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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6
votes
1answer
126 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
votes
1answer
132 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
votes
1answer
170 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} ...
12
votes
4answers
346 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 ...
1
vote
1answer
62 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 ...
1
vote
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 ...
3
votes
0answers
115 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
votes
1answer
387 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 ...
0
votes
1answer
57 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?
1
vote
2answers
160 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 ...
0
votes
0answers
165 views

Schwinger's Oscillator model of Angular Momentum and irreducilbe representations

I don't quite understand the connection between the final result attained by Schwinger in his oscillator model for angular momentum. Can any one help me understand how we get an " irreducible ...
3
votes
0answers
76 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
votes
4answers
223 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 ...
1
vote
0answers
140 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
votes
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
votes
1answer
37 views

Determinant Expression for Grover operator

I want find the characteristic polynomial for the grover quantum operator $U^{N\times N}$ $$\begin{align*} U=(2|D\rangle\langle D| -I_N)(2|M^{\perp}\rangle\langle M^{\perp}| -I) \end{align*}$$ ...
1
vote
1answer
65 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 ...
1
vote
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
votes
1answer
153 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
votes
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
votes
1answer
350 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 ...
3
votes
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 ...
1
vote
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 ...
1
vote
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
votes
0answers
288 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
votes
1answer
498 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
votes
3answers
297 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 ...
1
vote
0answers
64 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
votes
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. ...
-1
votes
3answers
191 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 ...
1
vote
2answers
856 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
votes
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 ...
1
vote
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 ...
2
votes
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 $A$ ...
1
vote
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
votes
1answer
179 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 ...
1
vote
1answer
107 views

Translation of an article

I need to read this article "On the spectrum of an energy operator for atoms with fixed nuclei in subspaces corresponding to irriducible representations of permutation groups" authors:G.Zhislin, A. ...
1
vote
0answers
28 views

References for three body problems with Fermi statistic

I'm studying the three body problem with two fermions of unitary mass and another different particle. I need references of the HVZ theorem in this case. Is there someone who knows them?
3
votes
1answer
125 views

Is the partial trace congruent under a change of basis?

My intuition tells me that the partial trace should be congruent under a change of basis. That is, if I have some matrix $A$ in the space of linear operators acting on a joint hilbert space: $A \in ...
0
votes
1answer
59 views

Dirac's notation? (QM)

I have a question regarding Dirac's notation in quantum physics. As far as I understand: $\langle a|b\rangle=(a1^*,a2^*)*(b1,b2)^T$ But what does $\langle1/2,1/2|J|1/2,-1/2\rangle$ mean?
1
vote
0answers
71 views

Fourier transform of integral kernel of the free resolvent

The free resolvent in $\mathbb{R}^3$ has this rapresentation $$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$with $\Im \sqrt{z}>0$. Then its integral kernel is ...
4
votes
1answer
106 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 ...
-1
votes
1answer
95 views

Qubit state finding [closed]

Suppose we have two qubits in the state $x|00\rangle+y|11\rangle $. What is the resulting state of the second qubit in that case? Use and to denote and respectively.
2
votes
1answer
64 views

Is multiplying by a measurable function $V$ always self-adjoint?

There are a handful of results establishing conditions on the measurable real-valued function $V(x)$ under which the operator: $$-\Delta + V(x)$$ Is (essentially) self-adjoint on ...
3
votes
1answer
124 views

A strange quantum potential: $V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$

So I have a strange quantum potential I have been playing with: $$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$ where $\mu$ is the Möbius function. This is what ...
2
votes
1answer
203 views

Question on complex number calculation for transmission coefficient of finite potential well

This is actually in my quantum mechanics textbook (pure math question though), and I just cannot see why this equality is true. Any help would be greatly appreciated! Let $F$ and $A$ be nonzero ...
6
votes
1answer
256 views

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 ...
3
votes
1answer
32 views

Essential selfadjointness preserved under unitarily transfomration?

I am wondering if essential selfadjointness of an operator in a Hilbert space is preserved under unitarily transformations. In other terms: let $H,H'$ be two isomorphic Hilbert spaces, with an ...
4
votes
5answers
118 views

General solution of $\frac{d^{2}u}{d\rho^{2}}=\frac{l\left(l+1\right)}{\rho^{2}}u$

In his discussion of the radial wave function of hydrogen Griffiths (Introduction to Quantum Mechanics, 2nd ed, p.146) gives the general solution ...
1
vote
2answers
87 views

Show $e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$

How do you show that $$e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$$ where $\sigma_i$ are the Pauli matrices.