All Questions
Tagged with quantum-mechanics abstract-algebra
27
questions
1
vote
1
answer
62
views
Eigenvalues of superoperators and their Choi matrices
It is well known that $\Phi$ is a completely-positive and trace-preserving (CPTP) map if and only if the corresponding Choi matrix $C_\Phi:=\sum_{i,j} E_{i,j}\otimes \Phi(E_{i,j})$ is positive semi-...
- 393
1
vote
1
answer
44
views
How to represent elements from $\bar{E}=E/Z(E)$ in the form $(a|b)$
From https://arxiv.org/abs/quant-ph/9608006
Background
The group $E$ of tensor products $\pm w_{1} \otimes \dots \otimes w_{n}$ and $\pm i w_{1} \otimes \dots \otimes w_{n}$, where each $w_{j}$ is one ...
- 329
1
vote
2
answers
551
views
Quantum physics and principle of non-contradiction
We all know that mathematics is based on axioms in the science of logic. These axioms we proceed through to prove the validity or falsehood of any statement. One of the most important of these axioms ...
- 360
5
votes
0
answers
75
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Commuting operators vs. commutative rings, and localization in the quantum mechanical sense vs. commutative algebra sense.
One of the answers in this MO post https://mathoverflow.net/questions/7917/non-commutative-algebraic-geometry draws a connection between localization in commutative algebra (or the failure thereof), ...
- 8,945
2
votes
1
answer
52
views
How to find define the relative phase of a qubit
Given a complex, two-dimensional vector space, let the vector $ |V\rangle = r_0 e^{i\theta_0} |0\rangle + r_1 e^{i\theta_1} |1\rangle $ correspond to the state of a qubit where $r_0,r_1,\theta_0,\...
- 6,099
5
votes
1
answer
153
views
Show That Wigner’s Theorem Defines a Surjective Homomorphism $\operatorname{U}(2) \rightarrow \operatorname{SO}(3)$
Preliminary Knowledge
We are working on the finite dimensional Hilbert space $\mathbb{C}^2$. The projective Hilbert space is given by
$$\mathbb{P}(\mathbb{C}^2)=\big(\mathbb{C}^2 \backslash \{0\}\big)...
- 252
1
vote
0
answers
35
views
Mathematical structure of time-dependent Hamiltonians
In quantum mechanics the states of a physical system live on a Hilbert space $\mathcal{H}$ and the Hamiltonian operator can be seen as a map $H_0: \mathcal{H} \to \mathcal{H}$. Algebraically this ...
- 2,239
0
votes
1
answer
77
views
Determinant of a differential operator of the action variations
I have to compute the determinant of a differential operator that comes from an action variation: $\frac{\delta^2S}{\delta q(\tau)\delta q(\tau)}=-M\frac{\partial^2}{\partial \tau^2}+V''(q)$.
It is ...
- 19
3
votes
2
answers
170
views
Proof using taylor series
I am currently trying to solve a quantum mechanics problem in which i need to prove that $e^A e^{-A} = 1$ where $A$ is an operator and the exponent function is defined by a taylor series. However, I ...
- 123
2
votes
1
answer
245
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Finding an Operator from its Commutator
I'm trying to figure out if it's possible to derive a general form for the linear operator $L_z$(rotational momentum in Quantum mechanics) which is a combination of the four linear operators
$$X, Y, ...
- 1,072
4
votes
0
answers
279
views
Quantum representation of a system of identical particles
I'm studying mathematics and I began a course in quantum statistics, in which I got to the discussion related to indistinguishibility of particles. My professor's notes are not very clear and ...
- 2,611
0
votes
0
answers
29
views
Sub-algebra such that Wigner (-Weyl) transform is a homomorphism
The Wigner distribution of an operator $A$ is given by
$$
W_A(x,p) :=\frac{1}{2\pi} \int_\mathbb{R} \! dy \, \langle x+y/2| A | x-y/2 \rangle \, e^{ipy},
$$
and associates a function in phase space ...
- 2,624
2
votes
1
answer
167
views
From the condition $[A,B]=A$, what can I say about $B$?
I'm struggling in understanding the meaning of this condition that I found in an operator equation:
\begin{equation}
[A,B]=A
\end{equation}
where both $A$ and $B$ are hermitian operators. What can I ...
- 123
0
votes
2
answers
948
views
Is it true that an arbitrary rotation $R\in SO(3)$ can be decomposed into rotations about any two fixed axes?
This is from exercise 4.11 in Nielsen and Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition), which I think might be in error.
Background
A quantum unitary operator on ...
- 596
1
vote
0
answers
63
views
Two particle operators
General form of two particle operators is
$$\hat{B} = \sum\limits_{ij,kl} B_{ij,kl} \hat{a}^{\dagger}_i\hat{a}^{\dagger}_j\hat{a}_k\hat{a}_l$$
and I am asked to compute the expected value of $\hat{B^...
- 1,622