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.
2
votes
1answer
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 $\rho_{123}\geq 0$ (...
1
vote
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 ...
0
votes
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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
votes
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 ...
2
votes
0answers
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 ...
1
vote
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 $V(x)=V_0 \delta(x)$, ...
0
votes
1answer
68 views
Substitution - mistake
Where do I mistake please? My computation differ from the result in the text about red terms. Thank you
2
votes
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 ...
13
votes
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 $$P_n=\{t_0,\dots,...
0
votes
1answer
40 views
General relationship between Green's functions and propagators?
Given a linear operator $\hat{L}$ and $x \in \mathbb{R}^n$, a Green's function of $\hat{L}$ is a function that satisfies $$\hat{L}_xG(x,x^\prime) = \delta^{(n)}(x-x^\prime)$$
A "propagator" $G(x,x^\...
3
votes
0answers
47 views
Technical operator theory question on Albeverio's “Solvable Models in quantum mechanics”
I'm currently studying S. Albeverio's book "Solvable models in quantum mechanics" where some technical things are used that I don't fully understand. It is a general technical operator theory question,...
0
votes
1answer
23 views
Occupation (or particle) number operator. Eigenvalues and eigenvectors. [closed]
https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf
Help me please. I made a screen (below) from the article above and highlighted what I did not understand.
Why is it true?
...
2
votes
0answers
30 views
How this definition of functional integral leads to the Wiener measure?
In Quantum Mechanics and Quantum Field Theory texts, one encounters a definition of functional integral in terms of a limiting process associated to a time-slicing procedure. This is the way I've ...
0
votes
0answers
60 views
Quantization of a Lie algebra of symmetries
In quantum mechanics we have quantization map $Q$ that maps classical observables to quantum observables. If symplectic manifold $(M, \omega)$ is the phase space of a classical system then classical ...
0
votes
0answers
32 views
Is this a “good enough” statement of Wigner's theorem from Quantum Mechanics?
I'm a fourth year physics and math student who is writing up a report on some quantum mechanical symmetries and their consequences. The "audience" for my paper are either senior physics majors or ...
0
votes
1answer
26 views
Are the SO(3) defining representation and Spin 1 the same?
Since the equation
$$\langle j,m^\prime|J_\pm|j,m\rangle =\sqrt{(j\mp m)(j\pm m+1)} \delta_{m^\prime,m+1}$$
holds both for spin 1 (the $\underline{1}$ rep for SU(2)) and angular momentum (SO(3)). Does ...
0
votes
0answers
39 views
Integral which involves two Hermit polynomials (Quantum forced harmonic oscillator)
I'm trying to understand the derivation of a formula which gives the probability of a quantum forced harmonic oscillator to transit to the state $n$, at instant $t$, if it was at state $m$, at instant ...
0
votes
0answers
17 views
Analogy of Exponential Map for Jordan Algebras
Today I gave a talk about this paper that constructs a Jordan algebra (more precisely, a JB algebra) to model (bounded) physical observables.
It cites this paper, that proves that every JB algebra $A$ ...
3
votes
0answers
27 views
Weak convergence in $H^1(\mathbb R^3)$ implies convergence of integrals
Suppose that $f_n \rightharpoonup f$ in $H^1(\mathbb R^3)$ (weak convergence). Then $$\int_{\mathbb R^3} \frac{\lvert f_n(x) \rvert^2}{\lvert x \rvert} dx \stackrel{n\to \infty}{\longrightarrow} \int_{...
0
votes
0answers
22 views
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\...
3
votes
2answers
59 views
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, $\...
2
votes
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.
1
vote
1answer
13 views
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 ...
0
votes
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 $\...
0
votes
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 ...
0
votes
0answers
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
$$\...
1
vote
1answer
26 views
If $\hat O f(x) = f(x^2+1)$, is $\hat O$ a linear operator?
If $\hat O f(x) = f(x^2+1)$, is $\hat O$ a linear operator? It seems this follows the condition being linear operator. But think in following way;
$\hat O f(x) = f(x^2+1) = g(x) \Rightarrow$
$\hat O ...
3
votes
0answers
52 views
Domain issues with Weyl quantization
Most algebras of observables from quantum mechanics are closed. For example, fix a separable Hilbert space $H$, and consider the algebra of bounded operators on it. This is a Banach space.
In ...
0
votes
1answer
34 views
Integrating with 3 products (Quantum Mechanics)
$$
\int_{-\infty}^\infty\frac{1}{\alpha}(1-\cos\alpha\pi)e^{i\alpha x}\,\mathrm{d}\alpha
$$
Solving this integral is part of solving for a Fourier transformation problem and I am stuck on this ...
0
votes
1answer
59 views
How do I prove that the angular momentum is a Hermitian operator?
Confirm that the operator $$\hat I_z= \left(\frac hi\right)\frac{d}{dφ},$$ where $\varphi$ is an angle, is Hermitian.
2
votes
1answer
29 views
Quantum mechanics: total orthonormal sets & position/momentum space
Consider the position and momentum vector sets
$$X= \text{{|x> | x $\in$ $R^3$}}$$
$$P=\text{{|p> | p $\in R^3$}}$$
By the assumption of quantum mechanics, both $X$ and $P$ are total ...
1
vote
2answers
52 views
Definition of Adjoint Operator for Quantum Mechanics
While learning about adjoint operators for quantum mechanics, I encountered two definitions.
The first definition is given by Shankar in The Principle of Quantum Mechanics:
Given a ket
$$ A\lvert ...
0
votes
2answers
36 views
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
votes
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 ...
1
vote
0answers
47 views
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 $q:\mathbb{Z}^d\rightarrow\mathbb{R}$ ...
1
vote
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 $...
-1
votes
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
2
votes
0answers
18 views
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 ...
0
votes
0answers
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 ...
4
votes
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 ...
0
votes
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, $U^+$...
0
votes
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 ...
0
votes
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 $I$ ...
7
votes
0answers
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
