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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relationship between the CNOT gate and the I and X single-qubit gates

I need to prove this relationship: $G_{CNOT} = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1|\otimes X$. So I think I need to show that both sides are a linear map on $H \otimes H \otimes H$ So ...
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1answer
280 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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0answers
31 views

Quantization and evaluating a complex function on multiple Riemann sheets

In Lecture 3 of his mathematical physics course, Carl Bender mentions that the evaluation of a complex function on multiple Riemann sheets can be used to describe the quantization of the laws of ...
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0answers
56 views

Axiom of choice in proof of Wigner's theorem?

In Appendix A of chapter 2 of "The Quantum Theory of Fields," vol. 1, Weinberg presents a proof of Wigner's theorem: given a symmetry transformation $T$ of rays, one can extend this to a symmetry ...
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1answer
23 views

Is the spectrum of a product of two operators, $AB$, invariant under $UAU^{\dagger}$ for unitary $U$?

This question is about linear operators on a Hilbert space. If necessary, the Hilbert space can be assumed to be finite dimensional. I have two Hermitian operators, $A$ and $B$. Do we have $$ ...
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1answer
79 views

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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1answer
59 views

Problem with a wavefunction in Quantum Mechanics (math) (Book solution possibly wrong?)

Well there is a problem in my book which lists this problem: Calculate the probability that a particle will be found at $0.49L$ and $0.51L$ in a box of length $L$ when it has (a) $n = 1$. Take the ...
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0answers
103 views

Tensor product of affine space and algebra

When reading about Quantum Mechanics, I always feel a bit disappointed when physicists consider that (for example) the 3-dimensional position of a particle must be decomposed into 3 coordinates $x, y, ...
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1answer
37 views

Complex Projective Line

How can I go about showing that a collection of all states is the complex projective line $CP^1$? All I understand at the moment is that an element in $CP^1$ is of the form ...
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0answers
57 views

Does $Z_A$ exist such that $\exp(X+A) = \exp(X) Z_A$?

I am considering an exponential on the following form: $$\exp(X + A \otimes I_B),$$ where $X$ is a Hermitian operator on a tensor Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$, $A$ is a ...
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0answers
51 views

Proving commutation relation in Algebraic Bethe Ansatz

I have a problem with proving a certain commutation relation. For my Bachelor's thesis I give a more mathematically rigurous 'treatment' of a select set of chapters of a paper by L.D. Faddeev. Noting ...
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1answer
50 views

Hermite Differential Equation - Non-integer values of $\lambda$

The Hermite differential equation, given by : $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ has solutions of the $$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} ...
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2answers
27 views

Hermitian - Using the Spectral Theorem

Ok so I am trying to show that if A is normal and has real eigenvalues, then A is hermitian. It was suggested that I try using the spectral theorem. So if we assume A is normal and has real ...
3
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2answers
76 views

Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $dom(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ...
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0answers
30 views

Algebraic formulation of quantum mechanics and unbounded operators

Posted in the physic site: In AQFT one specifies the structure of the observables as a $C^*$-algebra. This seems to excludes algebras that don't have a norm, such as the Heisenberg algebra. ...
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0answers
28 views

Homeomorphism between the space of all Ashtekar connections and spacetime?

This is a question I've asked in physics.stackexchange: Excerpt from an essay of mine: Let $\Psi(\varsigma)$ be the wavefunction in the loop representation, where $\varsigma:[0,1]\to\mathcal{M}$, ...
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0answers
65 views

Continuity in a physical context

I'm currently trying to solve an exercise for my quantum mechanics class and have run into a bit of a jam: Suppose we have the following potential : $V(x) = 0$ if $x > |a/2|$ but $V(x) = V_0$ if ...
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1answer
41 views

Can a unitary matrix be constructed from any doubly stochastic matrix?

Here is a question that came up while I was thinking about the foundations of quantum mechanics: Consider a unitary $n\times n$ complex matrix $U$, with elements $u_{ij}$. We know that the rows and ...
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93 views

How to solve analytically or simplify this coupled system of ODEs?

I have a coupled system of ODEs: $$\cases{ i\frac{\text{d}y_1}{\text{d}t}=A f(t)y_2(t)+E_1 y_1(t)\\ i\frac{\text{d}y_2}{\text{d}t}=A f(t)y_1(t)+E_2 y_2(t) }\tag1$$ Here $f(t)$ is a periodic function ...
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1answer
47 views

Angular Momentum Operators: A quick question [closed]

Given $L_+|l,m\rangle \propto |l,m+1\rangle$ and $L_-|l,m\rangle \propto |l,m-1\rangle$ Why isn't it the case that $\langle l,m|L_+L_-|l,m\rangle = 1$? Perhaps naively, but I assumed $\langle ...
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0answers
47 views

Question arising from quantum mechanics concerning groups and symmetries

I'm trying to understand a calculation my professor did in my quantum mechanics script. Here it is: Each rotation $R \in O(3)$ induces a unitary transformation in $L^2(R^3)$, i.e. the space of square ...
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1answer
79 views

Reference request for differential geometry/quantum chaos text

I'm looking for a differential-geometry based exposition of chaos theory and quantum chaos. Ideally, it would start with the Hamiltonian formalism (on symplectic manifolds) and discuss as many of the ...
2
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1answer
104 views

Finite element method for the 'Particle-In-a-Box' problem in quantum mechanics

(Apologies in advance for the lengthy question, but it really is needed for a precise description of what I've done!) In suitable units, the 'Particle-in-a-box' problem is described by the following ...
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1answer
51 views

eigenfunctions of hamiltonian in 'natural units'

Let $H=0.5(p^2+q^2)$ be the hamiltonian in natural units. Let $f_{n}$ be the eigenfunctions of H. Show that $<f_{n},f_{m}>=1$ if n=m, and equal to 0 otherwise. Do this by using the ...
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2answers
436 views

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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1answer
258 views

Integrals involving Hermite Polynomials

Could you please tell me, How to evaluate this integral which involve hermite polynomials, $\int_{-\infty}^\infty e^{-ax^2}x^{2q}H_m(x)H_n(x)\,dx=?$ where $H_n$ is the $n$-th Hermite polynomial ...
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1answer
45 views

the split of two quantum dice

We need to find the probabilities of the sum and the difference of two quantum dice. What is the probability of their sum to be 2? it can be accomplished only when both dice are 1. the probability of ...
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0answers
81 views

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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0answers
30 views

Expectation value in a Quantum derivation

I'm reading a physics paper (John Bell's 1964 paper on the EPR paradox if anyone is physics-curious) and I'm having an issue following his derivation. It's the probability distribution stuff -not ...
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1answer
47 views

Quantum Mechanics- Quantum Zero Paradox

Im struggling with this question after doing other question on my example sheet. I'm struggling with showing both results in the question and help will be very much appreciated. Thanks
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1answer
29 views

Continous and Discrete basis, Multiplication of Density Matrix and Hamiltonian.

Suppose I have a wave function $\psi(x)$ in position basis. I can make a density function by simply multiplying $\psi(x)$ and its conjugate $\psi^*(x)$. If I operate the density matrix ...
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0answers
68 views

Physical dimension of a complex wave function.

If the probability density function $\rho$ is: $$\rho=\psi^{*}\psi=|\psi|^2$$ where $\psi$ is a complex wave function that is a solution to the Schrodinger equation, what are the physical dimensions ...
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4answers
58 views

Is raising a value to the second power the same as multiplying it it's complex conjugate?

I was watching a video on YouTube on Quantum Mechanics Concepts and saw that if you wanted to convert a probability amplitude to a probability, you square it. In the video he said that this was ...
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1answer
39 views

Quantum measurement - How does this commutator correspond to the following?

From the book Quantum Measurement by Vladimir B. Braginsky and Farid Ya.Khalili How do they go from 5.18 to 5.19?
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1answer
68 views

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 ...
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0answers
31 views

Oscillator solutions with regard to TDSE Eigenvalues

Taking a shot at a QM harmonic oscillator problem tonight. Consider a 1-D harmonic potential: $$ V(x) = \frac {m\omega^{2}x^{2}} {2} $$ solve for the gen. solution to the TDSE, $ \psi(x,t)$ utilizing ...
2
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1answer
67 views

Delta function proof in QM

I'm actually working with some QM problems at the moment but I've hit a wall with a delta potential involved. The problem asks me to verify that $$ \frac{d \phi_{x=0^{+}}}{dx} -\frac{d ...
3
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3answers
98 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 ...
3
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1answer
65 views

Bra-ket multiplication

I'm studying a little bit of bra-ket notation and I found this property: $$\langle n| H_1 H_2|m\rangle=\sum_{k} \langle n|H_1|k\rangle \langle k|H_2|m\rangle$$ Is this property true? Why? Thank you! ...
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1answer
149 views

Evaluate an Integral involving Gaussian divided by square root of a quartic polynomial

Could you please tell me, How to evaluate the integral, $\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{e^{-a(x^2+y^2)}}{\sqrt{k^2+\beta^2(x^2-y^2)^2}}dx~dy$ I already have obtained a series ...
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0answers
57 views

Quantum Mechanics Angular Momentum Bra-ket notation

Show that if the state $$ \rvert\gamma\rangle $$ Is real then the expectation value of each component of the angular momentum is zero. Does this imply theangular momentum is zero? My Work: $$ ...
2
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2answers
211 views

Expected Values of Operators in Quantum Mechanics

I've recently started an introductory course in Quantum Mechanics and I'm having some trouble understanding what the expectation of an operator is. I understand how we get the formula for the ...
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1answer
89 views

Derive an algorithm to determine convex combinations

Problem statement Given is the density matrix of a spin-1/2 system which was set up in a state of superposition $$ \varrho = \begin{pmatrix} \frac{3}{4} & \frac{1}{4} \\ \frac{1}{4} & ...
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1answer
87 views

Quantum Hermiticity Bra-Ket notation please

If $A$ and $B$ are Hermitian operators, show that $$C~:=~i[A,B]$$ is Hermitian too. My work: $$\begin{gather} C=i(AB-BA) \\ \langle\psi\rvert C\lvert\phi\rangle = i\langle\psi\rvert ...
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0answers
60 views

Asymptotic behavior of $L^2$ norm for increased matrix dimensions

I am playing with matrices which are linear combinations of identity matrix, Pauli spin matrices, $\sigma_x$ and $\sigma_z$ or their tensor products. For example, let the matrix be $H$. So, $H$ could ...
2
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0answers
66 views

Change of basis and spectral theorem

I've been having trouble with such a rudimentary problem. Let us define a matrix $A$: $$A = \begin{pmatrix} 3 & 0 & -i \\ 0 & 3 & 0 \\ i & 0 & 3 \end{pmatrix}$$ A is a 3 by 3 ...
2
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1answer
30 views

calculating degenerancy

Given a function of two positive integers $n_x^2+n_y^2$. $n_x^2+n_y^2=50$ has three combinations of $n_x$ and $n_y$ that result in $n_x^2+n_y^2=50$: $$n_x=7,n_y=1$$ $$n_x=5,n_y=5$$ $$n_x=1,n_y=7$$ ...
3
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1answer
46 views

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 ...
3
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1answer
94 views

Solving a PDE arising from physics

Is there a way to find an analytic solution to the following PDE? $i \partial _t \psi = - \gamma \partial _x ^2 \psi - c x $cos$(\omega t) \psi $, where $\psi (x,t)$ is defined (in $x$) on the ...
3
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1answer
103 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 ...