All Questions
Tagged with quantum-mechanics complex-analysis
31
questions
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proving complex Integral relation from perturbation theory MQ
Someone can help me to prove this identity, it comes from a normalization in MQ. From perturbation theory time dependent we have
$$H(t)=H_0+H’(t)$$
$$|Ψ>=c_a(t)e^{-iE_at/\hbar}|Ψ_a>+c_b(t)e^{-...
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0
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31
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Hardy Hille type eigenfunction exapansion
I am trying a figure out Eigen-function expansion of the following kind.
$$
\exp\left[{-\frac{1}{4\alpha} \left(x^2 + y^2\right)}\right] I_{l + \frac{1}{2}} \left(\frac{x y}{2 \alpha}\right) = \sum_{n}...
- 65
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Proving limit exists and is positive for smooth function involving integral
Let $\mu$ be a function from $\mathbb R_+\rightarrow\mathbb R_+$ in $C^\infty$ with $\Upsilon:=\sup\{|p|:\mu(|p|^2)>0\}$. Suppose that $$\lim_{|p|\rightarrow \Upsilon}\frac{\mu(|p|^2)}{(\Upsilon-|p|...
- 461
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178
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Symmetric subspace dimension of Hermitian matrix
Is there a general formula for the dimension of the symmetric subspace of any Hermitian matrix?
The closest concept I was able to find is the following classification.
Is the complex dimension what I ...
- 559
1
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1
answer
547
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What is a "continuous" Hilbert space?
The book I'm reading about quantum mechanics uses the term "continuous" Hilbert space, whereby apparently in such a
space one has a "continuous" basis, e.g. $\{ |x\rangle \}_{x\in \...
- 307
1
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0
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62
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Integral over operators equal -When are operators equal?
This is a physics motivated question from quantum mechanics. When I have two operators $\hat{A}$ and $\hat{B}$ acting on the Hilbertspace $\mathbb{C}-L^2(\mathbb{R})$ with
$$\int\psi^*(x)\hat{A}\psi(x)...
- 320
1
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1
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110
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Is $\nabla^2 f/f\in\mathbb{R}$ for a complex-valued $f\in L^2$?
Given a complex-valued function $f: \mathbb{R}^N\rightarrow\mathbb{C}$ with $f\in L^2$, which vanishes at the domain boundaries, is twice differentiable, and antisymmetric upon two-coordinate exchange—...
- 145
2
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2
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187
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Is this difference of surface integrals zero? $\oint_S\bar{\psi}\nabla(x\cdot\nabla\psi)\cdot n dS-\oint_S(x\cdot\nabla\psi)\nabla\bar{\psi}\cdot ndS$
This is the follow-up question of When does this integral vanish, which appears in the derivation of the quantum virial theorem? and building on this answer.
Does the following difference of surface ...
- 145
5
votes
1
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233
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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
4
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824
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Given product and convolution of pair of functions can you find original pair?
Suppose you have two functions $f,g: \mathbb{R} \rightarrow \mathbb{C}$ and you have their product and convolution
$$ h_1(t) = fg$$
$$ h_2(t) = \int_{-\infty}^{\infty} f(t-\tau)g(\tau) d\tau $$
Is it ...
- 17.5k
1
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2
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112
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Why is $(\int dx)^2 = \int dx \int dy$
An exercise in a quantum mechanics book:
Show that
$$ G(a) = \int_{-\infty}^\infty e^{-\frac{1}{2}ax^2} = \sqrt\frac{2\pi}{a}$$
Solution
$$G(a)^2 = \int_{-\infty}^\infty dx \int_{-\infty}^\infty dy\...
- 260
0
votes
1
answer
102
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Quadrature for numerically evaluating Cauchy integral formula using unit circle as closed contour
I am currently trying to evaluate the derivatives of a function $F(z)$ in z=0, which is only known numerically on the unit circle ("$UC$") in the complex plane. My question is this:
given $z=...
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1
answer
73
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Deficiency index of symmetric operator is locally constant
Let $A: D_A \rightarrow H$ be a symmetric linear operator defined on $D_A \subseteq H$. Define the deficiency index of $A$ at $z \in \mathbb{C} \setminus \mathbb{R}$ to be $\dim( \ker (A^* - \bar{z}))$...
- 5,456
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1
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145
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Trouble Deriving the Canonical Commutation Relation from the Product Rule
From pg. 74 of No-Nonsense Quantum Mechanics, the author derives the canonical commutation relation from the momentum operator $\hat{p}_i$ as follows:
Question: How does the product rule (for ...
- 5,265
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1
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67
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Show $|f(z)|<1$ in open unit disc for Schur function $f(z)$
I'm a physics grad student who encountered the mathematics of spectral measures and orthogonal polynomials on the unit circle in the study of quantum time of arrival statistics. My question in ...
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