Questions tagged [quantum-mechanics]

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.

Filter by
Sorted by
Tagged with
12
votes
4answers
832 views

Orthogonal projections with $\sum P_i =I$, proving that $i\ne j \Rightarrow P_{j}P_{i}=0$

I am reading Introduction to Quantum Computing by Kaye, Laflamme, and Mosca. As an exercise, they write "Prove that if the operators $P_{i}$ satisfy $P_{i}^{*}=P_{i}$ and $P_{i}^{2}=P_{i}$ , then $P_{...
14
votes
4answers
2k views

Quantum mechanical books for mathematicians

I'm a mathematician. I have good knowledge of superior analysis, distribution theory, Hilbert spaces, Sobolev spaces, and applications to PDE theory. I also have good knowledge of differential ...
20
votes
4answers
1k 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 ...
4
votes
1answer
336 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 ...
20
votes
7answers
10k views

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 ...
17
votes
3answers
2k views

Quantum mechanics for mathematicians

I'm looking for books about quantum mechanics (or related fields) that are written for mathematicians or are more mathematically inclined. Of course, the field is very big so I'm in particular ...
5
votes
5answers
2k views

How to prove $e^{A \oplus B} = e^A \otimes e^B$ where $A$ and $B$ are matrices? (Kronecker operations)

How to prove that $e^{A \oplus B} = $$e^A \otimes e^B$? Here $A$ and $B$ are $n\times n$ and $m \times m$ matrices, $\otimes$ is the Kronecker product and $\oplus$ is the Kronecker sum: $$ A \oplus B =...
5
votes
2answers
880 views

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 ...
1
vote
3answers
103 views

Explanation of this integral

$$\int_{-\infty}^{\infty} e^{-\frac{i}{\hbar}(p-p')x} dx = 2\pi\hbar\delta(p-p')$$ I don't quite understand how this integration leads to the right hand side. Any explanation is appreciated.
4
votes
1answer
104 views

Tensor product in dual-space

I am a bid confused regarding the notation for tensor products when going into dual-space If $\left| \Psi \rangle \right. = \left| A \rangle \right. \left| B \rangle \right.$ is $ \left. \langle \Psi ...
29
votes
2answers
2k 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, but ...
15
votes
2answers
2k views

Prerequisite for Takhtajan's “Quantum Mechanics for Mathematicians”

I want to know the math that is required to read Quantum Mechanics for Mathematicians by Takhtajan. From the book preview on Google, I gather that algebra, topology, (differential) geometry and ...
4
votes
1answer
660 views

Probability and Quantum mechanics

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 ...
14
votes
0answers
312 views

What is the essential difference between classical and quantum information geometry?

This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory. I have a ...
13
votes
1answer
261 views

Why do physicists get away with thinking of the Dirac Delta functional as a function?

For instance they use it for finding solutions to things like Poisson's Equation, i.e. the method of Green's functions. Moreover in Quantum Mechanics, it's common practise to think of the delta ...
3
votes
0answers
3k views

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

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) + \sinh^2(z)}{(\...
5
votes
3answers
3k 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 ...
6
votes
2answers
145 views

How does one define the complex distribution $1/z$?

I have read the following formula in a quantum mechanics book, supposedly attributed to Dirac $$ \lim_{y\,\searrow\, 0} \frac 1 {x+iy} = \operatorname{p.v.} \left(\frac 1 x\right) - i \pi\delta (x)$$ ...
4
votes
0answers
93 views

Eigenvalue of an Euler product type operator?

Background Let us have the following orthonormal basis such that: $$ \langle m | n \rangle = \delta_{mn}$$ Consider the following operators defined as: $$ \hat 1 = | 1 \rangle \langle 1 | + | 2 \...
3
votes
2answers
776 views

Dirac Delta function identities

I'm using the definition of the one dimensional dirac-delta function, $\delta(x)$, being, $$\int_{-\infty}^{+\infty}\delta(x)f(x)dx = f(0) \tag1$$ and I'm doing a question which asks me to (via ...
2
votes
1answer
112 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 ...
9
votes
2answers
230 views

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 ...
3
votes
1answer
147 views

Geometric quantization: not understanding the curvature form and Weil's theorem

I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...
2
votes
1answer
101 views

Prove that complex exponentials $(\phi_n(x)=\frac{1}{\sqrt{a}}e^{i\frac{2\pi nx}{a}})_{n=0, \pm 1, \pm2,…}$ are complete and orthonormal on $[0,a]$

I want to prove that the complex exponentials $(\phi_n(x)=\frac{1}{\sqrt{a}}e^{i\frac{2\pi nx}{a}})_{n=0, \pm 1, \pm2,...}$ form a complete orthonormal system for the space of square-integrable ...
2
votes
0answers
131 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 ...
2
votes
1answer
664 views

Quantum translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: $$T_\...
2
votes
1answer
412 views

Eigenvalue equation and separable solutions

Recently, studying Quantum Mechanics I found a doubt regarding separable solutions and eigenvalue equations for differential operators. Suppose we are considering some space $\mathcal{H}$ of functions ...
2
votes
0answers
44 views

Quantum Harmonic Oscillator Eigenvalues Confusion

In standard PDE theory, one generates eigenvalues to Sturm-Liouville problems over a finite domain. So, for a wave equation, we have an infinite number of eigenvalues $𝜆_𝑛$ for a Dirichlet problem, ...
1
vote
1answer
266 views

Properties of the solution of the schrodinger equation

I'm considering the Free Schrodinger equation: $$ i\partial_{t}u+\Delta{u}=0,~\mathbb{R}\times\mathbb{R}$$ with $$ u(0,\cdot)=u_{0} \in C^{\infty}_{0}(\mathbb{R}).$$ The solution is given by: $$ u(t,x)...
4
votes
0answers
107 views

Evaluating Gibbs state in the second quantized formalism

First, let us fix our notation. If $A:\mathcal H \to \mathcal H$ is a linear operator on a single-particle Hilbert space $\mathcal H$, we can lift $A$ on the Fock space $\mathfrak F_{s/a}(\mathcal H)$ ...
4
votes
2answers
935 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
412 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!
2
votes
1answer
237 views

Simplex integral in connection with time ordered exponential

In Quantum Mechanics, one often defines the time ordered exponential like e.g. here. Now my question is how the factor of $N!$ arises. I know the simplex volume as the following integral: \begin{...
2
votes
0answers
130 views

Definition of irreducible representation requires the invariant subspace to be closed?

From Simon's "Operator Theory" A representation of a topological group $\rho:G\to\mathcal{L}(H)$ on an Hilbert space it is said to be irreducible if the only subspaces $H'\subset H$ such that $\rho(g)...
2
votes
0answers
64 views

Contour integration with 2 poles in Scattering theorey

I am studying Scattering theory but I am stuck at this point on evaluating this integral $G(R)={1\over {4\pi^2 i R }}{\int_0^{\infty} } {q\over{k^2-q^2}}\Biggr(e^{iqR}-e^{-iqR} \Biggl)dq$ Where $ ...
1
vote
0answers
37 views

How boundary conditions for a green function is same a original function

I am trying to study green's function.suppose we have a differential equation, $D_x f(x) =g(x)$ , We also provided with some boundary conditions for $f(x)$, But we supply this boundary conditions in ...
1
vote
0answers
30 views

Time evolution of a finite dim. quantum system

We will study the time-evolution of a finite dimensional quantum system. To this end, let us consider a quantum mechanical system with the Hilbert space $\mathbb{C}^2$. We denote by $\left . \left | ...
1
vote
2answers
4k views

What is the meaning of “Hermitian”?

Google search-bar gives the definition of Hermitian as: Hermitian: denoting or relating to a matrix in which those pairs of elements that are symmetrically placed with respect to the principal ...
1
vote
1answer
356 views

Position operator is self adjoint

Let $H=L^2(\Bbb{R})$ with the linear (unbounded) operator $P(f)(x)=x\cdot f(x)$ for each $x\in\mathbb{R}$. Have a look at the following domain: $$D(P)=\{f\in L^2(\mathbb{R}):\int_{\mathbb{R}}{x^2\cdot|...
1
vote
2answers
151 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
1
vote
6answers
249 views

Meaning of the notation $L^2(\mathbb{R}^3)$ and its generalization

I'm not sure whether this question really belongs to this website. In quantum physics texts, and physics stackechange website, I have often seen the notation $L^2(\mathbb{R}^3)$. My glossary of ...
1
vote
1answer
229 views

Question concerning trace over tensor products

I am trying to prove a certain theorem, which requires me to understand a certain manipulation. Let $\{E_n\}_{n\ge0}$ denote the eigenvalues, in increasing order, of an operator $H$ on a Hilbert ...
0
votes
1answer
91 views

Use of the symbolic operator $\left[e^{\frac{\partial}{\partial x}}\right]$ in Taylor's expansion

We know that Taylor's series expansion of a generic function $f(x)$, in the abscissa point $x_0\in\mathbb{R}$ is given by: $$f(x)=\sum_{n=0}^{\infty}\frac{(x-x_0)^n}{n!}f^{(n)}(x_0) \tag{1}$$ $$f(x)=...
0
votes
1answer
53 views

Determinant Expression for Grover operator

I want find the characteristic polynomial for the grover quantum operator $U^{N\times N}$ $$\begin{align*} U=(2|D\rangle\langle D| -I_N)(2|M^{\perp}\rangle\langle M^{\perp}| -I) \end{align*}$$ ...
0
votes
0answers
78 views

Finding the eigenvalues and eignevectors of this operator.

The operator $$\mathcal{\hat{G}} = (\xi - 1) \sum_{j=1}^N \int dk_j \; k_j \hat{a}^\dagger_j(k_j)\hat{a}_j(k_j),$$ is physically similar to the momentum operator in quantum mechanics. It has the ...