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Questions tagged [quantum-mechanics]

For questions on quantum mechanics, a branch of physics dealing with physical phenomena at microscopic scales, where the action is on the order of the Planck constant.

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35 views

Trace of a matrix exponential with tensor products, and Von Neumann entropy

$\def\T{\operatorname{Tr}}$ $\def\1{\mathbb{1}}$ Let $H=H_1\otimes H_2\otimes H_3$ be a finite dimensional Hilbert space, and let $\rho_{123}$ be a self-adjoint matrix with $\rho_{123}\geq 0$ (...
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1answer
24 views

Integral representation of log of operators

$\def\1{\mathbb{1}}$ Suppose we're in a "good enough" (finite for example) space, and we have positive (semi)-definite operators $P$ and $Q$. Let $\log{(P)}$ and $\log{(P)}$ be logarithms of $P$ and ...
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2answers
24 views

Evaluation of the commutator $[\frac{\partial}{\partial x_j},\frac{x_ix_j-r^2\delta_{ij}}{r^3}]=-2x_i/r^3$.

I am trying to evaluate a commutator of the form $[\frac{\partial}{\partial x_j},\frac{x_ix_j-r^2\delta_{ij}}{r^3}]=-2x_i/r^3$. Bt acting it on a function $f(\vec{r})$, I get that the commuator is ...
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23 views

Bases in Hilbert Space 2019

Prove that the set of functions $C$ = { $f_{0}(x)= \sqrt{\frac{1}{\pi}}$, $f_{n}(x)= \sqrt{\frac{2}{\pi}}. \cos(nx)$, $n$ $\in$ $N$ } is an orthonormal basis in $L^{2}(0, \pi)$. Hint: Use the facts ...
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31 views

Using Sylvesters theoren in Quantum Mechanics

2-d system consists of $N$ identical cells arranged linearly in series. The transfer matrix of a single cell is a unitary Hermitian 2x2 matrix with eigen values $Exp(\pm i\times\theta)$ Need to use ...
5
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1answer
56 views

Is the set $\{\delta_x\}_{x \in [a, \ b]}$ a basis for the set of distributions on $C^{\infty}_c([a, \ b])$?

$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1}\def\ket#1{#1\rangle}$ Is the set $\{\delta_x\}_{x \in [a, b]}$ a basis for the set of distributions on $C^{\infty}_c([a, \ b])$? Below are ...
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17 views

Commutator of the Hamiltonian and Parity Operator - evaluation of derivatives

I was studying the commutator of the Hamiltonian and parity operators in the $L^2$ space from Quantum mechanics and came upon the following: To show that the two operators commute, assuming we have a ...
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2answers
54 views

Delta distribution and the Schrödinger equation

While studying the lecture notes of my quantum mechanics course I came across something that seemed a bit odd. There we want to solve the Schrödinger equation for the potential $V(x)=V_0 \delta(x)$, ...
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1answer
68 views

Substitution - mistake

Where do I mistake please? My computation differ from the result in the text about red terms. Thank you
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1answer
69 views

Is it possible to take square root of the operator $e^{a\frac{d}{dx}}$?

Is it possible to take square root of the operator $e^{a\frac{d}{dx}}$ where $a$ is a real or complex constant? Actually in physics one can take the square root of $(\Box +m^2)$ operator associated ...
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1answer
149 views

In the Physicists' definition of the path integral, does the result depend on the choice of partitions?

The standard definition of the path integral in Quantum Mechanics usually goes as follows: Let $[a,b]$ be one interval. Let $(P_n)$ be the sequence of partitions of $[a,b]$ given by $$P_n=\{t_0,\dots,...
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1answer
40 views

General relationship between Green's functions and propagators?

Given a linear operator $\hat{L}$ and $x \in \mathbb{R}^n$, a Green's function of $\hat{L}$ is a function that satisfies $$\hat{L}_xG(x,x^\prime) = \delta^{(n)}(x-x^\prime)$$ A "propagator" $G(x,x^\...
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0answers
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Technical operator theory question on Albeverio's “Solvable Models in quantum mechanics”

I'm currently studying S. Albeverio's book "Solvable models in quantum mechanics" where some technical things are used that I don't fully understand. It is a general technical operator theory question,...
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1answer
23 views

Occupation (or particle) number operator. Eigenvalues and eigenvectors. [closed]

https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf Help me please. I made a screen (below) from the article above and highlighted what I did not understand. Why is it true? ...
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How this definition of functional integral leads to the Wiener measure?

In Quantum Mechanics and Quantum Field Theory texts, one encounters a definition of functional integral in terms of a limiting process associated to a time-slicing procedure. This is the way I've ...
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0answers
60 views

Quantization of a Lie algebra of symmetries

In quantum mechanics we have quantization map $Q$ that maps classical observables to quantum observables. If symplectic manifold $(M, \omega)$ is the phase space of a classical system then classical ...
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0answers
32 views

Is this a “good enough” statement of Wigner's theorem from Quantum Mechanics?

I'm a fourth year physics and math student who is writing up a report on some quantum mechanical symmetries and their consequences. The "audience" for my paper are either senior physics majors or ...
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1answer
26 views

Are the SO(3) defining representation and Spin 1 the same?

Since the equation $$\langle j,m^\prime|J_\pm|j,m\rangle =\sqrt{(j\mp m)(j\pm m+1)} \delta_{m^\prime,m+1}$$ holds both for spin 1 (the $\underline{1}$ rep for SU(2)) and angular momentum (SO(3)). Does ...
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0answers
39 views

Integral which involves two Hermit polynomials (Quantum forced harmonic oscillator)

I'm trying to understand the derivation of a formula which gives the probability of a quantum forced harmonic oscillator to transit to the state $n$, at instant $t$, if it was at state $m$, at instant ...
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17 views

Analogy of Exponential Map for Jordan Algebras

Today I gave a talk about this paper that constructs a Jordan algebra (more precisely, a JB algebra) to model (bounded) physical observables. It cites this paper, that proves that every JB algebra $A$ ...
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Weak convergence in $H^1(\mathbb R^3)$ implies convergence of integrals

Suppose that $f_n \rightharpoonup f$ in $H^1(\mathbb R^3)$ (weak convergence). Then $$\int_{\mathbb R^3} \frac{\lvert f_n(x) \rvert^2}{\lvert x \rvert} dx \stackrel{n\to \infty}{\longrightarrow} \int_{...
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Gauge invariance of the Hamiltonian for particle on external electric field

Let us assume we consider a problem of free electrons in an external electric field $$\hat{H}(\mathbf{r})=-\frac{\hbar^2}{2m}\nabla^2-e\Phi(\mathbf{r},t),$$ where $-\nabla \Phi(\mathbf{r},t) =E\...
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2answers
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Is this differential operator Hermitian?

The operator is $$\hat{A} = -i \left(x \frac{d}{dx} + \frac{1}{2} \right).$$ Is it true that $$\langle \hat{A} \psi_1(x)|\psi_2(x)\rangle = \langle \psi_1(x)|\hat{A}\psi_2(x)\rangle\ ?$$ Here, $\...
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1answer
62 views

Quantum Mechanics Book [closed]

I'm a postgraduate student and I'm looking for a Quantum mechanics book which has a great level of detail. Does anyone have any recommendations? Thanks.
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1answer
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Qualities of a discontinuous function that can be written as a finite/infinite sum of continuous functions

My question is given a discontinuous function, can it be written as an infinite or finite sum if continuous functions. Here, “continuity” is of course relative to a point in the domain. Also, do the ...
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0answers
22 views

sympy - manipulate expressions containing commutator

Let's consider this simple example: a particle in a one-dimensional parabolic potential. In the following, $p$ is the particle momentum operator, $m$ is the mass, $x$ is the position operator, and $\...
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2answers
39 views

Proving that every diagonalizable operator is normal

This question is insipired by the proof of the Spectral Theorem in Nielsen and Chuang's book (page 72). It says: Theorem: Any normal operator $M$ on a vector space $V$ is diagonal with respect to ...
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20 views

Show the expectation of the spin component $\langle \hat S_x \rangle=\hbar\,a_1a_2\cos\left(\frac{2\mu_BB\, t}{\hbar}\right)$ for real $a_1$ & $a_2$

Using $$\langle \hat S_x \rangle ={\chi_{s}}^{\dagger}(t) \hat S_x \chi_s(t)=\frac{\hbar}{2}{\chi_{s}}^{\dagger}(t)\sigma_x\chi_s(t)\tag{1}$$ where the time dependent $2$ - component spin vector is $$\...
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1answer
26 views

If $\hat O f(x) = f(x^2+1)$, is $\hat O$ a linear operator?

If $\hat O f(x) = f(x^2+1)$, is $\hat O$ a linear operator? It seems this follows the condition being linear operator. But think in following way; $\hat O f(x) = f(x^2+1) = g(x) \Rightarrow$ $\hat O ...
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0answers
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Domain issues with Weyl quantization

Most algebras of observables from quantum mechanics are closed. For example, fix a separable Hilbert space $H$, and consider the algebra of bounded operators on it. This is a Banach space. In ...
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1answer
34 views

Integrating with 3 products (Quantum Mechanics)

$$ \int_{-\infty}^\infty\frac{1}{\alpha}(1-\cos\alpha\pi)e^{i\alpha x}\,\mathrm{d}\alpha $$ Solving this integral is part of solving for a Fourier transformation problem and I am stuck on this ...
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1answer
59 views

How do I prove that the angular momentum is a Hermitian operator?

Confirm that the operator $$\hat I_z= \left(\frac hi\right)\frac{d}{dφ},$$ where $\varphi$ is an angle, is Hermitian.
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1answer
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Quantum mechanics: total orthonormal sets & position/momentum space

Consider the position and momentum vector sets $$X= \text{{|x> | x $\in$ $R^3$}}$$ $$P=\text{{|p> | p $\in R^3$}}$$ By the assumption of quantum mechanics, both $X$ and $P$ are total ...
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2answers
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Definition of Adjoint Operator for Quantum Mechanics

While learning about adjoint operators for quantum mechanics, I encountered two definitions. The first definition is given by Shankar in The Principle of Quantum Mechanics: Given a ket $$ A\lvert ...
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2answers
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Finding the Eigenvectors from a matrix with known Eigenvalues

I have a matrix called Ω. This is the matix: $$\frac 1 2\begin{bmatrix}2 & 0 &0\\0 & 3 &-1 \\0 & -1 & 3\end{bmatrix}$$ It's eigenvalues are known (I have calculated them ...
3
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1answer
31 views

Reference request for quantum Teichmuller space

I would like to ask for some detailed reference for quantum Teichmuller theory, better in a mathematical taste. I read a little bit on Kashaev's or Chekhov and Fock's, but find that I need to fill ...
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0answers
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No embedded point spectra for (discrete) Schrodinger operators with compactly supported potential

Consider the lattice $\mathbb{Z}^d$ and let $H_0$ denote the (negative) Laplacian on $l^2(\mathbb{Z}^d)$ with spectrum $[-2d,2d]$. Suppose that I add a potential $q:\mathbb{Z}^d\rightarrow\mathbb{R}$ ...
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1answer
34 views

Matrix exponentials, computing the product of $\exp(-iBt)$ and $\exp(-iB^{-1}t)$.

Suppose I have some matrix exponential $U(t)=\exp(-iAt)$ where $t$ is some real valued number, $A$ is a hermitian matrix (so $U(t)$ is unitary) where $A=B+B^{-1}$ and $B$ is unitary. Because $B$ and $...
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1answer
26 views

proof of Delta dirac sifting [closed]

I need to proof this expression : $\int_{-\infty}^\infty \delta(x-a)f(x) \, dx=f(a) $ Starting with this one: $\int_{-\infty}^\infty \delta(x)f(x) \, dx=f(0) $ Thanks in advance
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0answers
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Quadratic Taylor coefficient for free by normalizing; when is second Taylor coefficient real?

Let $f: \mathbb{R} \to \mathbb{C}$ and suppose that we know $|f(\lambda)| = 1$ for all $\lambda$. Consider the Taylor series around $0$: $$ f(\lambda) = a + b\lambda + c\lambda^2 + \cdots. $$ Instead ...
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30 views

How to determine basis functions partners of $x$, $x^2$, and $yz$ in $D_6$?

I have a character table for $D_6$ and I am trying to understand how to find the partners of basis functions. I am starting with $x$, $x^2$ and $yz$. I am currently working on the $x$ function but am ...
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2answers
132 views

Why is the magnetic Schrödinger operator positive?

In the book 'Schrödinger Operators' by Cycon et al. they prove that the magnetic Schrödinger operator (as well as the Pauli operator) have essential spectrum $\sigma_{ess} = [0,\infty)$ if $B$ has ...
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1answer
27 views

How to express a Hermitian operator with a Unitary operator and a diagonal matrix?

I'm stuck in this problem, and is the last one!. I have a hermitian operator A with its eigen-everything and I have to prove that it can be writen as $UDU^+$ where U is a unitary transformation, $U^+$...
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1answer
24 views

Is square of probability $p^2$ is less than probability $p$?

suppose we have $N$ possible states in system. There is a probability $p_n$ that system is in state $|n\rangle$, and the sum of all probabilities is one. Is there any general rule in math or physics ...
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0answers
34 views

naive spectral inequality

Consider the operator $H_0=-\Delta$ acting on $L^2(\mathbb R^n)$. The spectrum of $H_0$ is $\sigma(H_0)=\sigma_{ac}(H_0)=[0,+\infty)$. I've been said that it is straightforward to show that if $I$ ...
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113 views

Has the Hamiltonian Path Integral Been Made Rigorous?

It is well known that the Lagrangian formulation of the path integral has been made rigorous, via the Wiener measure and/or the Trottier product formula. I haven't seen mathematicians discuss the ...
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1answer
37 views

Spherical basis tensors

I was studying spherical basis as part of a physics course, and I stumbled the spherical representation of tensors. In Wikipedia (https://en.wikipedia.org/wiki/Spherical_basis) I found that any ...
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1answer
15 views

Characteristics of three hermitian matrices

Consider three hermitian matrices $L$, $K$ and $X$ such that $[L,K]=0$, $[X,K]=0$, but $[L,X] \not =0$, What can we say about the characteristics of each of the matrices? I understand that since the ...
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1answer
27 views

Squaring the impulse operator

$$\hat p_r^2 = \hat p_r \hat p_r = (-i \hbar)^2\frac1 r \frac d {dr} r\frac1 r \frac d {dr} r=-\hbar^2\frac {d^2} {dr^2}r$$ This probably means: $$\hat p_r =-i\hbar \frac1 r \frac d {dr} r$$ I'm ...
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0answers
26 views

Why does “noncommutative probability” capture quantum probability?

In this article, Terry Tao states that non-commutative probability can be used for quantum probability. However, he then goes on to explain non-commutative probability without explaining how it ...