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
30 views

Definite Gaussian/exponential integral

I've been revising some quantum mechanics, and I was wondering how I would calculate a specific standard deviation. The wave function I am working with is ...
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0answers
15 views

Transformation of inner product of wave functions under transformation of metric

Assume that we have a wave function $\psi(x)$ in the coordinate system $x$ in the Hilbert space $H_1$. The inner product of two states $\psi_1$ and $\psi_2$ are given as ...
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0answers
8 views

Gaussian 09 input of a QST2 calculation for the reaction C2H6->C2H5+H [closed]

Please, let somebody show a Gaussian 09 input of a QST2 calculation for the reaction C2H6->C2H5+H. Thank you for your help!
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0answers
9 views

Quantum Mechanics State Proportional to x [migrated]

I have a particle in a 1D infinite potential well. i.e $V(x)= \left\{ \begin{array}{c} V(x)=0 \text{ for } |x|<a \\ V(x)=\infty \text{ for } |x| \geq a \\ \end{array} \right. $ I ...
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0answers
12 views

Considering $Res^G_{H_\rho}$ instead of $G$ in quantum Fourier sampling

I am going through the proof of theorem 4 in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. Here, they are trying to calculate the probability of measuring the ...
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0answers
27 views

Normalize wave function and check if it is in Hilbert space

Consider a particle of mass m freely propagating within the box x ∈ [0, R]. Prepare the particle in the state corresponding to the wave function $ \psi (x) Asin(\frac{3\pi x}{2R}) ...
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1answer
25 views

Quantum mechanics, operators acting on generic state

For a quantum mechanical harmonic oscillator of constant real mass m and frequency ω, define the following operators on the Hilbert space: $ h = a^{+}a $ $e = [\sqrt{-1 + a^{+}a}]a^{+} $ $ f ...
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1answer
15 views

Why is the Hermitian conjugate of the Fourier transform of an operator not the transform of the Hermitian conjugate?

It is defined that: \begin{align} O(\omega)&=\frac{1}{\sqrt{2\pi}}\int O(t)e^{-i\omega t} \mathrm{d}t \tag{1} \\ O^{\dagger}(\omega)&=\frac{1}{\sqrt{2\pi}}\int O^{\dagger}(t)e^{-i\omega t} ...
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2answers
48 views

Norm of operator $A$ st. $A^2 = I$?

I'm wondering what can be said about the norm $||A||$ of an operator which squares to identity. All I can think of is that $$1=||AA|| \leq ||A||^2$$ so that $||A|| \geq 1$. But can anything else be ...
2
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1answer
21 views

Computate the commutator $[p^n,x]=-ihnp^{n-1}$

Computate the commutator of $[p^n,x]=-ihnp^{n-1}$. With $p=-ih \frac{\delta}{\delta x}$ the impulse operator. $h$ stands for $\frac{h}{2\pi}$. Answer: I do it with induction over $n$. For $n=1$ ...
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1answer
20 views

If $L$ is a diagonalizable linear operator, why is $f(L)$ well-defined for $f: \mathbb{C} \rightarrow \mathbb{C}$?

In a book on quantum mechanics, I encountered a statement equivalent to the following. Suppose $V$ is a finite dimensional inner product space over $\mathbb{C}$. Let $L : V \rightarrow V$ be a normal ...
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1answer
34 views

Find the eigenvectors of a hermitian matrix as a function of angles

I have a problem which seems flawed to me. Consider the hermitian matrix M \begin{pmatrix} a/2 & b^* \\ b & -a/2 \end{pmatrix} with a>0 real and b complex. Let $\theta$,$\phi$ ...
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1answer
27 views

How do outer products differ from tensor products?

From my naive understanding, an outer product appears to be a particular case of a tensor product, but applied to the "simple" case of tensor products of vectors. However, in quantum mechanics there ...
3
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1answer
44 views

Quantum Mechanics - Eigenvalue and Eigenvector of a Matrix

I'm attempting to find the eigenvalues and eigenvectors from the following matrix : \begin{pmatrix} -3\cos\theta&\sqrt{3}\sin\theta e^{iφ}&0&0\\\sqrt{3}\sin\theta ...
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1answer
29 views

counter example of sum of closable operators

Let $A$, $B$ and $A+B$ are closable operators. I am not sure the relation $\overline{A+B}\supseteq \overline{A}+\overline{B}$ is true or not(with equality if one operator is bounded. ) And I have a ...
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0answers
18 views

Intuitive interpretation of negative probabilities

I have heard that in quantum physics negative probabilities show up in certain distributions. Could you give an example that aids int he intuitional interpretation of a negative probability? For ...
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0answers
19 views

Commutative diagram for hidden subgroup representation of graph automorphism

The hidden subgroup representation of the graph automorphism problem is defined in the section 10.2 of QUANTUM ALGORITHMS FOR PROBLEMS IN NUMBER THEORY, ALGEBRAIC GEOMETRY, AND GROUP THEORY. It is as ...
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1answer
34 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 ...
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0answers
31 views

How to find the ground energy state solution in a quantum harmonic oscillator?

Recently, I came across a question which asks to solve the Schrödinger equation for a harmonic oscillator on $ [a, b] $ : $-\frac{\hbar^2}{2m}\frac{d^2\psi}{d x^2} + \frac{1}{2} m \omega^2 x^2 \psi = ...
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0answers
30 views

Existence of a canonical map of quadratic forms

For $X=\mathbb C^N\oplus \mathbb C^N$ equipped with a real structure $J^2=1$ and symplectic structure $S$ satisfying $$J(z_1,z_2)=(\bar z_2,\bar z_1),~~~~~S(z_1,z_2)=(z_1,-z_2)$$ we see that $X$ has ...
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0answers
10 views

Separation of Variables PDE on Klein Gordon Equ

I have to use separation of variables on the 3-D Klein-Gordon equation: $ c^2 \bigtriangledown^2 \Psi - \frac{\partial ^2}{\partial t^2}\Psi + (\frac{mc^2}{\hbar})\Psi = 0 $ where $ \Psi (r,t) = ...
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2answers
56 views

N-Representation of an Operator

Calculating $\langle{n} \; {| \; \hat{X}^2 \; |} \; m\rangle$ in the N-representation, where $| \; m\rangle$ and $| \; n\rangle$ are harmonic oscillator states and $\hat{X} = \sqrt{\frac{\hbar}{2mw}}( ...
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0answers
10 views

Positive semi-definite vs. semi-positive definite?

I've heard and read the phrase positive semi-definite in many places. However, the only place I can recall seeing semi-positive definite is in my quantum mechanics text, John S. Townsend's ...
2
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1answer
78 views

Quantum Mechanics Project Ideas!!! [closed]

I am in my first year in uni and I have to write a project in Quantum Mechanics. But I have been struggling with an idea for the project since I have recently started studying quantum mechanics and my ...
2
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1answer
40 views

A question regarding Eigenvalues

Note: $\psi,\psi^{\dagger} :\Bbb{R} \to \Bbb{C}$ and $x, \lambda_i , \hbar, m \in \Bbb{R}$ Say we know that $\lambda_1$ is a solution to the eigenvalue equation: $$\hat{\Pi}\psi(x)= \lambda_1 \psi(x) ...
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0answers
35 views

Dirac Notation: What Do $\langle u|v \rangle$ and $\langle u|T|v \rangle$ Represent?

I post this hoping for clarification, and particularly in a context of linear algebra without too much mention of matrices. (1) Let $V$ be a (perhaps infinite dimensional) Hilbert space and $V'$ its ...
3
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1answer
41 views

Singular value decomposition of sum of single particle operators

here is my question. Suppose to have an operator $L$ in a composite Hilbert space $A\otimes B$ which can be written as sum of single particle operator as: \begin{equation} L = (L_0\otimes \mathbb{I} + ...
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0answers
33 views

How to prove the given property of spherical harmonics?

How to prove: $$ \int^\pi_0\int^{2\pi}_0 Y_{l'',m''_l}(\theta,\phi) Y_{l',m'_l}(\theta,\phi) Y_{l,m_l}(\theta,\phi) \sin\theta \,d\theta \,d\phi = 0 $$ unless $l, l',$ and $l''$ are integers that can ...
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0answers
70 views

Pauli Matrices as Representations of Reflection Operators?

How does one derive, using, say, the Householder transformation $$ R(r) = (I - nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + ...
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0answers
21 views

Probability distribution obtained by repeatedly sampling $S_x,S_y$ on a spin-$S$ system

While trying to rework an upcoming quiz problem for a quantum physics course, I came up with the following question which turned out to be harder than I expected. (Note: I take $\hbar =1$ in all ...
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0answers
21 views

Unitary transformation with eigenvectors of fourier basis

I'm trying to understand this statement made in https://users.cs.duke.edu/~reif/courses/randlectures/UVnotes/lec18.pdf in the last paragraph: "Since U is multiplication by a group element, the ...
4
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2answers
32 views

Expectation of Complex Operators

Given an operator $\hat{\alpha}$, how do we obtain, $$ \sqrt{ \left\langle \left( \hat{\alpha} - \left\langle\hat{\alpha}\right\rangle \right)^2 \right\rangle } = \sqrt{ ...
4
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1answer
88 views

Intuition behind definition of spinor

Some time ago I searched for the definition of spinors and found the wikipedia page on the subject. Although highly detailed the page tries to talk about many different constructions and IMHO doesn't ...
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1answer
19 views

Complex conjugate part of TISE

i would like to ask about complex conjugate part of this equation why ? I know that and so
3
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1answer
98 views

Introductory book on probability for physicists

I'm a physics student looking to start learning more about probability. Is there some introductory book on measure theoretical probability theory that includes sections on quantum probability? To ...
4
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3answers
44 views

Two operators $X$ and $Z$ in an infinite dimensional Hilbert space satisfying $X^2=Z^2=I$ and $\{X,Z\}= 0$

I am seeking to extend the following theorem to the case of infinite dimensional Hilbert space: Suppose we have two Hermitian operators $X$ and $Z$ in a finite dimensional Hilbert space $\mathcal H$. ...
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2answers
27 views

Dirac Notation Confusion [closed]

I am trying to express the vectors $0\choose 1$ and $1\choose 0$ in dirac notation wrt the basis {|$0\rangle,$|$1\rangle$} How do I distinguish between the above two vectors given that all vectors ...
2
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3answers
41 views

Example of a Projection Operator in $\mathbb{R^3}$

I'm looking for an operator $\hat P$ in $\mathbb{R^3}$ such that $\hat P^2=\hat P$ that is also Hermitian
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0answers
13 views

Book Recommendation for Poisson Manifold and Deformation Quantisation

Can someone please recommend a basic introduction to the concept of Poisson Manifolds and Deformation Quantization. I'm new to Theoretical Physics and had to go through a lot of books before I even ...
0
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1answer
57 views

Riesz representation theorem finite dimnesional case

I am taking a Quantum Mechanics course not a Functional Analysis course so I have only had a very basic introduction to Hilbert Spaces. I don't understand where the if and only if statement arises ...
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1answer
17 views

$\frac{d^2}{dx^2} = \lim_{\delta x\to 0} [f(x-\delta x) - 2f(x) + f(x+\delta x))/\delta x^2]$ find a matrix

Given that $$\frac{d^2}{dx^2} = \lim_{\delta x\to 0} [f(x-\delta x) - 2f(x) + f(x+\delta x))]\cdot\frac1{\delta x^2}$$ find an appropriate matrix that could represent such a derivative operator, in a ...
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1answer
18 views

Showing a orthogonal basis is complete

$\psi_1 = \frac{1}{\sqrt{2}}$ $\psi_2 = \sqrt{\frac{3}{2}}x$ By shwoing that any arbitrary function $f(x)=ax+b$ can be represented as linear combination of $\psi_1$ and $\psi_2$, show that ...
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1answer
38 views

general two-state system

Consider a two-state system: The Hamiltonian takes the general form $$ H= \begin{pmatrix} a_1 & c-id \\ c+id & a_2 \\ \end{pmatrix} $$ where $a_1, a_2, c, d \in\mathcal{R}$ $H$ can be ...
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0answers
15 views

Berry's curvature equation

The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation [1] $$ V_{m} = {- 1 \over B^2 } * i * ...
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0answers
115 views

Tridiagonal matrix w/trigonometric eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots ...
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1answer
40 views

How to construct a complete set in $L^2(\mathbb{R}^3)$ starting with the Spherical Harmonics?

The Spherical Harmonics form a complete set of functions on the sphere $S^2$, so that any function of $f: S^2\to \mathbb{R}$ can be written uniquely as $$f(\theta,\phi)=\sum_{l=0}^\infty ...
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0answers
29 views

Expressing a solution to a differential equation in a more compact form

We know that $Ae^{ikx}$ is a solution for $k \geq 0$ and $Be^{ikx}$ is a solution for $k \leq 0$. So does the $N$ here depend on whether or not $k \geq 0$ or $\leq 0$? Also I do not understand how ...
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1answer
41 views

Tensor product distributive property?

Is it true that for vectors $a$, $b$, $c$, $d$ we have $$|a\rangle \otimes |b\rangle \langle c| \otimes \langle d|= |a \rangle \langle c| \otimes |b \rangle \langle d|?$$ So does this kind of ...
1
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1answer
42 views

How to define properly the radial and angular dependence of a function?

Recently I've came across the following situation when studying Quantum Mechanics: suppose we have two operators $A,B$ on the space of functions $L^2(\mathbb{R}^3)$ and suppose they have the following ...
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1answer
44 views

How to write this down in a rigorous manner?

I'm studying Quantum Mechanics and there's something I'm in doubt in how to write it down in a rigorous way. The idea is: consider a Hilbert space $\mathcal{H}$ and one hermitian operator $A\in ...