Operators Definition and 1000 Threads

Now I am starting to learn Quantum Mechanics. In the class I am taught about operators, postulates and all other basic stuff. I understand operators to be +, -, /, etc; but quantum mechanical operators are entirely different; to understand them, I think, I need to know the historical...

Homework Statement True or false? If T: ℙ8(ℝ) → ℙ8(ℝ) is defined by T(p) = p', so exists a basis of ℙ8(ℝ) such that the matrix of T in relation to this basis is inversible. Homework EquationsThe Attempt at a Solution So i think that my equations is of the form: A.x = x' hence A is...

Homework Statement Suppose a linear operator L satisfies <A|L|A> = 0 for every state A. Show that then all matrix elements <B|L|A> = 0, and hence L = 0. Homework Equations ##<A|L|A>=L_{AA} and <B|L|A>=L_{BA}## The Attempt at a Solution It seems very straight forward and I don't know how...

Homework Statement The demonstration for the momentum operator in Quantum Mechanics goes something like this <v>=\frac{d}{dt}<x>=\frac{d}{dt} \int x \Psi^* \Psi dx and then one ends up with <p>=m<v>=\int \Psi^* (-i \hbar \frac{d}{dx}) Psi dx however, if you swap the congugates you get...

Homework Statement Consider the following state constructed out of products of eigenstates of two individual angular momenta with ##j_1 = \frac{3}{2}## and ##j_2 = 1##: $$ \begin{equation*} \sqrt{\frac{3}{5}}|{\tiny\frac{3}{2}, -\frac{1}{2}}\rangle |{\tiny 1,-1}\rangle +...

First, I have a question regarding the conservation of probability. The book shows (quite elegantly) that $$ \frac {d}{dt} \int_{-\infty}^{\infty} |\Psi (x, t)|^2dx = \frac {i\hbar}{2m} \Big{(}\Psi ^* \frac{\partial \Psi}{\partial x} - \frac{\partial \Psi ^*}{\partial x} \Big{)} \Big...

Why is it required that interactions between fields must occur at single spacetime points in order for them to be local? For example, why must an interaction Lagrangian be of the form \mathcal{L}_{int}\sim (\phi(x))^{2} why can't one have a case where \mathcal{L}_{int}\sim\phi(x)\phi(y) where...

Hi! If I have understood things correctly, in a multi-electron atom you have that the spin operator ##S## commutes with the orbital angular momentum operator ##L##. However, as these operators act on wavefunctions living in different Hilbert spaces, how is it possible to even calculate the...

Puting a minus in front of the momentum in the field expansion gives ##\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde p} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} + a_{\bf{p}}^ + {e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right){\rm{ }}\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde...

Hi, This is a question regarding Example 3.6 in Section 3.5 (p.35) of 'QFT for the Gifted Amateur' by Lancaster & Blundell. Given, [a^{\dagger}_\textbf{p}, a_\textbf{p'}] = \delta^{(3)}(\textbf{p} - \textbf{p'}) . This I understand. The operators create/destroy particles in the momentum state...

Hello, so I have a couple of related questions. 1) If you have a wavefuction Ψ, and act on it with some operator, does it have to give you the same wavefunction back (ie. does the wavefunction have to be an eigenfunction of the operator)? Could you have a wavefunction like e-iħtSin(x)? Since...

I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality δ(dq/dt) = d(δq)/dt Where q is some coordinate, and δf is the first variation in...

How does one show that eAeB=eA+Be[A,B]/2 where A,B are operators and [ , ] is the commutator. The QM book I am using states it as a fact without proof, but I would like to see how it is proved. I've muddled around with the series expansion, but can't get farther than a few term by term products...

How does one show that eAeB=eA+Be[A,B]/2 where A,B are operators and [ , ] is the commutator. The QM book I am using states it as a fact without proof, but I would like to see how it is proved. I've muddled around with the series expansion, but can't get farther than a few term by term products...

Hello every one . If the derivative is a linear operator ( linear map ) Then what is the example of non-linear operator Thanks .

Hello. I'm having trouble understanding what is required in the following problem: Find the relation between the matrix elements of the operators $\widehat{p}$ and $\widehat{x}$ in the base of eigenvectors of the Hamiltonian for one particle, that is, $$\widehat{H} = \frac{1}{2M} \widehat{p}^2...

How can the fact that ##\hat x## and ##\hat p## are Hermitian be used to prove that ##\hat x - \frac{i}{m \omega} \hat p## and ##\hat x + \frac{i}{m \omega} \hat p## are adjoints of each other?

Hi, I am new here. I don't know how can I type mathematical operators or symbols here. Can anyone help me out?

Hello! I'm currently making my way through the book "Quantum Field Theory of Point Particles and Strings" and on page 13 they talk are talking about quantization of the classical versions momentum and position. The first part to quantizing these is turning them into operators. The books goes on...

I am currently reading "Differential Equatons with Applications" by Ritger and Rose, and I need some clarification about some notation and convention that they are using. I think it all stems from a lack of clarity of the difference between the operator d/dx and the "object" (I don't know what...

I'm reading Alvarez-Gaume review on Seiberg-Witten theory: http://arxiv.org/abs/hep-th/9701069. Around page 23 you can find the following claim: "This is a Clifford algebra with 2N generators and has a 2N-dimensional representation. From the point of view of the angular momentum algebra, a^I...

Homework Statement Solve the system using differential operators. Determine the # of arbitrary constants and then compare to your solution. Homework Equations D substitution: replace x' with Dx and y' with Dy The Attempt at a Solution I have the solution to this one, but I'm working...

Say I have a hamiltonian with fermion creation / annihilation operators like this: \sum_{k_1,k_2,k_3,k_4} c_{k_1,\uparrow}^{\dagger} c_{k_2,\downarrow}^{\dagger} c_{k_3,\downarrow} c_{k_4,\uparrow} where the k's are momenta and the arrows indicate spin up / spin down. Can I commute operators...

Look at the following attached picture, where they prove the coherent states are eigenfunctions of the annihiliation operators by simply proving aexp(φa†)l0> = φexp(φa†)l0>. I understand the proof but does that also prove that: aiexp(Σφiai†)l0> = φiexp(Σφiai†)l0> ? I can see that it would if you...

I have difficulty in understanding the Density Operator. Please see attached file. (From the Book " Quantum Mechanics Demystified Page 250) Most grateful if someone could help! Peter Yu

In case of quantum LHO in eigen state of the system ##|n \rangle## \langle \hat{T} \rangle=\langle \hat{U} \rangle=\frac{1}{2}(n+\frac{1}{2})\hbar \omega What will happened in some superposition of states? Does Ehrenfest theorem can tell me something more general? Is it possible to say that...

Homework Statement Consider a particle in an energy eigenstate ##|n\rangle.## Calculate ##\langle x\rangle## and ##\langle p_x\rangle## for this state. Homework Equations ##x = \sqrt{\frac{\hbar}{2m\omega}}(a+a^{\dagger})## The Attempt at a Solution ##\langle x\rangle =...

- a|n>=C|n-1> - a+|n>=D|n+1> And because |n-1> is normalized, <n-1|n-1>=1: (<n|a+)(a|n>)=C2 Thus, <n|a+a|n>=C2 Where a is the annihilation operator and a+ is the creation operator I don't understand this as...

α I have been studying translation operators of the type T = exp ( -ipx0/ hbar) where p is the momentum operator which leads to T+xT = x+x0. I am ok with that but then I came across the following equation concerning raising and lowering operators exp(-alpha a+) a exp(alpha a+) = a + alpha. Is...

Hello, I'm wondering, is it possible to define an operator on a Hilbert space that gives information about the "distinctness" of superposition components? As a simple example, imagine that we have two particles. Let |3> designate the state in which they are 3 meters apart, let |5> designate...

Hello, I'm wondering, is it possible to define an operator that gives information about the "distinctness" of superposition components? As a simple example, imagine that we have two particles. Let |3> designate the state in which they are 3 meters apart, let |5> designate the state in which...

Let's denote ## \mathbf{p} ## and ## \Pi ## as the momentum and parity operators respectively. It's known that ## \mathbf{p} ## doesn't commute with ## \Pi ##, so they do not share the same set of eigenkets (plane wave doesn't have parity). But I just calculated that ##[\mathbf{p}^2,\Pi] = 0##...

Can anyone explain to me why the following operators are rotation operators: \begin{align*}R_x(\theta) &= e^{-i\theta X/2}=\cos(\frac{\theta}{2})I-i\sin(\frac{\theta}{2})X= \left(\!\begin{array}{cc}\cos(\frac{\theta}{2}) & -i\sin(\frac{\theta}{2}) \\ -i\sin(\frac{\theta}{2})&...

Homework Statement I will denote operators by capital letters. The question is calculate <p | XXPP | x> / <p | x > Homework Equations X |x> = x |x> P |p> = p |p> P |x> = -i(hbar)d/dx X |p> = i(hbar)d/dp The Attempt at a Solution If I start on the RHS and take PP out I get...

from the relativity forum https://www.physicsforums.com/threads/spacetime-in-qm-or-qft.802721/ Sonderval stated (transferred here so not off topic): http://scienceblogs.de/hier-wohnen-drachen/artikelserien/[/QUOTE'][/PLAIN] So the standard Schroedinger Equation can be used for both particles...

Hey! How does the operator of angular momentum operates in exponential form? $$ e^{-i\theta J}\vert l, m \rangle = ?? $$ where $$J\vert \Psi \rangle = J\vert l, m \rangle$$ and $$J^2\vert \Psi\rangle = \hbar^2 l(l+1)\vert \Psi\rangle $$ Also, how do you operate $$J_-$$ and $$J_+$$...

Hi, We know the convergence of a series but what does it mean to say that "an operator converges or diverges"?

(First of all I never saw Hilbert spaces in a mathematical class, only used it in intro QM so far, so please don't assume I know that much when answering.) Let's consider the Hilbert space on the interval [a,b] and the operator ##\textbf{L} = \frac{d^{2}}{dx^{2}} ##. Then ##\textbf{L}## is...

Hello, I was just watching a youtube video deriving the equation for the Hamiltonian for the harmonic oscillator, and I am also following Griffiths explanation. I just got stuck at a part here, and was wondering if I could get some help understanding the next step (both the video and book...

Homework Statement Use the spin##-1## states ##|1,1\rangle, \ |1,0\rangle, \ |1, -1\rangle## as a basis to form the matrix representations of the angular momentum operators. Homework Equations ##\mathbb{\hat{S}}_+|s,m\rangle = \sqrt{s(s+1)-m(m+1)}\hbar|s,m+1\rangle##...

Hello; I'm reading "principles of quantum mechanics" by R.Shankar. I reached a theorem talking about Hermitian operators. The theorem says: " To every Hermetian operator Ω,there exist( at least) a basis consisting of its orthonormal eigenvectors.Its diagonal in this eigenbasis and has its...

Hello, I juste don't know how this was done it is on the solutionnary of a very long exercise and i am not getting this calculation 1. Homework Statement <1,0| ax+ay++ax+ay+axay++axay|0,1> = <1,0|1,0> Homework Equations 3. The Attempt at a Solution We have that |0,1> = ay+ |0,0> I don't...

Homework Statement The state of an electron is, |Psi> =a|l =2, m=0> ⊗ |up> + Psi =a|l =2, m=1> ⊗ |down>, a and b are constants with |a|2 + |b|2 = 1 choose a and b such that |Psi> is an eigenstate of the following operators: L2, S2, J2 and Jz. The attempt at a solution I am really not sure...

Spin or angular momentum in my book is formulated in the basis of eigenstates of the operator that measures the angular momentum along the z-axis. But in principle I guess this could just as well have been done in the basis of eigenstates of Ly or Lx. Will that change anything in the equations...

Homework Statement Given the operator A = d/dx and B = x and the function f(x) = xe^(-ax) evaluate : ABF(x) and BAF(x) Do these operators commute (yes/No) Homework Equations [A,B]F(x) = ABF - BAF = 0 ; means they commuteThe Attempt at a Solution [A,B]F(x) = ABF - BAF = 0 =d/dx(x^2e^-ax) -...

I'm trying to re-derive a result in a paper that I'm struggling with. Here is the problem: I wish to calculate (\nabla \otimes \nabla) h where \nabla is defined as \nabla = \frac{\partial}{\partial r} \hat{\mathbf{r}}+ \frac{1}{r} \frac{\partial}{\partial \psi} \hat{\boldsymbol{\psi}} and...

I know for two linear operators $$H_1, H_2$$ between finite dimensional spaces (matrices) we have the relations (assuming their adjoints/inverses exist): $$(H_1 H_2)^* = H_2^* H_1^*$$ and $$(H_1 H_2)^{-1} = H_2^{-1} H_1^{-1}$$ but does this extend to operators in infinite dimensions? Thanks.

HI, Suppose there are two operators A and B , We have to find A /B - Will it equal to AB-1 OR B-1 A , Because i have read that it equals to AB-1 , BUT i could not find reason for that. thanks

Hello there. I am trying to proove in a general way that [Lx2,Lz2]=[Ly2,Lz2]=[Lz2,Lx2] But I am a little bit stuck. I've tried to apply the commutator algebra but I'm not geting very far, and by any means near of a general proof. Any help would be greatly appreciated. Thank you.

Homework Statement I'm reading Peskin and Schroeder to the best of my ability. Other than a few integration tricks that escaped me I made it through chapter 2 with no trouble, but the beginning of chapter three, "Lorentz Invariance in Wave Equations", has me stumped. They are going through a...