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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1answer
35 views

Pauli Spin Operator General Rotation

I would like to calculate the Pauli spin operator rotation $$ U^{\dagger } \overset{\rightharpoonup }{\sigma } U$$ where $$\overset{\rightharpoonup }{\sigma }=\sigma _x \overset{\rightharpoonup ...
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1answer
51 views

Quantum Fourier Transform and roots of unity.

I need to find $QFT_{6}$ for the state quantum state $\frac{1}{\sqrt2}(|0\rangle + |3\rangle)$. I received a very sufficient answer recently on simplifying nth roots of unity, but I am having a lot of ...
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2answers
103 views

How does one simplify exponents for complex primitive nth roots of unity?

Let us define a complex primitive N-th root of unity, omega: $$ \omega = \cos(\theta) + i\sin(\theta) \\ = e^{\frac{2\pi}{N}} $$ By the definition of an nth root of unity, ω is the second solution to ...
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2answers
121 views

Geometric meaning of block-diagonalization of a matrix

some times we need to do block-diagonalization in favor of easy computation. For instance, for a matrix like this $$ \begin{bmatrix} A_{11} & A_{12} & A_{13} & 0 & 0 & 0\\ A_{21} ...
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0answers
43 views

How many projectors do two commuting self-adjoints have in their common spectral decomposition?

If $A$ and $B$ are two commuting observables on a Hilbert space of dimension $n$ say. So, $$A = \sum_{j \leq a} \lambda_j P_j $$ $$B = \sum_{i \leq b} \mu_i Q_i $$ $$I_n = \sum_{i \leq b} Q_j = ...
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1answer
64 views

Commutation of abstract o(3) generators and vectors.

I've been given the following problem, and I'm quite lost with it - any help would be fantastic!! Let $L_1$, $L_2$, and $L_3$ denote the abstract o(3) algebras. You are given that $\vec{A} = (A_1, ...
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3answers
235 views

Why does the Method of Successive Approximations for a Differential Equation work?

Time dependent perturbation theory in quantum mechanics is often derived using the Method of Successive Approximations for a Differential Equation. I have not seen an explanation or a more rigorous ...
2
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1answer
194 views

1D Schrodinger/Laplace equation via finite differences: incompatible eigenvalues

I need to solve a variant of the 1D Schrodinger's equation equation using finite differences, so I decided to play a little bit with the real-space representation of some operators. Using the ...
5
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1answer
192 views

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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1answer
41 views

Probability distribution of two party quantum states

I am going through a blog post written by Thomas Vidick. It states following three assumptions by Bell. Measurement independence (“free will”): the state $\lambda$ is independent of ${x,y}$ (since ...
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0answers
58 views

Quantum mechanics/Probability Question?

I have a 6 question homework from my Quantum Mechanics Class and I solved most of it (or at least attempted most of it). This one however is tripping me up. Any help would be appreciated. A 3D ...
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1answer
60 views

What is the notation for separable states or independent variables?

Is there any specific notation that two quantum states are separable or that two random variables are independent?
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1answer
127 views

Simon's algorithm for n = 3

The Simon's problem is as follows: Suppose we are given a function $f : \{0, 1\}^n \to \{0, 1\}^m$, with $m \ge n$, and we are promised that either $f$ is 1-to-1, or there exists a non-trivial s such ...
2
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1answer
229 views

Arbitrarily using Sin and Cos as eigenfunctions of a Hamiltonian?

In the context of quantum optics, the rotating wave Hamiltonian can be written: $\hbar\begin{pmatrix} -\Delta & \Omega/2\\ \Omega/2 & 0 \end{pmatrix}$ The eigenvalues can then be calculated ...
4
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1answer
217 views

Uniqueness of solutions to Schrödinger's equation

Consider \begin{cases} u_t(x,t)=\sqrt{-1} u_{xx}(x,t), \quad (x,t)\in[0,2\pi]\times[0,\infty)\\ u(x,0)=f(x),\quad x\in[0,2\pi] \end{cases} where $f(\cdot) \in C^\infty$ is periodic with period ...
6
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1answer
128 views

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 ...
3
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1answer
143 views

To show a set of projectors summing to the identity implies mutually orthogonal projectors

The general setting is the study of positive operator measures in quantum mechanics http://en.wikipedia.org/wiki/Positive_operator-valued_measure instead of the projector operator measures. Going ...
3
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1answer
176 views

A question about Moyal product

In Geometric quantization theory the Moyal product is one of main tools. We know Moyal product for the smooth functions $f$ and $g$ on $ℝ^{2n}$ takes the form $f\star g = fg + \sum_{n=1}^{\infty} ...
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4answers
358 views

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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1answer
63 views

Fourier transformations in Simon's quantum paper

I am reading this paper by Simon. This is one of the earliest quantum algorithm papers. In the paper he presents a routine starting at the end of page six. The first step runs a Fourier transformation ...
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0answers
37 views

Statement about density operators, proof feedback

I'm trying to solve the following exercise: Let $V,W$ be finite dimensional inner product spaces over $\mathbb{C}$. Show that for every $\psi \in V\otimes W$ with $\langle \psi, \psi ...
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0answers
116 views

Solving Schrodinger for harmonic oscillator(griffiths analytic method)

I was just getting into quantum mechanics. But I'm having a bit of trouble following griffiths for the analytic method... it goes like so The Schrodinger equation $$ ...
4
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1answer
418 views

Dirac Orthonormality Proof - Can't Make Sense of Complex Integral

I'm having trouble rationalizing a particular statement that is, surely, present in many quantum mechanics textbooks. The following statement comes from the orthnormalization condition for ...
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1answer
58 views

pre-quantization on cotangent bundle $T^*M$

Let M be a manifold. Is there any pre-quantization on cotangent bundle $T^*M$?. In which case this pre-quantization is unique?
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2answers
173 views

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just ...
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0answers
80 views

The intuition behind a matrix of a Hamiltonian?

We have derived an elegant partition function for a problem which resembles a quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: ...
3
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4answers
224 views

Which book to read on quantum-related mathematics

Recently I watched the "Big Bang Theory" and decided to google about quantum mechanics. It really intrigued me. But I also understood that I am too stupid to understand even the basic mathematics in ...
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0answers
145 views

How to Diagonalize an Extremely Large Sparse Matrix in SLEPc/PETSc

Dear Friends, Recently I have started with learning SLEPc/PETSc, but I didn't find a way to solve my problem. I have to solve a big sparse matrix which is a two dimensional quantum ...
6
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3answers
111 views

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 ...
0
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1answer
37 views

Determinant Expression for Grover operator

I want find the characteristic polynomial for the grover quantum operator $U^{N\times N}$ $$\begin{align*} U=(2|D\rangle\langle D| -I_N)(2|M^{\perp}\rangle\langle M^{\perp}| -I) \end{align*}$$ ...
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1answer
67 views

Grover Algorithm Orthogonal vectors

I'm study the Grover algorithm. About this picture my lecture say that the expression $|s'\rangle$ in the computational basis is $$|s'\rangle = \dfrac{1}{\sqrt{N-1}}\sum_{x\neq w}|x\rangle ...
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1answer
32 views

Expecation for tensor products

We are told that $$Z|0 \rangle = | 0\rangle \\ Z | 1\rangle =-|1\rangle \\ X|0\rangle =|1\rangle \\ X|1\rangle =|0\rangle$$ and we have the state $$|\psi\rangle =|0\rangle |1\rangle +|1\rangle ...
2
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1answer
154 views

Left-ratio and right-ratio in (not necessarily commutative) field

I am reading Birkhoff–Neumann (1936) about (not necessarily commutative) field $F$. The authors use terms right-ratio and left-ratio in section 13. Right-ratio is denoted as $[x_1, x_2, ... ...
2
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0answers
53 views

To construct a Schrödinger wave with prescribed mean position and momentum

Let $\lambda_x, \lambda_\xi>0$ be fixed parameters which we will call position lengthscale and momentum lengthscale respectively. Fix a (position-)point $x_0\in \mathbb{R}$ and a (momentum-)point ...
7
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1answer
354 views

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 ...
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2answers
182 views

Can the $0$-norm represent determinism?

In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm. Call $\{v_1,\ldots,v_N\}$ a unit vector ...
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1answer
77 views

Dense in the special unitary group

I was going through the PhD thesis of Daniel Nagaj. In the last paragraph on page 30, there is this following sentence. A universal gate set must be dense in the group $SU(n)\ldots$ My question ...
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0answers
36 views

Demonstrate basic property of Hermitian

I want to demonstrate that $(A|v\rangle)^* = \langle v|A^*$ Given only this $(AB)^*=A^*B^*$ and $(|v\rangle, A|w\rangle) = (A^*|v\rangle, |w\rangle)$ Is it possible? The difficulty is that i don't ...
2
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0answers
291 views

Is quantum game theory reducible to classical game theory? [closed]

Modnote: This question was manually migrated (closed and crossposted) to MathOverflow by request of the OP. Quantum game theory is an extension of classical game theory to the quantum domain. It ...
4
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1answer
517 views

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 ...
5
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3answers
301 views

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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0answers
65 views

Pure Phase Number

I am read a solution (4.9) Here say: ... both $a, d$ are pure phases, so that it is always possible to find (non unique) real numbers $\alpha, \beta, \delta$ such that $a = ...
0
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1answer
72 views

Wave-function with boundary conditions

I'm given a complex wave-function $\psi(t, x)$, in one spatial dimension, which satisfies $i \partial_t \psi = \partial_x^2 \psi$, a simplified form of Schrodinger's equation in one spatial dimension. ...
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3answers
193 views

Quantum uncertainty can explain the Riemann Hypothesis?

In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
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2answers
926 views

Bessel function recursion relation

I'm reading a paper and the following set of radial equations is derived: $ -i \lbrack \partial_r + \frac{1}{r} \left( \frac{1}{2} - \nu \right) \rbrack u(r) = \pm k v(r) $ $ -i \lbrack \partial_r ...
2
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0answers
109 views

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
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1answer
37 views

Expectation value of pure state in quantum mechanics

It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle ...
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0answers
49 views

Doubt about the spectrum of an operator

I consider the Laplacian operator $$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
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0answers
34 views

Three body problem with point interactions

I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials). Is ...
2
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1answer
186 views

Tensor and Kronecker product

I have a follow exercise. Let $\left|\Psi\right\rangle = \tfrac{1}{\sqrt{2}}\left(\left|0\right\rangle + \left|1\right\rangle\right)$. Write out $\left|\Psi\right\rangle^{\otimes 2}$ xplicitly, both ...