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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Point groups where the tensor square of two-dim. irreps (over $\mathcal{O}(3)$) does not contain a two-dim. irrep in its decomposition
Which are the point groups where the tensor square of a two dimensional irreducible representation does not decompose into a sum that contains a two-dimensional irreducible representation?
For ...
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0answers
19 views
Dual Space to a Lie Algebra. Problem from Takhtajan. Finding the center of a Poisson Algebra
So I am trying to solve Problem 2.19 from the book "Quantum mechanics for mathematicians" by Takhtajan.
The problem is the following:
Let $g$ be a finite-dimensional Lie Algebra with a Lie bracket $[,...
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0answers
15 views
Spectral decomposition of the resolvent map
Let $P_\Omega$ be a projection valued measure and let $R_A(z)=(A-z)^{-1}$ be the resolvent map.
It can be shown that $$R_A(z)=-\sum_{j=0}{\frac{A^j}{z^{j+1}}}$$
whenever this series is defined.
My ...
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0answers
25 views
Retrieving a function of many complex variables from its manifold of zeros
Physical background:
In the Bargmann-Segal representation, the states of bosonic quantum systems are holomorphic functions of $N$ complex variables where $N$ is the number of degrees of freedom. The ...
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0answers
29 views
Particle in a 3-D Box
Reading through my quantum mechanics book I've stumbled on a question any help would be great.
Suppose we had a 3-D square well i.e. $$V(x,y,z)=\begin{Bmatrix}
0 \ \text{if}\ 0\leq x \leq a \ , \ 0\...
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0answers
15 views
Quantum analog of a cell in classical phase space [closed]
While going through the "Ideal Gas" and "Gibbs paradox" in statistical mechanics, I found a line mentioning that "A cell in phase space in classical discussion is analogous to a state in Quantum ...
4
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1answer
37 views
Relationship between a metric and quantum mechanical matrices
Let $ds:=\sigma_x dx+ \sigma_y dy + \sigma_z dz$. Then squaring $ds$, we get
$$
ds^2=\sigma_x^2dx^2+ \sigma_y^2dy^2+\sigma_z^2dz^2 + (\sigma_x\sigma_y+ \sigma_y\sigma_x)dxdy + (\sigma_x\sigma_z+\...
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0answers
25 views
Operator identity in Quantum mechanics
The question is :
If an operator $\hat{A}$ follows the property that $\hat{A}.\hat{A}=\mathbb{I}$ , where $\mathbb{I}$ is the identity operator, then prove that: $\exp(\theta \hat{A})=\cosh (\...
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0answers
17 views
Sub-algebra such that Wigner (-Weyl) transform is a homomorphism
The Wigner distribution of an operator $A$ is given by
$$
W_A(x,p) :=\frac{1}{2\pi} \int_\mathbb{R} \! dy \, \langle x+y/2| A | x-y/2 \rangle \, e^{ipy},
$$
and associates a function in phase space ...
3
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1answer
55 views
Why is the definition of sin and cos in terms of exponentials is similar to the definition of $L_x$ & $L_y$ in terms of raising & lowering operators?
I noted a similarity outlined below:
The angular momentum operators in $x$ and $y$ direction can be written:
$$L_x=\frac{1}{2}(L_++L_-)$$
$$L_y=\frac{1}{2i}(L_+-L_-)$$
$cos(x)$ and $sin(x)$ can be ...
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1answer
25 views
Expressing Equations in Lagrange Subsidiary Form
If we consider the Hamiltonian for the simple harmonic oscillator given by,
$$H(p,x) = \frac{p^2}{2m}+\frac{kx^2}{2}$$
where $m$ is the mass, $k$ is the stiffness and $p$ is the momentum, then the ...
0
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1answer
29 views
Schrödinger's Equation with square potential
I have written some code to solve and plot the time independent Schrödinger's equation with potential x^2, which has a bound state with odd integral energy eigenvalues. My code plots the graphs up to ...
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2answers
58 views
How to generate a rotation matrix given an angular momentum matrix
In 3 dimensions, the total angular momentum (for $z$) matrix is given. It generates the rotation matrix around $z$ by $e^{-i\theta J_3/h}.$ My question is how do we actually go about doing this? I ...
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1answer
19 views
eigenstates of linear operators and their powers
In algebra, it has come up a number of times that if $A$ is a linear operator, then, for any integer $k>0$, $A^k$ inherits the eigenvectors of $A$. This is a very straightforward proof. However, $A$...
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0answers
20 views
essentially self adjointness of Laplacian with the inverse square singular potential.
Let define $H_c: L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ as a densly defined linear operator as follow:
$$ H_c u= \Delta u - \frac{c}{|x|^2} u$$
In reading a paper I encountered to the following ...
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1answer
26 views
Heisenberg Picture for 1D Simple Harmonic Oscillator
The Halmiltonian for 1D simple harmonic oscillator is
$$
H = \frac{1}{2m}(P^2 + m^2 \omega^2 X^2).
$$
Show that in the Heisenberg picture, the sum of expectation
$$
\langle X_{t+\pi/2\omega}^2 \...
4
votes
1answer
31 views
Definition of outer product
I am trying to understand the concept of outer product in quantum mechanics. I read "Quantum Computing explained" of David MacMahon.
I can understand the transition in (3.12):
$$(|\psi\rangle \...
1
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0answers
31 views
Rutherford Scattering - Annular Detector in the Far Field [closed]
I have been tasked to find the rate at which scattered electrons will be detected on an annular detector in the far-field. The exact question I'm working with is:
Suppose that 1keV electrons, ...
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3answers
32 views
When is maximising a definite integral the same as maximising the integrand?
When is maximising
$$\int_a^b f(x) \text{d}x$$
the same as maximising $f(x)$?
Context: I was trying to find the most probable location of an electron in the ground state hydrogen atom, where the ...
3
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3answers
71 views
Reference request: representation focused view of quantum mechanics
I'm interested in learning a bit more about quantum mechanics from a more Lie algebra/representation theoretic perspective, hopefully including but not going past quantum field theory.
I was ...
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0answers
81 views
Quantum Info in math departments
I'm currently a graduate student in mathematics getting my master's degree. I am interested in a bit probability and partial differential equations, and, secondarily, a bit of mathematical physics; ...
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163 views
Is this physical model exactly solvable?
There exists a popular model in the Physics of heavy quark bound systems, called the Cornell potential model, in which the inter-quark potential is modeled to vary with radial distance $r$ as
$$V(r) ...
7
votes
3answers
211 views
Computing the product $(\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n$
I want to compute the product
$$
(\frac{d}{dx}+x)^n(-\frac{d}{dx}+x)^n,
$$
for a natural number $n$. For $n$ equal to 0 or 1, the computation is very simple obviously, but for such a low number as 2 ...
2
votes
1answer
82 views
Hilbert space inequality in series expansion coefficients
In the context of some quantum-mechanical problem, there occurs a series development which gives the dependence of an "energy-type" eigenvalue $\varepsilon$ on some parameter $Q$. $\Psi_i$ are ...
2
votes
1answer
30 views
CNOT quantum gate using tensor products
In textbook, it states that the CNOT gate with the X gate applied on second qubit is
\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&0&1\\
0&0&1&...
2
votes
1answer
47 views
Solving a differential equation for the propagation of Heller's Gaussian wave packet
I am an undergraduate student working on a research project in theoretical chemistry, but I figure this particular obstacle is primarily a mathematical one. I will begin with a pretty long context, ...
0
votes
1answer
30 views
Common eigenfunctions of 0 eigenvalue for the angular momentum components
I need to demonstrate (considering the natural representation in $L^2(S^2)$) that if two non zero vectors in $H^2(S^2)$ satisfy $J_a\phi=0$$ \forall a \in {1,2,3}$ (where $J_a$ are the components of ...
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0answers
34 views
Examples of prime ideals for Lie algebras
I am looking for examples of prime ideals for Lie algebras. In particular, I am interested in examples involving the Lie algebra given by the commutator of endomorphisms of a complex vector space and ...
2
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1answer
62 views
Does the Time Evolution Operator Satisfy the Schrodinger Equation?
I would just like to confirm my solution to the following question. I'm a bit hesitant on my solution because of a specific step. I would just like confirmation if that step, which I will point out, ...
0
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1answer
57 views
Clarification on notation from a probabilty/quantum mechanics question
I'm working through Stephen Barnett's book on quantum information and have come across the following question (1.5, for anyone keeping track at home)
A particle counter records counts with an ...
0
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1answer
64 views
Some questions on coherent states and corresponding Hilbert spaces. Mathematical formalism.
I have a few questions related to coherent states.
I'm trying to understand the topic using this source.
Does the set of all coherent states form a Hilbert space?
Or it forms the dense subset only? ...
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2answers
35 views
Time evolution of a finite dim. quantum system part 2
This is an extension to the previous post.
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 $\...
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Time evolution of a finite dim. quantum system
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 | ...
2
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0answers
34 views
Von Neumann ergodic theorem for purely continuous spectrum
$\newcommand{\1}{1\negthickspace{\mathrm{I}}}$
Von Neumann ergodic theorem states that, if $U(t)$ is a one-parameter group of unitaries acting on a Hilbert space $\mathcal{H}$, we have
$$
\lim_{T\to ...
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1answer
36 views
Need serious help with notation representing matrices as “scalars”
Here, $\hat{S_{j}}$ are Spin-1/2 system spin matrices. For those who don't know, it's just a 2x2 matrix.
I am not familiar with representing matrices as "scalars" when doing multiplying with $e_{j}$,...
1
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1answer
48 views
Inequality relating the probabilities of a quantum state to the euclidean distance of states.
My professor has provided us with the following proposition (without proof).
I am trying to prove this. i'm having quite some trouble proving the first inequality, right under the first sentence. ...
1
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0answers
26 views
Probabilistic solution of Hydrogen atom PDE?
I know that the Feynman-Kac formula gives a representation of the solutions of PDEs of the form
$$ \partial_t u = Lu + Vu$$
for some differential operator $L$ and a bounded potential $V$.
From the ...
0
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0answers
21 views
Why does the position operator have continuous spectra?
I am new to QM and was wondering how it is deduced that the position operator has a continuous spectrum of eigenvalues and eigenvectors. In particular, is the equation $\hat{x}|x\rangle=x|x\rangle$ ...
3
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1answer
51 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$ (...
2
votes
1answer
33 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
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2answers
28 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 ...
0
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0answers
38 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
67 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
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0answers
45 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
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2answers
61 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
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1answer
69 views
Substitution - mistake
Where do I mistake please? My computation differ from the result in the text about red terms. Thank you
2
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1answer
73 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
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1answer
175 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
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1answer
48 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
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0answers
50 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,...
