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
24 views

Why the principal quantum number (n) is maximum 7? [closed]

Can you explain why actually the principal quantum number is maximum 7?
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0answers
26 views

Properties of Hermitian matrices

Consider hermitian matrices M1, M2, M3, M4 that obey the property Mi Mj + Mj Mi = 2δij I where I is the identity matrix and i,j=1,2,3,4 a) Show that the eigenvalues of Mi=+/- 1 (Hint: Go to the ...
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2answers
47 views

Compatible Hilbert space subspaces - need help understanding a statement made in a book

A book I'm reading has the following in a section on lattices formed by subspaces of a Hilbert space : Two subspaces $M$ and $N$ are compatible if there exist three mutually disjoint subspaces ...
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1answer
41 views

Double-commutator $[f,[f, - \Delta]] = -2 |\nabla f|^2.$

This book (proof of Theorem 3.2) in chapter 3.1 claims click me that an easy computation shows that $$[f,[f, - \Delta]] = -2 |\nabla f|^2.$$ where $[.,.]$ denotes the commutator. Unfortunately, I ...
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1answer
30 views

Perturbation theory, why are the assumptions of the method satisfied?

I am a undergrad student interested in math taking quantum mechanics. Yesterday I was introduced to what physicists call perturbation theory, non-degenerate case. According to authors Griffiths, ...
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1answer
32 views

Entanglement and linear algebra

How do you represent entanglement of two particles in quantum mechanics using linear algebra? How does measure of one particle affecting the state of other quantum mechanically captured linear ...
3
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0answers
49 views

Any closed form for this expression?$ \sum_{k=0,\,l=0}^{k=n,\,l=m}\frac{\lambda^{l+k}}{k!\,l!}\sqrt{\frac{n!\,m!}{(n-k)!(m-l)!}}\delta_{n-k,\,m-l}$

I am looking for a closed form of this expression. If you have seen something like this or remember something similar, please let me know. My sincere thank! $$ ...
2
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1answer
39 views

The distribution solution to $L^{+} u=0$ is u=0, where $L^{+}=-\frac{d}{dx}+x$?

Consider the creation operator $L^{+}=-\frac{d}{dx}+x$. If $u\in L^2(\mathbb{R})$ and is a distribution solution to the equation $L^{+}u=0$, then for any $\phi\in C_0^{\infty}(\mathbb{R})$ we have ...
2
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1answer
38 views

How to calculate this Delta Dirac Integral?

I'm trying to understand the result of this integral. $$\int_{-\infty}^a\delta(x-b)\,\mathrm dx$$ The book gives me the following answer. If $a>b$, $1$; If $a<b$, $0$; I'm not sure but ...
0
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1answer
25 views

Probability of a qubit state

Everything I've read says that you can get the probability of a qubit state by squaring the state's component in the amplitude vector. For instance $[1/\sqrt{2}, 1/\sqrt{2}]$ and $[1/\sqrt{2}, ...
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1answer
38 views

How to plot a qubit on the Bloch sphere?

I've been reading pages such as this one: http://comp.uark.edu/~jgeabana/blochapps/bloch.html Which talk about the Bloch sphere, but I've been unable to figure out how to plot states on the sphere ...
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0answers
22 views

Reference request for complex scalar field, propagators worked out with path integral approach? [closed]

In quantum field theory, the Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^*\phi.$$Can anyone supply me a reference to where the ...
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0answers
14 views

Prove that the Gram–Schmidt procedure produces an orthonormal basis for V . [duplicate]

Suppose |w1> , . . . , |wd> is a basis set for some vector space V with an inner product. Gram–Schmidt procedure, which can be used to produce an orthonormal basis set |v1>, . . . , |vd> for the ...
5
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2answers
88 views

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 ...
4
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0answers
64 views

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 ...
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1answer
69 views

What does $\operatorname{supp}(A)$ mean?

I'm looking at a paper (specifically this one). In the paper, we have a positive operator $A$, and the operator $\operatorname{supp}(A)$ is supposed to be a projection operator. Does anybody know ...
1
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1answer
31 views

eigenenergies when Hamiltonian is $\hat{H}^2$ − $\hat{H}$

If the eigenenergies of the Hamiltonian $\hat{H}$ are $E_n$ and the eigenfunctions are $\psi_n(r)$ , what are the eigenvalues and eigenfunctions of the operator $\hat{H}^2$ − $\hat{H}$ ? Attempted ...
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0answers
10 views

Proof of lineal closure

I'm looking for a proof that the subset of a Hilbert space containing all state vectors assigning probability 1 to some event E is a subspace, that is, is lineally closed. I have a proof of it that ...
4
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1answer
79 views

proving a quadratic form is closed

I'm trying to show that, given a spectral measure $d\mu_\psi(\lambda)$ for a self-adjont operator $A$, for the following quadratic form $$q_\lambda(\psi)=\int_{\mathbb ...
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0answers
16 views

Deutsch–Josza algorithm in Hilbert spaces

What would be the construction of Deutsch–Josza algorithm in Hilbert spaces, e.g. With white noise analysis?
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1answer
54 views

How to rigorously understand continuous bases?

In Quantum Mechanics it is quite common to see the idea of a continuous basis of a Hilbert space. In truth if $\mathcal{H}$ is the state space of a quantum system and if $X : U\subset \mathcal{H}\to ...
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0answers
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How is the uncertainty of fractional computation 4 events per 10 million?

I have observed a systematic four-event uncertainties in a series of particle physics computations among selection size of up to 10 000 000 events. Now, this term came up with the computation of the ...
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1answer
46 views

Spectral theorem for momentum operator

I'm trying to apply the spectral theorem to the explicit example of the momentum operator: \begin{equation} p:= i\frac{d}{dx} \end{equation} on the domain $D=\{\psi\in H^1(0,2\pi)\ |\ ...
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0answers
24 views

The probability of measuring the control qubit in zero in a quantum circuit

I’m working on an assignment where I have to solve some questions about a quantum circuit. In particular, I have a quantum circuit with three qubits: $|0\rangle$(referenced to as the control qubit), ...
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2answers
21 views

Approximation to series for quantum harmonic oscillator

For the function $$h(\xi) = C \sum \frac 1 {(j/2)!} \xi^j$$ Griffith's makes the following approximation at large $\xi$: $$h(\xi) = C \sum \frac 1 {(j/2)!} \xi^j \approx C \sum\frac 1 {j!} \xi^{2j} ...
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2answers
47 views

How to verify the completeness of a given Hilbert space?

Hilbert space is defined as a complete inner product space. It is also said that a finite dimensional vector space with inner product is trivially complete. I have two questions. How can I verify ...
4
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1answer
68 views

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
23 views

Simple example of a not strongly continuous operator on a Hilbert space?

Let $\cal H$ be a Hilbert space. Let $U(t)$ with $t\in \mathbf R$ be a one-parameter family of linear operators on $\cal H$. Strong continuity for $U(t)$ is defined as the condition that $$ \lim_{t\to ...
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0answers
25 views

Matrix whose eigenvectors are Hermite polynomials

I first constructed a symmetric matrix as the Laplacian operator, and its eigenvectors are a series of harmonics functions as expected. I programmed it and convinced myself. The matrix looks like: $$ ...
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0answers
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Covariance of nonlinear sde

My problem is to compute the covariance of the following Ito process $$ dX_t=AX_t+\sum_{k=1}^{n}B_kX_tdW_k, $$ where $A,B_k$ are nonlinear operators defined on a complex separable Hilbert space $H$. ...
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1answer
73 views

Show the first Chern class of a $U(1)$ bundle is integral.

I am working from John Baez's book: "Gauge Fields, Knots and Gravity". So I will stick to the notation used in that book. I am stuck at exercise 122 of part II (page 283), it reads: Show that if ...
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1answer
35 views

Showing that $HTH = e^{-i \frac{\pi}{8} \sigma_x}$ (quantum gates)

I'm trying to prove that an arbitrary single qubit unitary (read: unitary two by two matrix, and thus rotation up to a phase) can be composed from Hadamard and T gates, given by $ H = ...
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0answers
29 views

Construct bivariate symmetric (polynomial) nonnegative functions (distributions) over the unit square with certain properties

Construct bivariate symmetric polynomials $f(x,y) = f(y,x) \ge 0$ over $[0,1]^2$, with $f(1,y) = f(x,1)=0$, such that the univariate marginal distributions are both proportional to $$(1-u^2)^4$$, ...
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0answers
31 views

qubit measurment, i need to calculate the probabiliy

Let $$\operatorname {qubit} = \frac{1}{6^{0.5}}\times\big(\left.i\, \lvert 00\right\rangle + \left\lvert 10\right\rangle - 2\left\lvert 11\right\rangle\big)$$ If we measure only the second qubit in ...
2
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1answer
42 views

Commutation of a partial trace with an operator

Let the partial trace $\mathrm{tr}_B$ be a mapping from an endomorphism End$\left( H_A\otimes H_B \right)$ onto an endomorphism End$\left( H_A \right)$. Then the partial trace is defined as $$ ...
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1answer
110 views

quantum mechanics violate Bell's inequality

I have this function $$ \begin{aligned} F\big(\theta_a,\theta_b,\phi_a,\phi_b\big) = \ & – \big[\cos \theta_a \cos \theta_b \big] – \big[\sin\theta_a \sin\theta_b \sin\phi_a \sin\phi_b\big] \\ ...
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0answers
41 views

How to Construct $N$-dimensional Unitary Matrix Basis

Galitski's Exploring Quantum Mechanics says on its page 29, (There are $N^2$ linearly) independent Hermitian matrices of rank $N$. The number of independent unitary matrices is also $N^2$, since ...
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1answer
68 views

Quantum probability and quantum measure theory

Do quantum probability and free probability mean the same thing - that is, they deal with noncommutative random variables? What about quantum measure theory? Is quantum measure theory the foundation ...
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2answers
66 views

A question on use of square integrable functions

I'm approaching this from a physicist's perspective, so apologies for any inaccuracies (and lack of rigour). As far as I understand it, a square-integrable function $f(x)$ satisfies the condition ...
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0answers
33 views

Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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1answer
75 views

Projection of a 3D spherical function to a carteasian axis

I have a 3D function defined in a spherical coordinate system $(r,\theta,\phi)$, which is written as a product of a radial function $R_{nl}(r)$ and a spherical harmonic $Y_{lm}(\theta,\phi)$ I.e $$ ...
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1answer
59 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
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0answers
35 views

On block density matrices and CPTP maps

Consider a $n\times n$ density matrix $\rho$ and decompose it as $$ \rho=\left[\begin{array}{c|c} \rho_A & \rho_B \\ \hline \rho_B^\dagger & \rho_C\end{array}\right], $$ with $\rho_A$, ...
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0answers
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Questions to the virial theorem

Let $H = H_0 + V$ be the Hamiltonian of the single electron where $H_0 = - \Delta, V = - \frac{\gamma}{|x|}$. Now one defines the dilation group $U(s) \psi(x) = e^{-ns/2} \psi(e^{-s}x), s \in \mathbb ...
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A Question about Quantization and Partition Function

I have a question about quantization and partition function, which sound a little bit inappropriate. But I still want to ask for help. I think that quantization is a unitary representation of the ...
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0answers
66 views

Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let ...
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0answers
224 views

Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output.

I've been trying to solve the following Schrödinger equation numerically, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + ...
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1answer
39 views

x-momentum operator $p_x$ expressed as multiple of Translation operator

On this page https://en.wikipedia.org/wiki/Rotation_operator_%28quantum_mechanics%29 under "The translation operator," they use Taylor expansion. As part of that proof they state $p_x = ih * ...
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Why $|j_1 - j_2 | \leq j \leq j_1 +j_2 $ holds for $J= J_1 +J_2 $, the addition of angular moment?

I wonder why the total angular momentum $$J=J_1 +J_2 $$ is given in the range of $$ |j_1 - j_2 | \leq j \leq j_1 +j_2 $$ Of course we can verify this in the course of finding Clebsh-Gordan ...
5
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1answer
65 views

Schrödinger operator with delta (zero range) interaction.

I am reading the book of Albeverio named Solvable models in quantum mechanics. In the first chapter it is explained how to realize the operator $"-\Delta+\delta_0"$ as a self adjoint operator on ...