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.
6
votes
0answers
91 views
Is the failure of $\mathcal{B}(H)\simeq H\otimes H^*$ in infinite dimensions the reason for non-normal states in quantum information?
In the algebraic formulation of quantum physics/information, states $\omega: \mathcal{A}\rightarrow \mathbb{C}$ are defined as linear functionals on a $C^*$-algebra $\mathcal{A}$ (algebra of ...
0
votes
0answers
27 views
Show that don't exist a linear representation for the Galilei's group. [on hold]
Let G Galilei's group and R a projective representation, show that don't exist $\phi : G\longrightarrow \mathbb(C) $ such that $V: G \longrightarrow U(H) $, $V(g)=\phi(g) R(g) $ such that $V(g_{1}g_{2}...
3
votes
2answers
62 views
Schrödinger equation involving the Dirac-Delta
I am taking a course on quantum mechanics and I try to understand the time-independent Schrödinger-equation with the Delta-potential:
$$\frac{-\hslash^2}{2m}\psi''(x)-V_0\delta(x)\psi(x)=E\psi(x)$$
...
1
vote
1answer
89 views
Why can linear operators have $[A,B]=I$ in infinite dimensions when it is impossible in finite dimensions?
For any $n\times n$ matrices $A,B$, we have $\text{Tr}(AB)=\text{Tr}(BA)$. This can be easily proved by algebraic summation. Thus $\text{Tr}([A,B])=\text{Tr}(AB-BA)=0$. Also, $\text{Tr}(kI)=kn$. ...
1
vote
0answers
25 views
Ehrenfest Theorem and correlation among observables at the quantum scale
I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
1
vote
2answers
33 views
Is $|a_1\rangle\langle a_2|\otimes |b_1\rangle\langle b_2| = |a_1b_1\rangle \langle a_2b_2|$? Why or why not?
I'll be using the Dirac bra-ket notation. Is $|a_1\rangle\langle a_2|\otimes |b_1\rangle\langle b_2|$ the same as $(|a_1\rangle\otimes|b_1\rangle)(\langle a_2|\otimes \langle b_2|)$? I'm not sure ...
0
votes
1answer
42 views
Confusion regarding $|A|^2 \int\limits_{-\infty}^\infty e^{i(p-p')x/h}=|A|^2 2\pi h\delta(p-p')$
The book "Introduction to Quantum Mechanics (Second edition)" by Griffiths says the following on pg 103:
$$|A|^2 \int\limits_{-\infty}^\infty e^{i(p-p')x/h}=|A|^2 2\pi h\delta(p-p')$$
Here $\delta$...
3
votes
0answers
33 views
Hamiltonian Flows and Heisenberg picture of quantum mechanics
I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial)
In particular what I know is that:
...
0
votes
0answers
41 views
Coarsenings In Deutsch Et Al's Constructor Theory
Disclaimer: I posted a questions on constructor theory here a few days ago but received two closing votes, I guess because it consisted of several subquestions, so I deleted it and now try to focus on ...
2
votes
1answer
47 views
What are the automorphisms on the strucuture consisting of the nonzero vectors of a Hilbert space with the orthogonality relation?
Let $V$ be an infinite-dimensional complex Hilbert space. With this space we can associate a relational structure $V^+ = (V^+, \bot)$, where $V^+$ is the set of non-zero vectors in $V$, and $\bot$ ...
3
votes
1answer
88 views
When is $\exp(-iHt)$ well-defined?
If $H$ is a linear operator, what restrictions should be put on $H$ in order for $\exp(-iHt)$ to be well defined?
How do you define $\exp(-iHt)$ when $H$ is infinite-dimensional? (If it is possible)
0
votes
1answer
17 views
Orthonormality of Hermite function
I was wondering if someone could tell me when the following relation holds?
where $H_{n}(x)$ are Hermite polynomials and $\delta(x-x')$ is Dirac delta function:
$$
\sum_{n=0}^\infty
\frac{1}{\sqrt{\pi}...
0
votes
1answer
51 views
Lifting representation Heisenberg algebra
I (think) I've found the Heisenberg Lie algebra representation through quantization. Where we have $q \mapsto q$ and $p \mapsto -i \hbar \frac{\partial}{\partial q}$.
So this is only a Lie algebra ...
1
vote
1answer
27 views
Fundamental property of Green's function is violated
$$\langle x| D|x'\rangle=D_x\langle x|x'\rangle=D_x\,\delta(x-x')$$
$DD^{-1} = I$ , Where '$I$ ' represents the Identity
position representation of this equation is
$\langle x|D|x'\rangle \langle ...
1
vote
0answers
25 views
How boundary conditions for a green function is same a original function
I am trying to study green's function.suppose we have a differential equation,
$D_x f(x) =g(x)$ , We also provided with some boundary conditions for $f(x)$, But we supply this boundary conditions in ...
0
votes
0answers
19 views
Green function derived from the idea of position representation in quantum mechanics
I am trying to study green's function by using a lecture by NPTEL its link is
https://www.youtube.com/watch?v=ZJ7v6VZQ32k&t=439s
I don't get this step by him at the minute of ten.
$D_x G(x,x^{'}...
0
votes
1answer
52 views
Position representation of an operator
$$\langle x| M|x'\rangle=M(x)\langle x|x'\rangle=M(x)\,\delta(x-x')$$
I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general operator $M $?
$\...
4
votes
0answers
81 views
Quantum representation of a system of identical particles
I'm studying mathematics and I began a course in quantum statistics, in which I got to the discussion related to indistinguishibility of particles. My professor's notes are not very clear and ...
3
votes
2answers
38 views
Does adding a constant to the potential of Schödinger's equation always make the solution pick up a factor $e^{-iV_0t/\hbar}$?
I've been going over problem 1.8 of [Griffiths, Introduction to Quantum Mechanics, 2nd edition], where it asks to prove the following: Say $\Psi(x,t)$ satisfies the time-dependent Schrödinger equation
...
3
votes
1answer
35 views
Probability difference scattering potentials
Let $V_1(x)$ and $V_2(x)$ be two real potential functions of one space dimension, and
let $m$ be a positive constant. Suppose $V_1(x)\le V_2(x) \le 0$ for all $x$ and that
$V_1(x) = V_2(x) = 0$ for ...
0
votes
0answers
21 views
prequantization euclidiean space
In general the prequantum hilbert space of some manifold $M$ will be the sections of the complex line bundle. I ommitted a lot of details of course.
Locally, a section of a complex line bundle ...
0
votes
0answers
14 views
Discrete spectrum of potential with negative integral
I have a Schrödinger operator $H=-\frac{d^{2}}{dx^{2}}+V$. Let's assume that $V$ is compactly supported and that $\int_{-\infty}^{\infty}V(x)dx<0$. Does this guarantee that $\sigma_{p}(H)\subset [-\...
2
votes
1answer
27 views
How to find the value with which I need to divide a unitary matrix such that its first component lies within the range of cosine?
In quantum mechanics, we generally deal with unitary matrices. In the IBM cloud computers and Qiskit, the general $2\times 2$ unitary is defined as
$$U(\theta, \phi, \lambda) =
\begin{pmatrix}
\...
4
votes
2answers
103 views
Mathematically rigorous Quantum Mechanics
I am a student of mathematics attending a course in Quantum Mechanics. This course is held by a physicist, and it is really confusing for me to follow his reasonments. With this, I do not mean to be ...
0
votes
1answer
35 views
Is this the correct definition of a discretization of a functional?
Physicists define path integrals by limits of discretizations, instead of using measure theory and Wiener measures.
Now, the issue is that most texts do not give clear definitions, so it seems almost ...
-1
votes
1answer
23 views
Showing that any linear operator can be written as a sum of Hermitian matrices [duplicate]
Let $(V, \mathbb{C})$ be a complex - valued vector space. Let $A$ be any linear operator acting on this vector space. Suppose that $B = \{|v\rangle_{k}\}_{k=1}^{n}$ is a basis set for $(V, \mathbb{C})$...
2
votes
1answer
38 views
Is there a construction of the Wiener measure by discretization and limits which parallels the Physics ideas?
In Physics one constructs the path integral by a limiting process together with a discretization procedure.
Now, in order to better paralell with the Wiener measure, consider this in Euclidean ...
0
votes
1answer
22 views
How to properly define the “discretization of a functional”?
In the derivation of the path integral formulation of quantum mechanics, most Physics books end up finding the following (or similar) expression:
$$K(q',t';q,t)=\lim_{N\to \infty}\int\left[\prod_{k=1}...
0
votes
1answer
37 views
Momentum operator in the position basis
J.J Sakurai shows in the section of ' Momentum operator in the position basis' as
$P$$\lvert\alpha\rangle$=$\int dx^{'}\lvert\ x{'}\rangle\Bigl(-i{h\over 2\pi}$ $\partial\over\partial x{'}$$ \...
3
votes
0answers
49 views
Oscillations of an Energy Eigenstate
Energy eigenstates of a 1-dimensional particle are given by solutions to differential equations of the form
$$
\left(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right) \psi(x) = E\psi(x)
$$
where $V$ ...
1
vote
0answers
16 views
Different behavior of the Klein-Gordon equation
By Fourier transform, it is well known that the Klein-Gordon, wave and Schrödinger flows are $e^{it\sqrt{1-\Delta}}$, $e^{it \sqrt{-\Delta}}$ and $e^{it\Delta}$, respectively.
I saw a phrase "the ...
0
votes
0answers
47 views
Maximising linear function over a specific convex set of density matrices
All matrices being discussed in this question are density matrices, so they have the following properties:
Hermitian
Positive Semidifinite
Trace = 1
We are currently in the space of all 4*4 density ...
0
votes
0answers
25 views
quantization clarification
I'm wondering something small about the link between representations and quantization. For quantization you start with some phase space (symplectic manifold) $M$, and you have a classical observable $...
0
votes
1answer
35 views
Determine the matrix representation for an operator written as an outer product
Suppose $|v_{i} \rangle$ is an orthonormal basis for an inner product space $V$. What is the matrix representation for the operator $|v_{j}\rangle \langle v_{k}|$, with respect to the $|v_{i}\rangle$ ...
1
vote
1answer
51 views
Proving a vector state is pure
Let $\mathcal{M}$ be a until $*$-algebra of 3x3 complex matrices. We have the general form of a vector state $\omega_{\psi} : \mathcal{M} \to \mathbb{C}$ over $\mathbb{C}^3$ as given by
$$\omega_{\psi}...
1
vote
0answers
29 views
Proofing equation containing time-ordering operator
Preparing for a presentation at university (I'm a Bachelor physics student) I have come across the formula below containg the time-ordering operator $T$. Although i have now understood the action of ...
0
votes
2answers
19 views
Why is the set of 2-norm preserving matrices richer than set of 4-norm preserving matrices
First, I seek a general characterization of 2-norm and 4-norm preserving matrices. Second, I seek to understand, using this characterization, why preserving the 2-norm can be described to have a far ...
0
votes
0answers
10 views
how can i determine the equations of the instantaneous helical axis relative
A point O of a rigid solid describes a plane O1X1Y1 a radius b of angular velocity ω const n. The body rotates around O by executing a regular precession motion (φ' = ν = const, θ = θ0 = const.). The ...
0
votes
1answer
19 views
Why is the spectrum of the hamiltonian for an infinite square well just a point spectrum?
Consider the Hamiltonian $H = -\Delta + V$ where $V$ is the potential conrresponding to an infinite square well:
$$V(x) = \begin{cases}0,&\text{if } 0, \leq x \leq L;\\\infty,&\text{otherwise}...
0
votes
0answers
11 views
Geometric phase in magnetic field
$\newcommand{\ket}[1]{\left|#1\right>}$
$\newcommand{\bra}[1]{\left<#1\right|}$
$\newcommand{\dv}[2]{\frac{d #1}{d#2}}$
$\newcommand{\braket}[2]{\left<#1\middle|#2\right>}$
Following ...
0
votes
2answers
43 views
Commuting Operators Have the Same Eigenvectors, but not Eigenvalues.
The following problem is out of my quantum mechanics textbook.
Assume that two operators $H$ and $\Gamma$ commute. Show that if $|\psi\rangle$ is a non-degenerate eigenvector of $H$, that is, $H|\...
2
votes
2answers
35 views
Does the Trotter Product Formula apply to the Simple Harmonic Oscillator?
We can write the Hamiltonian for the simple harmonic oscillator as $\hat{H} = \hat{T} + \hat{V}$ where $\hat{T} = \dfrac{-\hbar^2}{2 m} \dfrac{\partial^2}{\partial x^2}$ and $\hat{V} = \dfrac{1}{2} m \...
0
votes
0answers
63 views
Why does the “shooting” or “wag the dog” method give a bound state?
I am using numerical methods to solve Schrodingers equation. I have identified an interval for E (energy) in which one solution tends to infinity, but on the other side the other solution tends to ...
1
vote
1answer
78 views
Application of the operator $\exp\left( \alpha \frac{\partial^2}{\partial q^2}\right)$
I need to apply the operator
$$\exp\left( \alpha \frac{\partial^2}{\partial q^2}\right) \tag{1} \label{1}$$
To the function
$$M(x) N(y +C_{1}p)\mathcal{F}[f(q)](p) \tag{2} \label{2}$$
where $M(x)$ ...
-1
votes
1answer
28 views
exponentiating a matrix with complex elements
$$\exp(i\pi/4*
\begin{matrix}
0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0 \\
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
\end{matrix}
)$$
pretty ...
0
votes
0answers
16 views
Can embeddings be categorized into pre-embeddings and post-embeddings?
Consider a spin $1/2$ particle that moves on a two-dimensional spherical surface, and examine its orbital angular momentum in quantum mechanics.
Approach 1: Take the surface as a Riemann surface, ...
1
vote
0answers
40 views
Understanding the Following Integral Notation
I'm a little confused on the notation my professor used for the following integral.
\begin{equation}
\int \bar{Y}_{l_f}^{m_f} \left( \dfrac{-Y_1^1 + Y_1^{-1}}{\sqrt{2}}, \dfrac{iY_1^1 + iY_1^{-1}}{\...
2
votes
1answer
50 views
Deeper understanding of the adjoint of a linear operator
My undergraduate classes in Q.M describes the adjoint of a linear operator purely as a mathematical formality.
At this point, I'd like a deeper and heuristic understanding of it.
My questions are ...
1
vote
0answers
18 views
The action of tensor product over N terms on a ket.
Equation (6) of the paper titled, Multi-player and Multi-choice quantum game has left me puzzled-after many hours-as to how it is being derived. My working begins from the generic form seen just after ...
0
votes
0answers
12 views
Is this general form of Von Neumann's reduction postulate correct?
I have had a look at a book on 'Quantun Measurement' by Braginsky and Khalili$^1$. In it appears an equation that I would like confirmation of. The equation seems odd, in that it sets a probability ...
