the momenta $\pi$, $\pi^*$. I can't see how should arise the ...
3
votes
1answer
140 views
commutation of operator product expansion
In CFT, when we have an OPE:
$$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 ...
1
vote
1answer
54 views
Can I prove boundedness of an operator without checking it for its whole domain?
(I don't have a direct reference so this is a little fishy and I'll delete it if nobody recognises what I'm talking about, but I though for starters I'll ask anyway)
I've heard at university that if ...
