All Questions
Tagged with linear-algebracommutator
31 questions
1vote
2answers
56views
Canonical commutation relation between raising/lowering operators not satisfied [duplicate]
We know that the canonical commutation relation between the raising and lowering operator $\hat a_I$, $\hat a^\dagger_J$ should result in the identity $\delta_{IJ}$: $$[\hat a_I, \hat a^\dagger_J] = \...
- 283
1vote
2answers
535views
If a Hermitian and Unitary operator commute, do they have a simultanous eigenbasis?
I was learning about Hamiltonian with discrete translational symmetry. We showed that there is a simultaneous eigenbasis of the Hamiltonian $H$ and the discrete translation symmetry being the bloch ...
- 23
1vote
1answer
165views
Commuting normal operators have common eigenbasis?
One often finds in quantum mechanics textbooks (for physicists) a "proof" that self-adjoint and commuting operators $A,B$ have a common eigenbasis. However, in the standard proof of Bloch's ...
- 1,301
1vote
2answers
131views
Question on the space of quantum operators and its algebra
The vector space of quantum states $|\psi\rangle$ is a hilbert space $\mathcal{H}$. Now, since the middle 20's of the past century, the quantization procedure states that one of the quantization ...
- 335
0votes
1answer
475views
Is the evolution of a density matrix linear with respect to density matrices?
The evolution of a density matrix in quantum mechanics is given by $$i\hbar\dot\rho=[H,\rho]=H \rho-\rho H.$$ If it is linear, can be written the rhs as $A \rho$ for a linear operator $A$?
0votes
1answer
141views
Condition of the product of 2 operators being an observable [duplicate]
I'm trying to understand a bit the conditions of operators commuting, or themselves being an observable. Here I have the operator $\hat{A}$ which has Eigenvalues $-1,+1$ and Eigenstates $|u_1\rangle, |...
- 149
0votes
1answer
226views
Diagonalizing Operators Simultaneously [duplicate]
Suppose we have a Hamiltonian operator $\hat{H}$ and another operator $\hat{A}$ such that $[\hat{H},\hat{A}]=0$. Then, if the spectrum of $\hat{H}$ is non-degenerate, from my understanding the ...
- 97
0votes
0answers
153views
Eigenbasis of Hamiltonian and momentum operators
I was taught that, if two Hermitian operators commute, they share the same eigenbasis. Since the Hamiltonian and momentum operators commute, am I right in concluding that they share the same basis of ...
0votes
1answer
243views
Commutation relation confusion of ladder operators in Quantum Mechanics
Suppose that $X$ and $N$ are operators such that they follow the commutation relation $$[N,X]=cX$$ for some scalar c. In this Wikipedia article it is shown that if $|n \rangle$ is some eigenstate of ...
- 405
0votes
1answer
855views
Degeneracy and Simultaneous diagonalization in Quantum Mechanics
I used to know that two operators can be simultaneously diagonalized, given they commute, they are hermitian and are non-degenerate (By simultaneous diagonalization, I mean they share a complete set ...
0votes
1answer
123views
How to know if an operator is compatible with the Hamiltonian?
So, in a question I am given the orthonormal eigenvectors of an operator, $A$, and its non-degenerate eigenvalues - I am not told which eigenvalue corresponds to which eigenvector. The eigenvectors of ...
- 41
1vote
1answer
669views
Sum of commutator and Anticommutator
Suppose $A$ and $B$ are Hermitian Operators. Then what will be the nature (purely real/purely imaginary/complex number of form $a + ib$, $a,b \in \mathbb{R}, a, b \neq 0$) of eigenvalues of $[A, B] + ...
2votes
2answers
546views
Eigenvalues of Product of 2 hermitian operators [closed]
Let $A$ and $B$ be two Hermitian operators. Let $C$ be another operator such that $C = AB$. What can we say about Eigenvalues of $C$? Will they be real/imaginary/complex? What I did was to search for ...
0votes
1answer
108views
If $\big[[\hat{A},\,\hat{B}],\, \hat{A}\big] = 0$ does that mean $\hat{A}$ and $\hat{B}$ commute?
Let's say we have the following identity: $$\Big[\big[\hat{A},\,\hat{B}\big],\, \hat{A}\Big] = 0.$$ Expand the LHS: \begin{align} \Big[\big[\hat{A},\,\hat{B}\big],\, \hat{A}\Big] &= 2\hat{A}\hat{B}...
- 48
0votes
2answers
163views
Eigenstates for $\vec{L}^2, L_z, L_x$ and $L_y$? [closed]
I am asked to find states $|j,m\rangle$ that are simultaneously eigenstates for $\vec{L}^2, L_z, L_x$ and $L_y$. I know that the $L_i$ operators do not commute and hence you cannot have a state $|\phi\...
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