All Questions
Tagged with quantum-mechanics analysis
22
questions
2
votes
1
answer
161
views
Stochastic differential equations with only time integrals
I want to reason about the stochastic differential equation
$$ dX_t = A_t X_t dt $$
Where $A_t$ is a matrix valued stochastic process, and hence $X_t$ is a vector valued stochastic process. Are there ...
1
vote
1
answer
74
views
Why harmonic functions does not belong to basis while it's eigenvector of Laplacian operator?
When we study quantum mechanics, teacher often says that $H = -\Delta$ is a Hermitian operator (more precisely, a self-adjoint operator), and then reference linear algebra to note that its ...
- 1,231
1
vote
0
answers
94
views
Using density argument of Schwartz functions in $L^2$ to show an operator is symmetric
The position operator $X$ is defined as multiplication by $x$, i.e.
$$(X\psi)(x):=x\psi(x).$$
We take the domain $D(X)\subset L^2(\mathbb R)$ for $X$, which we can take
$$D(X):=\{\psi \in L^2(\mathbb ...
- 5,601
4
votes
1
answer
82
views
An $\ell^qL^p$ inequality: $\|\nabla\langle\nabla\rangle^{-2}\varphi\|_{\ell^2L^4}\leq \|{\varphi}\|_{L^2}$.
An $\ell^qL^p$ inequality: $\|\nabla\langle\nabla\rangle^{-2}\varphi\|_{\ell^2L^4}\leq \|{\varphi}\|_{L^2}$. The dimensions I care about are $d=2$ and $d=3$.
We define $\ell^qL^p$ in the following way....
- 461
3
votes
0
answers
40
views
Deducing the Fermi sea is ground state for the kinetic energy operator $\sum_j(-\Delta_{x_j})$
I am reading a paper and have no background in quantum mechanics. Any help/references would be appreciated.
The ground state for a system of $N$ non-interacting fermions on the unit torus $\mathbb T^d$...
- 461
-1
votes
1
answer
58
views
If $T\geq 0$ then $|\langle x,STx\rangle|\leq\|S\|\langle x,Tx\rangle$ [closed]
I am trying to prove that on a complex Hilbert space, if $T\geq 0$ and $S$ is any bounded linear operator, then $T\geq 0$ then $|\langle x,STx\rangle|\leq\|S\||\langle x,Tx\rangle|$.
- 461
0
votes
2
answers
92
views
Details of a Fourier transform
I'm currently reading this article about how coherence properties of light influence the dynamics of absorption in molecules. In the article, equation 6 describes a chaotic light source whose time ...
- 72
0
votes
1
answer
24
views
Does the commutator $[-\Delta+ (w\ast T_K), K]$ make sense?
Suppose that $T$ is abounded self adjoint operator on $L^2(\mathbb R)$ and $K$ be the integral kernel of $T$. Define a function $T_K(x):=K(x,x), x \in \mathbb R.$
My question: Can the following ...
- 3,837
1
vote
2
answers
348
views
Question about Schrodinger equation [closed]
I meet this problem when I am reading an introductary PDE textbook. I wonder if you could give a rigorous proof of it.
The wave function $u(t,\mathbf{x})$ is governed by the $Schr\ddot{o}dinger$ ...
- 27
2
votes
1
answer
83
views
Time to find particle left of the origin maximal
We study a quantum particle conned to a
one-dimensional box with walls at the positions ±1.
The Hilbert-space of this system in
the Schrödinger representation is again given by $L_2(-1,1)$. In this ...
- 558
3
votes
1
answer
101
views
Find Heisenberg evolution (matrix products with complex numbers)
We will study the time-evolution of a finite dimensional quantum system. To this end, let us consider a quantum mechanical system with the Hilbert space $\mathbb{C}^2$. We denote by $\left . \left | ...
- 558
5
votes
1
answer
233
views
Use Schrödinger's equation in order to derive a differential equations
We will study the time-evolution of a finite dimensional quantum system. To this end, let us consider a quantum mechanical system with the Hilbert space $\mathbb{C}^2$. We denote by $\left . \left | ...
- 558
3
votes
0
answers
225
views
Partition function for a general Ising model as Gaussian integral
Consider the Hamiltonian of Ising model
$$H= - \sum_{x,y} J_{x,y}\sigma_x \sigma_y - \sum_x h\sigma_x$$
where $\sigma_x$ can be $1$ or $-1$ and the number of sites is finite ( i.e. there are a finite ...
8
votes
1
answer
9k
views
Deriving Rodrigues Formula and Generating function of Hermite Polynomial from $H_n(x)= e^{x^2/2}(x-\frac{d}{dx})^ne^{-x^2/2}$
There are a variety of ways of first defining the Hermite Polynomials in a certain way and then to derive alternative representations of them. For example in Mary Boas' Mathemmatical methods (p. 607, ...
- 1,517
3
votes
1
answer
90
views
Analysis of a function: Showing that Quantum Mechanics violates relativity
Consider the Hilbert space $L^2(\mathbb{R}^d)$ and a self-adjoint, positive operator $H$ (Hamiltonian).
Let $\psi_t$ be a solution to the Schrödinger equation (with $\psi_0$ the initial condition), ...
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