All Questions
Tagged with differentiation mathematical-physics
28 questions
1
vote
1
answer
142
views
Can a wave function discontinuous in the time variable be a solution of the Schrödinger equation?
It is well known that wave functions that are discontinuous in the space variable cannot be solutions of the Schrödinger equation, because the Schrödinger equation is a second-order differential ...
- 29
3
votes
3
answers
579
views
How does Kirchhoff's voltage law relate to the spatial derivative of voltage?
I'm reading this libretexts article on the basics of transmission line theory. In it, they include this circuit diagram as a model of a uniform transmission line:
They then say that applying ...
- 2,373
2
votes
1
answer
191
views
Decay of the First Derivative of the Quantum Wave Function
I understand that the Hilbert space of all physical solutions of the Schrodinger equation have the property where
$$
\lim_{x\to\infty}\Psi=0
$$
For one of my assignments, I wanted to use
$$
\lim_{x\to\...
- 65
1
vote
1
answer
54
views
Proof of differentiate form of dynamical semigroups
I am studying some basics of the pure mathematical background for open quantum systems from Angel Rivas`s book which is "Open quantum systems, an introduction".
Here is a theorem (Page 6, ...
- 119
2
votes
2
answers
465
views
Variation of a time-ordered exponential
Consider the time-ordered exponential (Wilson line):
$$
U(t_{f},t_{i})
=
\mathcal{T}\text{exp}\left(-i\int_{t_{i}}^{t_{f}}\mathcal{A}(t)dt\right)\tag{1}
$$
Where $\mathcal{A}(t)$ is some matrix-valued ...
- 587
3
votes
2
answers
814
views
D'Alembertian of a Dirac delta function of a spacetime interval (i.e. with support on the 3+1D light-cone)
How one differentiates a delta-function of a spacetime interval? Namely,
$$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$
Somewhere I saw that the result ...
-1
votes
1
answer
163
views
Statistical physics is unable to prove that $TdS=d\overline{E}$
I will pose $k_B=1$.
Suppose a system of statistical physics with the constraints:
$$
\begin{align}
1&=\sum_{q\in\mathbb{Q}}\rho(q)\\
\overline{E}(\beta)&=\sum_{q\in\mathbb{Q}} E(q)\exp(-\...
- 1,558
2
votes
0
answers
240
views
Torsion form and exterior covariant derivative
The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
- 258
2
votes
2
answers
270
views
Does it make sense to speak in a total derivative of a functional? Part II
I am trying to derive the Noether theorem from the following integral action:
\begin{equation}
S=\int_{\mathbb{\Omega}}d^{D}x~\mathcal{L}\left( \phi_{r},\partial_{\nu}%
\phi_{r},x\right) , \tag{II.1}\...
- 387
6
votes
1
answer
2k
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Functional derivative commutes with total derivative
I have a question about a rule from the calculus of variations.
Assume we consider the space of differentiable functions on $C^1(\mathbb{R})$ (or for the sake of simplicity the smooth functions $C^{\...
- 1,555
5
votes
1
answer
909
views
General derivative of the exponential operator w.r.t. a parameter
I am interested in the calculation of the general $N$th derivative w.r.t. a parameter $\lambda$ of a quantum mechanical exponential operator with the following structure:
\begin{equation*}
\frac{\...
- 385
1
vote
0
answers
68
views
Have fractional order differential models been explored as an alternative to standard gravitational field theory?
Since Einstein introduced his field equations and general theory of relativity, experimental evidence, at least on the cosmic scale has repeatedly supported the theory. Nevertheless, many seeking to ...
- 11.7k
1
vote
0
answers
63
views
Fractional derivatives in physics? [duplicate]
Fractional deriatives are interesting and all, but are there and scenarios in physics (either confirmed or hypothetical) where fractional calculus is part of a model?
- 1,707
0
votes
0
answers
88
views
Physical meaning of the eigenfunctions and eigenvalues of the Fractional Laplacian
What is the physical meaning of the eigenfunctions and eigenvalues of the Fractional Laplacian?
user170351
0
votes
0
answers
54
views
Mathematical Description of Time Speeding Up?
People are able to experience time speeding up or slowing down. This is confusing to me on a mathematical level because dT/dT is 1. Is there some way that makes sense for this not to be 1? The speed ...
Alephnull
3
votes
1
answer
2k
views
How does one properly define the derivative of one operator-valued function?
In Quantum Mechanics we usually consider operator-valued functions: these are functions that take in real numbers and gives back operators on the Hilbert space of the quantum system.
There are ...
- 37.4k
4
votes
2
answers
993
views
Why do we consider potential energy function $U(x)$ differentiable?
Recently when skimming through my physics-text I encountered an interesting definition of Force
$$F(x) = -\frac{\mathrm dU(x)}{\mathrm dx}$$
We were taught that some functions are continuous but not ...
- 373
10
votes
1
answer
806
views
Physical intuition/interpretation of fractional derivatives/integrals?
Oftentimes, when the derivative and integral operations are introduced within the realm of physics, we are taught some physical interpretation of them:
Velocity is the derivative of position
...
- 421
6
votes
2
answers
2k
views
Relationship between Connection and Material Derivative
Suppose $D\subset \Bbb R^3$ contains a fluid and that $f : D\times \mathbb{R}\to \mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined ...
- 37.4k
12
votes
1
answer
1k
views
Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction
The Schrodinger equation in Hilbert space is expressed as : $$\frac{\partial}{\partial t} \psi(t) = \frac{-i}{\hbar}H\psi(t). $$
Here $\frac{\partial}{\partial t} \psi(t) \equiv \psi'(t) \equiv\lim \...
- 421
12
votes
1
answer
900
views
Can You Obtain New Physics from the use of Fractional Derivatives?
I was curious if anyone could give me an example of the use of fractional derivatives in physics and explain what they offer that "conventional" mathematics does not (in terms of new physics and not ...
- 808
8
votes
1
answer
712
views
When motion begins, do objects go through an infinite number of position derivatives?
This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative ...
- 83
18
votes
1
answer
3k
views
Is there a "covariant derivative" for conformal transformation?
A primary field is defined by its behavior under a conformal transformation $x\rightarrow x'(x)$:
$$\phi(x)\rightarrow\phi'(x')=\left|\frac{\partial x'}{\partial x}\right|^{-h}\phi(x)$$
It's fairly ...
2
votes
1
answer
623
views
Differentiation and delta function
Need help doing this simple differentiation.
Consider 4 d Euclidean(or Minkowskian) spacetime.
\begin{equation}
\partial_{\mu}\frac{(a-x)_\mu}{(a-x)^4}= ?
\end{equation}
where $a_\mu$ is a constant ...
- 73
6
votes
5
answers
4k
views
What is the meaning of following expression $C=\frac{\delta Q}{dT}$ mathematically?
Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics):
Many text books (even Wikipedia) writes wrong expressions (from ...
- 2,101
15
votes
4
answers
3k
views
What is the relation between (physicists) functional derivatives and Fréchet derivatives
I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books:
$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\...
- 2,526
16
votes
5
answers
9k
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Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation)
We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is
$$\nabla^2\frac{q}{r}~...
- 3,802
11
votes
2
answers
1k
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The derivation of fractional equations
Recently I saw some physical problems that can be modeled by equations with fractional derivatives, and I had some doubts: is it possible to write an action that results in an equation with fractional ...
- 1,247