# Tagged Questions

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, ...
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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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### 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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### 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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### Takhtajan's “Quantum Mechanics for Mathematicians”

I want to know the math that is required to read Takhtajan's "Quantum Mechanics for Mathematicians". From the book preview on Google, I gather that algebra, topology, (differential) geometry and ...
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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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### 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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### 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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### 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 ...
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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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### Quantum Mathematics?

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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### 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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### 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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### Complementary text for mathematical Quantum Mechanics lectures

I'm looking for a text to complement Frederic Schuller's lectures on QM. His approach is very mathematical -- in fact it looks like the first 12 of 21 lectures are just about the mathematical ...
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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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### Regarding Ladder Operators and Quantum Harmonic Oscillators

When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator: Suppose that are two operators $L^{+}$ and $L^{-}$ and ...
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### Defining conditional quantum probability

My knowledge of quantum mechanics is very limited, but I will try to ask a purely mathematical question here. If there is a text or resource that explains this, I would definitely appreciate any ...
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### Physical interpretation of $q$-deformation

I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a ...
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### Why do we use square in measuring a qubit with probability?

A superposition is as follow: $$\vert\psi\vert = \alpha\vert 0\rangle + \beta\vert 1\rangle.$$ When we measure a qubit we get either the result 0, with probability $\vert \alpha\vert^{2},$ or ...
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### Spectral theorem in Quantum Mechanics

In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
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How does one derive, using, say, the Householder transformation $$R(r) = (I - nn^*)(r),$$ the reflection representation of a vector $$R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + yR(\... 0answers 100 views ### Standard deviation of a quantum walk? The standard deviation of a classical random walk with n steps is \sqrt n - Standard deviation of a random walk. I have read in many places that the standard deviation of a quantum walk n with a ... 3answers 330 views ### Unitary invariance Why is it that for any non-negative matrix M and unitary matrix U, we have$$\sqrt{UMU^\dagger}=U\sqrt{M}U^\dagger$$? This question has to do with Problem 2c from this sheet. I think I am ... 3answers 298 views ### Geometric Algebra/ Calculus for Physics I don't know if this would be a better question for physics.SE, but I'll try here first: There is at least one good book on classical mechanics using the geometric algebra/ calculus (GA): New ... 3answers 47 views ### Two operators X and Z in an infinite dimensional Hilbert space satisfying X^2=Z^2=I and \{X,Z\}= 0 I am seeking to extend the following theorem to the case of infinite dimensional Hilbert space: Suppose we have two Hermitian operators X and Z in a finite dimensional Hilbert space \mathcal H. ... 1answer 170 views ### If the expectation \langle v,Mv \rangle of an operator is 0 for all v is the operator 0? I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ... 2answers 197 views ### Basic Quantum Mechanics Concepts with Continuous Spectra The following are a couple excerpts of the first chapter of Sakurai and Napolitano, Modern Quantum Mechanics, 2nd edition: Prior to these formulas, the text discusses the fundamental mathematics ... 1answer 183 views ### General theory behind ladder operators To derive the representation of SO(3) one uses the ladder operator method. What is the theoretical basis for this method? Often the ladder operators are simply stated in the textbooks of quantum ... 1answer 80 views ### Lagrangian invariant under left and right multiplication by unitary matrices, slick way to see? Is there a slick way to see that the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G\partial_\mu G^{-1}),where $G$ is an $N \times N$ unitary matrix, is invariant under left and right ...
I don't quite understand how the probability language of sample spaces, $\sigma-$algebra, random variables, etc, fit into the quantum mechanics' formalism. To wit, we usually say that an observable ...