To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.
To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.:
Questions tagged [linear-algebra]
1121 questions
96
votes
9 answers
Are matrices and second rank tensors the same thing?
Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified:
Are matrices and second rank tensors the same thing?
If the answer to 1 is yes, then…
Revo
- 17,504
44
votes
2 answers
How to take partial trace?
$L$ is a linear operator acting on hilbert space $V$ of dimension $n$, $L: V \to V$. The trace of a linear operator is defined as sum of diagonal entries of any matrix representation in same input and output basis of $V$. But if $L$ is a linear…
advocateofnone
- 1,595
- 1
- 14
- 27
34
votes
7 answers
What is the actual use of Hilbert spaces in quantum mechanics?
I'm slowly learning the quirks of quantum mechanics. One thing tripping me up is... while (I think) I grasp the concept, most texts and sources speak of how Hilbert spaces/linear algebra are so useful in quantum calculations, how it's the…
Ringo Hendrix
- 561
30
votes
1 answer
What're the relations and differences between slave-fermion and slave-boson formalism?
As we know, in condensed matter theory, especially in dealing with strongly correlated systems, physicists have constructed various "peculiar" slave-fermion and slave-boson theories. For example,
For Heisenberg model, we have Schwinger-fermion…
Kai Li
- 3,794
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30
votes
5 answers
Is there a simple proof that Kirchhoff's circuit laws always provide an exactly complete set of equations?
Suppose I have a complicated electric circuit which is composed exclusively of resistors and voltage and current sources, wired up together in a complicated way. The standard way to solve the circuit (by which I mean finding the voltage across, and…
Emilio Pisanty
- 137,480
27
votes
3 answers
Use of 'complete' as in 'complete set of states' or 'complete basis'
Question. In the context of QM, I hear the phrases 'complete set of states' and 'complete basis' (among other similar expressions) thrown around rather a lot. What exactly is meant by 'complete'?
Further remarks. I understand the term 'complete…
gj255
- 6,525
27
votes
3 answers
Perturbation theory with degeneracy even after 1st order
Most textbooks on basic quantum mechanics tell you that when your initial Hamiltonian $H_0$ has degenerate states, then before you can do (time independent) perturbation theory with a perturbation matrix $V$ on it, you have to first diagonalize $H_0…
Lagerbaer
- 15,136
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27
votes
7 answers
Why do physicists use the Bra and Ket notation when mathematicians tend not to?
Please forgive me that this is a lay question. I realise it will not add any profound physics to stack exchange, and I ask only out of layman's curiosity.
I saw a question on google from a physicist asking (on Quora I think) why mathematicians do…
Rory Cornish
- 1,117
26
votes
5 answers
What exactly is a dimension?
How do you exactly define what is and isn't a dimension? I heard somewhere that it is "anything you can move through" but if that is right, why wasn't time and space considered a dimension before Einstein?
user72789
25
votes
2 answers
What information does the trace of a matrix give?
I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant will tell about the volume of the space enclosed…
Ayushi
- 251
24
votes
10 answers
What is an example of three numbers that do not make up a vector?
In one of Feynman's lectures he mentions that:
Of course it is generally not true that any three numbers form a
vector
In order for it to be a vector, not only must there be three numbers,
but these must be associated with a coordinate system in…
Zinn
- 351
23
votes
9 answers
Are force vectors members of a vector space?
Vectors in vector spaces depend only on their size and direction.
Force vectors, for example, depend also on their location. Opposite force at different locations, for example, do not annihilate each other but form torque.
Why this anomaly?
user4951
- 601
23
votes
7 answers
Linear algebra for quantum physics
A week ago I asked people on this site what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of Linear Algebra. Of course, I'm just 1 week in, but I…
kamal
- 235
23
votes
1 answer
Linear response theory for Gross Pitaevskii equation
I am trying to linearize the following GP eq:
\begin{equation}
i\partial_{t}\psi(r,t)=\left[-\frac{\nabla^{2}}{2m}+g\left|\psi(r,t)\right|^{2}+V_{d}(r)\right]\psi(r,t)
\end{equation}
The ansatz for the mean-field wavefunction…
Andrei
- 317
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18
votes
1 answer
Validity of Bogoliubov transformation
In condensed matter physics, one often encounter a Hamiltonian of the form
$$\mathcal{H}=\sum_{\bf{k}}
\begin{pmatrix}a_{\bf{k}}^\dagger & a_{-\bf{k}}\end{pmatrix}
\begin{pmatrix}A_{\bf{k}} & B_{\bf{k}}\\B_{\bf{k}} &…
leongz
- 4,174