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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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Why do odd dimensions and even dimensions behave differently?

It is well known that odd and even dimensions work differently. Waves propagation in odd dimensions is unlike propagation in even dimensions. A parity operator is a rotation in even dimensions, but ...
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Mathematics needed in the study of Quantum Physics

As a 12th grade student , I'm currently acquainted with single variable calculus, algebra, and geometry, obviously on a high school level. I tried taking a Quantum Physics course on coursera.com, but ...
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Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
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Quantum mechanics for mathematicians

I'm looking for books about quantum mechanics (or related fields) that are written for mathematicians or are more mathematically inclined. Of course, the field is very big so I'm in particular ...
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Quantum mechanical books for mathematicians

I'm a mathematician. I have good knowledge of superior analysis, distribution theory, Hilbert spaces, Sobolev spaces, and applications to PDE theory. I also have good knowledge of differential ...
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Prerequisite for Takhtajan's “Quantum Mechanics for Mathematicians”

I want to know the math that is required to read Quantum Mechanics for Mathematicians by Takhtajan. From the book preview on Google, I gather that algebra, topology, (differential) geometry and ...
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What is the essential difference between classical and quantum information geometry?

This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory. I have a ...
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1answer
689 views

Topological Quantum Field theories

I was wondering about the following on TQFTs. It is said that TQFTs have vanishing Hamiltonians $\hat{\mathcal{H}}$. Firstly, I would like to ask: Why is this so? Secondly, consider the ...
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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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Two Dirac delta functions in an integral?

For context, this is from a quantum mechanics lecture in which we were considering continuous eigenvalues of the position operator. Starting with the position eigenvalue equation, $$\hat{x}\,\phi(x_m, ...
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Dimensionality in quantum mechanics

I was reading Introduction to quantum mechanics by David J. Griffiths and came across following paragraph: $3$. The eigenvectors of a hermitian transformation span the space. As we have seen, ...
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1answer
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Why do physicists get away with thinking of the Dirac Delta functional as a function?

For instance they use it for finding solutions to things like Poisson's Equation, i.e. the method of Green's functions. Moreover in Quantum Mechanics, it's common practise to think of the delta ...
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Does this notion of morphism of noncommutative rings appear in the ring theory literature?

Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on ...
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1answer
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Quantization of angular momentum: is Dirac's proof wrong?

I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
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1answer
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Are GNS representations the way to build physical Hilbert spaces?

Consider a separable $C^*$ algebra $\mathcal A$. The space of states is also separable in the weak* topology, let $S$ be a countable dense subset. Denoting with $H_\omega$ the GNS representation of ...
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Orthogonal projections with $\sum P_i =I$, proving that $i\ne j \Rightarrow P_{j}P_{i}=0$

I am reading Introduction to Quantum Computing by Kaye, Laflamme, and Mosca. As an exercise, they write "Prove that if the operators $P_{i}$ satisfy $P_{i}^{*}=P_{i}$ and $P_{i}^{2}=P_{i}$ , then $P_{...
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Category Theory and Quantum Mechanics

I am wondering if particle interactions in quantum theory can be modeled as a morphism between $2$ categories. My reasoning is that since the states of particles are modeled as vectors in a Hilbert ...
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Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
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Studying quantum mechanics without physics background

I am a PhD math student, and I am wondering if I should study quantum mechanics while I don't have an undergrad background in physics. I posted this question in physics stackexchange, but there doesn'...
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Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.

I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces. I was wondering what is the physical ...
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1answer
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Derivation of Schrödinger's equation

I recall a famous quote of the late physicist Richard Feynman: Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger. This quote ...
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“Fun” question: anyone know why $e$ (Euler's Number) was chosen for wave functions?

First, let me say that this is merely something I have always wondered about, and can never seem to find a good reference for. I simply want to know... the geek in me. Why was $e$ (Euler's Number) ...
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Solving $y'' + (ax+b)y = 0$

This is a problem in quantum mechanics when one considers a linear potential; in physics-speak the equation would be written as $$\frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2}(E-ax)\psi = 0,$$ with $V(x)...
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1answer
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Relationship between dual space and adjoint of a linear operator

I am having a hard time understanding the concept of adjoint of a linear operator. Given a finite dimensional Hilbert space $H$ over a field $F$, I know the dual space is the vector space $H^*$ of all ...
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What are the conditions that ensure that a linear operator on a separable Hilbert space has a discrete spectrum and its eigenvectors form a basis?

I am particularly interesed in answers such as: symmetric, bounded, positive. self-adjoint and bounded essentialy self-adjoint and positive. (Those were invented) I'm interested in this question in ...
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Can epsilon be a matrix?

Question In the following expression can $\epsilon$ be a matrix? $$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + \...
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1answer
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A non-integrable representation of the Heisenberg Algebra

Let $\mathfrak h$ be the Heisenberg algebra in dimension 1, generated by vectors $P$, $Q$ and $I$ satisfying $[P,Q] = I$, $[P,I] = [Q,I] = 0$. A representation of $\mathfrak h$ on a Hilbert space $X$ ...
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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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Which Hilbert Space in Quantum Physics? [duplicate]

I'm a maths student who would also like to know a bit about quantum physics. I keep reading that possible states of a system are represented by elements of a certain Hilbert space $\mathcal{H}$. This ...
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How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done with this series: $$ \sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$ As hardmath says in his comment, the series ...
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3answers
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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 ...
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Applications of Operator Algebras to modern physics

I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics ...
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1answer
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'Quantum' approach to classical probability

Quantum mechanics defines a state of a system as a superposition of 'classical' states with complex coefficients, thus reducing many problems to linear algebra. Can classical statistics be approached ...
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2answers
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What is the difference between vector space and dual space?

I read that in Dirac notation, kets are elements of a vector space and bras are elements of the dual space. My question is, what is the difference between vector space and dual space, and why are bras ...
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1answer
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Do infinite dimensional Hermitian operators admit a complete basis of eigenvectors?

I'm currently taking a quantum mechanics course. We have proven that hermitian operators always have real eigenvalues, that we can choose the eigenvectors to be orthonormal, and that finite ...
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How to find interesting operators for a quantum system?

How can we find "interesting" operators for a quantum mechanical system? I can think of the following method: Given some system with an associated Hilbert space $V$ and Hamiltonian $H:V\rightarrow V$,...
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Baker Campbell Hausdorff formula for unbounded operators

Baker Campbell Hausdorff formula says that for elements $X,Y$ of a Lie algebra we have $$e^Xe^Y=\exp(X+Y+\frac12[X,Y]+...),$$ which for $[X,Y]$ being central reduces to $$e^Xe^Y=\exp(X+Y+\frac12[X,Y])....
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Do these two sets of matrices form groups?

Stimulated by some Physics backgrounds, consider the following two sets of matrices. Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ]\overset{\...
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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) ...
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Tridiagonal matrix w/trigonometric eigenvalues

Let $n$ be a natural number and $B$ be the $n\times n$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots & 0 \...
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Quantum Mathematics? [closed]

As per my last question, this has less to do with cold, hard, and fast calculations and more to do with the interplay between mathematics and philosophy...but as armchair philosophers aren't as hard ...
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How does one define the complex distribution $1/z$?

I have read the following formula in a quantum mechanics book, supposedly attributed to Dirac $$ \lim_{y\,\searrow\, 0} \frac 1 {x+iy} = \operatorname{p.v.} \left(\frac 1 x\right) - i \pi\delta (x)$$ ...
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Example for projective geometry used in quantum mechanics

In the book The Road to Reality by Roger Penrose, projective geometry as developed during the Renaissance is framed as (eventually) playing a pivotal role in quantum mechanics. (In fact, Penrose seems ...
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Quantum Mechanics state space

In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space $L^2(\mathbb{R}^3)$. On the other hand, on the book I'm ...
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1answer
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Why are Fokker–Planck equation and Feynman path integral formalisms equivalent?

Feynman path integral is equivalent to Fokker–Planck equation. This is mentioned here, but it's not clear why. (This page says Schrodinger equation is also equivalent to Fokker–Planck equation which ...
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A Weird Contradiction about angular momentum operator in quantum mechanics

I am starting with the standard definition of an angular momentum operator in quantum mechanics given as $$\mathbf{L} = k(\mathbf{r}\times\mathbf{p}) = k(\mathbf{r}\times\nabla),$$ where $k=-\mathbf{...
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1answer
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Derivative of a bra?

I understand that $$ \frac{\mathrm d}{\mathrm dt} \langle\psi|\psi\rangle =\left[\frac{\mathrm d}{\mathrm dt} \langle\psi|\right]|\psi\rangle + \langle\psi|\left[\frac{\mathrm d}{\mathrm dt}|\psi\...
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1answer
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Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
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1answer
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Dirac Gamma matrix identity

In my library's (old -- 1996) copy of Peskin and Schroeder, there's an identity I'm struggling to prove. In my copy it occurs on page 42, between equations 3.28 and 3.29, but I don't know how well ...
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Application of Schwartz Kernel Theorem to Quantum Mechanics

I am currently reading Quantum Mechanics for Mathematicians and have a question about a statement made in the book: Remark. By the Schwartz kernel theorem, the operator B can be represented by an ...