15

Were $\psi(x)$ not continuous at $x=0$ then $\psi''(x)$ would contain the derivative of a $\delta$-function, and there is nothing else in the equation $H\psi=E\psi$ that could cancel it, so a discontinuity is not allowed.


10

Let us assume the integral form $$ \psi(x)~=~ \frac{2m}{\hbar^2} \int^{x}\mathrm{d}y \int^{y}\mathrm{d}z\ (V(z)-E)\psi(z) \tag{1}$$ of the time independent 1D Schrödinger equation (TISE), where the potential $$V(x)~=~V_0\delta(x)\tag{2}$$ is a Dirac delta potential. Let us assume that the wavefunction $\psi\in {\cal L}_{\rm loc}^1(\mathbb{R})$ is a locally ...


8

If the operator $A$ belongs to $B(H)$ (the space of everywhere defined bounded operator on the Hilbert space $H$) and is normal: $$A^*A=AA^*$$ then it admits a spectral decomposition $$A = \int_{\mathbb{C}} z dP(z) = \int_{\sigma(A)} z dP(z)$$ and, with an obvious notation, $|\sigma(A)| \leq ||A|| <+\infty$. In this case (and also in the general case ...


2

A possible additional condition is that $\psi_2$ and $\psi_2'$ are bounded. Then $$\lim_{x\to\infty}\psi_1' \psi_2=\lim_{x\to\infty}\psi_1 \psi_2'=0$$ and since the Wronskian is a constant, this constant must be identically $0$. But if $\psi_2$ or $\psi_2'$ are unbounded, then it might be that $\lim_{x\to\infty} \psi_1'\psi_2\ne0$ and similarly for the ...


2

The algebra of (bounded) operators acting on the Hilbert space $\mathcal{B}(\mathcal{H})$ is a $C^{*}$-algebra. Therefore, its subalgebra generated by the observables of a particular quantum mechanical model is itself a $C^{*}$-algebra. Given a $C^{*}$-algebra and an algebraic state $\omega$, one can reconstruct a $*$-representation of the algebra on the ...


2

Note that you are working in the center-of-mass of the system. If the gas is say, a ballon, and you find that: $$ c_{12} \equiv \langle \vec v_1 \cdot \vec v_2 \rangle \ne 0 $$ then the system is moving. If $c_{12}$ is small, then it is drifting slowly. Imagine if the ballon is moving much faster than the average speed from Maxwell's distribution (say, it's ...


1

As the commenters suggest, compute the average value. $\theta$ takes on a uniform distribution from $0$ to $2\pi$. So by the Law of the Unconscious Statistician, $$\mathbb{E}[\vec{x}\cdot\vec{y}] = \mathbb{E}[|\vec{x}||\vec{y}|\cos\theta] \\ = \int_0^{2\pi}|\vec{x}||\vec{y}|\cos\theta\frac{1}{2\pi-0}\ d\theta \\ = \frac{1}{2\pi}|\vec{x}||\vec{y}|\int_0^{2\pi}...


1

Here are examples of solutions that have toroidal null infinities: Schmidt, B.G., Vacuum space-times with toroidal null infinities, Class. Quantum Grav.,13, 2811–2816, (1996), doi:10.1088/0264-9381/13/10/017, free pdf. Hübner, P., More about vacuum spacetimes with toroidal null infinities, Class. Quantum Grav.,15, L21–L25, (1998), doi:10.1088/0264-9381/15/...


1

The solution to the problem goes as follows: Observation one: The plane partition intersects the wall of any of the aforementioned containers along a Young tableaux, as the following figure illustrate: Let's write that tableaux as $\mu^{T}= (\mu^{T}_{1},...,\mu^{T}_{n})$ with the sequence $\mu^{T}_{1}...,\mu^{T}_{n}$ non-increasing (by definition). ...


1

The other answers are very good. I want to address one mis-think in the question. "... if $\psi$ is discontinuous at $x=0$, then we'll have trouble writing $\psi(0)$ ..." Not always. As a vast oversimplification, suppose that you have obtained $\psi$ as a Fourier transform, which has $\lim_{x \rightarrow 0^-} \psi(x) = -1$ and $\lim_{x \...


1

You can think of the delta potential $V(x)=g\delta(x)$ as representing the limiting form of a potential barrier (or well) of height $\Lambda$ and width $w$, with $w\Lambda=g$. For any finite $\Lambda$, $\psi(x)$ and $\psi’(x)$ must be continuous at both edges of the barrier. Within the barrier, Schrödinger’s equation implies that $\psi’’(x)\sim\Lambda$, so ...


1

Let $f$ denote a function. A function $y = f(x)$ associates a unique $y$ to every $x$. Some functions have other properties such as: continuity, one-to-one, and onto. A function is a special type of a relation. A function can be defined for more than one variable; for example, $y = f(a, b, c)$ provides a unique $y$ for a specific set of $\{a, b, y\}$ ...


1

For the "physics part" of my question: I have found the paper String Orbifolds and Quotient Stacks very useful and explicit. Apparently the best strategy to deal with stacks in the context of a non-linear sigma model is to think on the target space as locally an orbifold (exactly the way mathematicians divulge the idea of a stack). For the "...


1

There are no more kets than bras, nor less. We could say they are equal in number (though it is hard to imagine two uncountable infinite sets having the same number of elements), and that is because the space of kets and the space of bras are isomorphic topological vector spaces. Quoting from de la Madrid, Loosely speaking, a rigged Hilbert space (also ...


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