Operator Definition and 1000 Threads

I came across a previous exam question which stated: Do all physical states, ψ, abide to Hψ = Eψ. I thought about it for a while, but I'm not really sure.

I'm going through a derivation and it shows: (dirac notation) <una|VA-AV|unb>=(anb-ana)<una|V|unb> V and A are operators that are hermition and commute with each other and ana and anb are the eigenvalues of the operator A. I imagine it is trivial and possibly doesn't even matter but why does...

Homework Statement Prove that the states $$|z, \alpha \rangle = \hat S(z)\hat D(\alpha) | 0 \rangle $$ $$|\alpha, z \rangle = \hat D(\alpha) \hat S(z)| 0 \rangle $$ are eigenvectors of the squeezed amplitude operator $$ \hat b = \hat S(z) \hat a \hat S ^\dagger (z) = \mu \hat a + \nu \hat a...

Hello all, as far as I can see this question is not posted already, my apologies if it is and please provide a link. But I'm watching this video on youtube: And at 22:38 there's an expression given for the uncertainty of an arbitrary operator Q, however I'm concerned the expression is incorrect...

Homework Statement Hi, my task is to show that the momentum operator is hermitian. I found a link, which shows how to solve the problem: http://www.colby.edu/chemistry/PChem/notes/MomentumHermitian.pdf But there are two steps that I don't understand: 1. Why does the wave function approach...

Problem: Let $T$ be the linear operator on $\mathbb{R}^3$ defined by $$T(x_1, x_2, x_3)= (3x_1, x_1-x_2, 2x_1+x_2+x_3)$$ Is $T$ invertible? If so, find a rule for $T^{-1}$ like the one which defines $T$. Prove that $(T^2-I)(T-3I) = 0.$ Attempt: $(T|I)=\left[\begin{array}{ccc|ccc} 3 &...

Homework Statement I have a spin operator and have to find the eigenstates from it and then calculate the eigenvalues. I think I managed to get the eigenvalues but am not sure how to get the eigenstates.Homework Equations The Attempt at a Solution I think I managed to get the eigenvalues out...

Hey, here's a quick question: What is the Hamiltonian operator corresponding to a decaying Carbon-14 atom. Any insight is quite appreciated!

In the book that I read, an operator is defined to be a linear map which maps from a vector space into itself. For example, if ##T## is an operator in a vector space ##V##, then ##T:V\rightarrow V##. Now, what if I have an operator ##O## such that ##T:V\rightarrow U## where ##U## is a subspace...

As it says; I was looking over some provided solutions to a problem set I was given and noticed that, in finding the expectation value for the momentum operator of a given wavefunction, the following (constants/irrelevant stuff taken out) happened in the integrand...

Suppose ##V## is a complex vector space of dimension ##n## and ##T## an operator in it. Furthermore, suppose ##v\in V##. Then I form a list of vectors in ##V##, ##(v,Tv,T^2v,\ldots,T^mv)## where ##m>n##. Due to the last inequality, the vectors in that list must be linearly dependent. This...

Homework Statement verify that Ψ(x) = ( 1/a√π)½ exp(-(x2/2a2)) is a solution to the TISE for linear harmonic oscillator. Where a = √(hbar/mw). and V(x) = ½ mw2x2. Homework Equations HΨ=EΨ E_n = (n+½)hbar*wThe Attempt at a Solution I've started by differentiating the wave function twice to...

Hey, folks. I, on a whim today, started taking a MOOC quantum mechanics course that I have the functional math skills necessary to do but have virtually no background knowledge of quantum to start with and am incredibly rusty on stuff like PDE's; Quite frankly I'm out of my league, but the...

Dear all, I want to know how to convert operator matrix when using Dirac Bra-Ket notation when it must be converted into a new dimension. I am currently working on transition dipole moment operator matrix D which I am going to use the following one: D = er Where e is charge of electron, r is...

suppose that the momentum operator \hat p is acting on a momentum eigenstate | p \rangle such that we have the eigenvalue equation \hat p | p \rangle = p| p \rangle Now let's project \langle x | on the equation above and use the completeness relation \int | x\rangle \langle x | dx =\hat I we...

Homework Statement Given the following k-space representation of the wave function: Ψ(k,t) = Ψ(k)e-iħk2t/2m use the wave number representation to show the following: <x>t=<x>0 + <p>0t/m <p>t=<p>0 Homework Equations <x>=∫Ψ*(x,t)xΨ(x,t)dx <p>=∫Ψ*(x,t)(-iħ ∂/∂x)Ψ(x,t)dx The Attempt at a...

Homework Statement Homework EquationsThe Attempt at a Solution I'm fine with parts a) and b) However I don't understand what part c) is asking me to do. How do I 'measure' an operator? There are only two things I can think to do: 1. Find the expectation values of A for <Φ1|A|Φ1> and...

I want to calculate the commutator ##{\Large [p_i,\frac{x_j}{r}]}## but I have no idea how I should work with the operator ##{\Large\frac{x_j}{r} }##. Is it ## x_j \frac 1 r ## or ## \frac 1 r x_j ##? Or these two are equal? How can I calculate ##{\Large [p_i,\frac 1 r]}##? Thanks

Hi, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions! On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you...

Can these tensor be seen as operators on two elements. So given two elements of something they produce something, for instance a scalar ?

What does it mean by "In the position representation -- in which r is diagonal" in the paragraph below? How can we show that? Does it mean equation (3) in http://scienceworld.wolfram.com/physics/PositionOperator.html? (where I believe the matrix is in the ##|E_n>## basis)

I know that the time-evolution operator in quantum mechanics is ##e^{-iHt}##. Is this also called the Schrodinger time-evolution operator? Also, can you guys explain why the amplitude ##U(x_{a},x_{b};T)## for a particle to travel from one point ##(x_{a})## to another ##(x_{b})## in a given...

Homework Statement In Zettili's QM textbook, we are asked to find the trace of an operator |\psi><\chi| . Where the kets |\psi> and |\chi> are equal to some (irrelevant, for the purposes of this question) linear combinations of two orthonormal basis kets. Homework Equations...

This is the key step to transform from position space Schrodinger equation to its counterpart in momentum space. How is the first equation transformed into 3.21? To be more specific, how to integral Laplacian term by parts?

Homework Statement I am trying to get the hamiltonain operator equality for a rigid rotor. But I don't get it. Please see the red text in the bottom for my direct problem. The rest is just the derivation I used from classical mechanics. Homework Equations By using algebra we obtain: By...

Hi all, Some processes can not happen at the tree level, but it happen via loops, like for Higgs decay to pair of glouns or pair of photons, (h -> gg), (h -> y y) . For instance, effectively h -> gg written as ##~ h~ G^a_{\mu\nu} G_a^{\mu\nu}~ ## which is Lorentz and gauge invariant .. Now if...

How is (5.240b) derived? I get {U^{-1}}^\dagger(t, t_0)\,U^{-1}(t, t_0)=I instead. My steps: \begin{align}<\psi(t_0)\,|\,\psi(t_0)>&=\,<U(t_0, t)\,\psi(t)\,|\,U(t_0, t)\,\psi(t)>\\ &=\,<U^{-1}(t, t_0)\,\psi(t)\,|\,U^{-1}(t, t_0)\,\psi(t)>\\ &=\,<\psi(t)\,|\,{U^{-1}}^\dagger(t, t_0)\,U^{-1}(t...

Why isn't the second line in (5.185) ##\sum_k\sum_l<\phi_m\,|\,A\,|\,\psi_k><\psi_k\,|\,\psi_l><\psi_l\,|\,\phi_n>##? My steps are as follows: ##<\phi_m\,|\,A\,|\,\phi_n>## ##=\int\phi_m^*(r)\,A\,\phi_n(r)\,dr## ##=\int\phi_m^*(r)\,A\,\int\delta(r-r')\phi_n(r')\,dr'dr## By the closure...

Homework Statement Define n=(x + iy)/(2)½L and ñ=(x - iy)/(2)½L. Also, ∂n = L(∂x - i ∂y)/(2)½ and ∂ñ = L(∂x + i ∂y)/(2)½. with ∂n=∂/∂n, ∂x=∂/∂x, ∂y=∂/∂y, and L being the magnetic length. Show that a=(1/2)ñ+∂n and a†=(1/2)n -∂ñ a and a† are the lowering and raising operators of quantum...

Homework Statement Let \mathcal{A}: \mathbb{R^3}\rightarrow \mathbb{R^3} is a linear operator defined as \mathcal{A}(x_1,x_2,x_3)=(x_1+x_2-x_3, x_2+7x_3, -x_3) Prove that \mathcal{A} is invertible and find matrix of \mathcal{A},A^{-1} in terms of canonical basis of \mathbb{R^3}. Homework...

Given a transformation ##U## such that ##|\psi'>=U|\psi>##, the invariance ##<\psi'|\psi'>=<\psi|\psi>## of the scalar product under the transformation ##U## means that ##U## is either linear and unitary, or antilinear and antiunitary. How do I prove this? ##<\psi'|\psi'>## ##= <U\psi|U\psi>##...

I would like to prove that the angular momentum operators ##\vec{J} = \vec{x} \times \vec{p} = \vec{x} \times (-i\vec{\nabla})## can be used to obtain the commutation relations ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##. Something's gone wrong with my proof below. Can you point out the mistake...

Homework Statement Homework Equations The Attempt at a Solution First substitute ##\Phi(p,t)## in terms of ##\Psi(r,t)## and similarly for ##\Phi^*(p,t)##, and substitute ##p_x^n## in terms of the differentiation operator ##< p_x^n>\,=(2\pi\hbar)^{-3}\int\int...

Homework Statement Let \phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2} be a linear operator defined as: (\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2 where B= \begin{bmatrix} 3 & -2 \\ 2 & -2 \\ \end{bmatrix} Find rank,defect and one basis of an image and kernel of linear operator...

The well-known eigen value expression A(a)=a(a) assuming the operator which represents a physical phenomena acts on a quantum state which is represented by an eigen vector, (a) corresponds to an observed value a. But I am wondering if the same operator A can act on (a) and produce another eigen...

According to Daniel Gillespie in A Quantum Mechanics Primer (1970), " . . . any observable which in classical mechanics is some well behaved function of position and momentum, f(x,p), is represented in quantum mechanics by the operator f ( \hat{x} , \hat {p} ) . That is, a = f (x,p) . . ...

Hi! First of all I want apologize for my bad english! Second, I'm doing a physical chemystry course about the main concepts of quantum mechanics ! The Professor has given to me this definition of "the adjoint operator": <φ|Aψ> = <A†φ|ψ> My purpose is to verificate this equivalence so i...

Probably trivial, but for matrices with different ranks, Det is not closed for addition? I think it is closed under multiplication? So really I must show Det not closed under addition for square matrices of the same order... $ D(A_n) = \sum_{j=1}^{n} a_{1j}C_{ij} $ and $ D(B_n) =...

Hi, I'm having difficulty with this program in a textbook. The instructions are as follows: Overload the + operator as indicated. Sample output for the given program: First vacation: Days: 7, People: 3 Second vacation: Days: 12, People: 3 This is the code that follows #include <iostream>...

I am trying to perform the operation a on a translated Gaussian, ie. the ground state of the simple harmonic oscillator (for which the ground state eigenfunction is e^-((x/xNot)^2). First, I was able to confirm just fine that a acting on phi-ground(x) = 0. But when translating by xNot, so a...

I need to find a matrix representation for operator A=x\frac{d}{dx} using Legendre polinomials as base. I would use a_{mn}=\int^{-1}_{-1}P_m(x)\,x\frac{d}{dx}\,P_n(x)\,dx, but I have the problem that Legendre polinomials aren't orthonormal \langle P_{i}|P_{l}\rangle=\delta_{il}\frac{2}{2i+1}. I...

Homework Statement The problem originally asks to evaluate ##exp(\frac{-i\pi L_x}{h})## in a ket |l,m>. So I am wondering if I can treat the operator as a parity operator or if I really have to expand that exponential, maybe in function of ##L_+## and ##L_-##. 2. The attempt at a solution If...

Homework Statement A function of a hermitian operator H can be written as f(H)=Σ (H)n with n=0 to n=∞. When is (1-H)-1 defined? Homework Equations (1-x)-1 = Σ(-x)n= 1-x+x2-x3+... The Attempt at a Solution (1-H)-1 converges if each element of H converges in this series, that is (1-hi)-1...

Hey all, I got some question referring to the interaction picture. For example: I have the Hamiltonian ##H=sum_k w_k b_k^\dagger b_k + V(t)=H1+V(t)## When I would now have a time evolution operator: ##T exp(-i * int(H+V))##. (where T is the time ordering operator) How can I transform it...

Hi can someone direct me to a free software to calculate eigenvalues and normalized eigenfunctions of a linear integral operator. I am trying to solve a fredholm integral equation with degenerate kernel using it instead of linear equations thanks sarrah

From P. Meystre's book elements of quantum optics (Many labels of equations are wrong:H) Page 83, the annihilation operator and creation operator, which are helpful to discuss harmonic oscillator, are defined as ## a=\frac{1}{\sqrt{2\hbar\Omega}}(\Omega q+ip),\\...

I have a linear integral operator $K\psi=\int_{a}^{b} \,k(x,s) \psi(s) ds$ $L\psi=\int_{a}^{b} \,l(x,s) \psi(s) ds$ both are continuous I know how to obtain the eigenvalues of each alone. But how can I calculate the eigenvalues of the operator $I-{L}^{-1} K$ or at least the norm...

Hi everyone, I tried to find the Eigenstate of the angular momentum operator myself, more specifically I tried to find a Function Y_{lm}(\theta,\phi) with L_zY_{lm}=mħY_{lm} and L^2Y_{lm}=l(l+1)ħ^2Y_{lm} where L_z=-iħ\frac{\partial}{\partial \phi} and...

I am a beginner in quantum mechanics and I am confused about the operator ΔA defined to be ΔA Ξ A - <A>. Can someone please tell me how to interpret <A>? From what I can understand, <A> is the expectation value and is defined to be <Ψ|A|Ψ>. But that is just a scalar correct? How do subtract a...

Hi, I have some trouble with the following problem: Let E be a Banach space. Let A ∈ L(E), the space of linear operators from E. Show that the linear operator φ: L(E) → L(E) with φ(T) = T + AT is an isomorphism if ||A|| < 1. So the idea here is to use the Neumann series but I can't really...