Operator Definition and 1000 Threads

In Griffiths book, he says (a+ + a-)2 = a+2 + a-2 + a+a- + a-a+. Why can you NOT do the same thing for a+2 = (-ip+mωx)2 ?! When I do this to find the 2nd excited state of SHO, it gives me wrong answer. I actually have to apply a+ two times to ψ0 in order to get ψ2. It is ridiculous that...

The problem asks to show that the kinetic energy operator is Hermitian. The operator is given as T= -h^2/2mΔ but I know I can also write it as p^2/2m which would be (- ih∇)(-ih∇). My main question is if I can prove this in 1-D so that T=(-h^2/2m)d^2/dx^2 does that generalize to...

Homework Statement Calculate the result of the transformation of the vector operator \hat{V_{y}} by rotation \hat{R_{x}} around an angle \alpha . Homework Equations I believe that \hat{R_{x}} = \begin{pmatrix} 1& 0& 0\\ 0& cos\alpha & -sin\alpha \\ 0 & sin\alpha & cos\alpha...

Homework Statement Let the operator G={|ψ1>,|ψ2>,|ψ3>}, be orthonormal base in the Hilbert space. Now we make another operator U where U|ψi>=|ψi+1> for i=1,2 and U|ψ3>=|ψ1>. Show that the operator U is unitary operator. 2. The attempt at a solution I'm trying to argue that if the G...

What operator acting on the vacuum state (vacuum state of the box?) gives a m^3 box of cosmic background radiation at 2.7K? As the temperature 2.7K slowly drops (wait a million years) must our operator above change in time? Do photons scatter via gravitions so that their energy changes...

Heyo. On page 4 of Srednicki's QFT text, the following equation is given (in an attempt to make the Schrodinger equation relativistic): ##i\hbar \partial_t \psi(x,t) = \sqrt{-\hbar^2c^2 \nabla^2 + m^2 c^4}\psi(x,t)## where ##\psi(x,t) = \left \langle x|\psi,t \right \rangle## is the position...

Again, from the Peskin and Schroeder's book, I can't quite see how this computation goes: See file attached The thing I don't get is how the term with (\partial^{2}+m^{2})\langle 0| [\phi(x),\phi(y)] | 0 \rangle vanishes, and also why they only get a \langle 0 | [\pi(x),\phi(y)] | 0 \rangle...

Hello MHB, I am stuck at this problem for quite a long time now. Problem. Let $F_p$ denote the field of $p$ elements, where $p$ is prime. Let $n$ be a positive integer. Let $V$ be the vector space $(F_p)^n$ over the field $F_p$. Let $GL_n(F_p)$ denote the set of all the invertible linear...

Hi, Is there any good books which explain/calculate Wigner rotation, tensor operator, two-particle helicity state and related stuff in detail? Thanks.

Homework Statement Hi guys, I've not started a course on QM yet, but we are currently learning the maths used in QM. Show, by taking the trace of both sides show that finite dimensional matrix representations of the momentum operator p and the position operator x which satisfy [p, x] =...

Homework Statement Hi guys, actually this isn't a homework question, but rather part of the working in a textbook on Linear Algebra. Homework Equations The Attempt at a Solution I'm not sure why it's U*li instead of U*il. Shouldn't you flip the order when you do a matrix...

Sorry for reopening a closed thread. But I have exactly the same doubt as this guy: https://www.physicsforums.com/showthread.php?t=346730 And the answer doesn't actually answer his question. I do get delta(p+p'), but they just help me in getting a_{p}a_{-p} and a_{p}^{\dagger}a_{-p}^{\dagger}...

How do these operations with Del operator work?? Homework Statement Let's say A and B are expressed by their cartesian components as: A = <P, Q, R> and B = <M, N, O> what would be the differente between (A.∇)B and B(∇.A) ? Homework Equations The Attempt at a Solution I tried...

The thing is that often in the problems on quantum mechanics I've found that an operator is given, but not the base on which it is represented. I'll give an especific example in a moment. So then the problem asks me to find the eigenvalues and eigenvectors for a given operator and to express it...

I am trying to understand the how the time evolution operator is used versus the Feynman propagate. My limited understanding is the following for which I am seeking clarity: 1. The time evolution operator is a unitary operator which enables us to calculate a probability amplitude of one...

If we have the function f : x \mapsto f(x) = 3x^2, I am used to Lagrange's prime notation for the derivative: f' : x \mapsto f'(x) = 6x. I'm fond of this notation. But it has been mostly abandoned in my engineering courses in favor of Leibniz's notation, using differential operators such...

This has been bugging me for a while, so I'm really hoping someone can give me a good answer. Please get as technical as necessary, I'm a 4th year HE physics grad student so I do know my stuff! Why is the time-reversal operator made anti-unitary in quantum mechanics? It is very straight...

Normal Operator Proof Homework Statement Prove an operator ##T \in L(V)## is normal ##⇔ ||T(v)|| = ||T^*(v)||##. Homework Equations (1) ##T \in L(V)## is normal if ##TT^*= T^*T##. (2) If T is a self-adjoint operator on V such that ##<T(v), v> = 0, \space \forall v \in V##, then...

Hello all, Is there any really good software, package or otherwise, that is helpful for simplifying long operator expressions, particularly with (anti-commuting) electron operators? For example, I'd like to be able to transform Hamiltonians, project onto subspaces with specific occupation...

Find its eigen values. Is this operator linear?

How this is defined? ##\vec{r}\cdot \vec{\sigma}##? where ##\vec{r}=(x,y,z)## and ##\vec{\sigma}=(\sigma_x,\sigma_y,\sigma_z)##. ##\sigma_i## are Pauli spin matrices.

Homework Statement Given the operators \hat{x}=x\cdot and \hat{p}=-i\hbar \frac{d}{dx}, prove that: [\hat{x}, g(\hat{p})]=i\hbar \frac{dg}{d\hat p}Homework Equations None. The Attempt at a Solution I have very little idea on how to begin this problem, but I don't want a solution, I simply...

Srednicki eqn. (2.23) and (2.24) states: We can make this a little fancier by defining the unitary spacetime translation operator T(a) \equiv \exp(-iP^\mu a_\mu/ \hbar) Then we have T(a)^{-1} \phi(x) T(a) = \phi(x-a) How do we get the second equation from the first equation?

While using the ∇ operator, most of the times we can treat it as a vector. I came across a few formulae(basically product rules).. ∇×(A×B)=(B.∇)A-(A.∇)B+A(∇.B)-B(∇.A) where A and B are vectors I wanted to know if there is any direct way of deriving it. By direct I mean assuming the basic...

I am studying an article http://arxiv.org/abs/quant-ph/9907069 and having some problems understanding it. Is self adjointness of an operator a sufficient or necessary and sufficient requirement for its eigen vectors with the generalized eigenvectors (i don't know what are these) to form...

Hello, Something I have some time wondering and still couldn't find the answer is to this question: if there is some relation between the Spectrum (functional analysis) and the Frequency spectrum in Fourier Analysis. Now that I think about it there seems to be a casuality the use of the...

I'm studying Shankar's book (2nd edition), and I came across his equation (15.3.11) about spherical tensor operators: [J_\pm, T_k^q]=\pm \hbar\sqrt{(k\mp q)(k\pm q+1)}T_k^{q\pm 1} I tried to derive this using his hint from Ex 15.3.2, but the result I got doesn't have the overall \pm sign on the...

I am confused about this. I have always thought that the modulo operator always has the result of a while number between 0 and the modulo divisor minus 1. I presume that the terms are called: a % b = c a : dividend b : divisor c : remainder a & b > 0 : a % b = b * [ ( a / b ) -...

I am finding it unintuitive to follow a calculation in a certain notation I am not too familair with. To write down the equation $$ z(t) = - y(t) + \tau \frac{\partial y}{\partial t} $$ the following notation is employed $$ z(t) = -(1-\tau \frac{\partial }{\partial t}) y $$ So far so gud...

Can someone please explain the below proof in more detail? The part in particular which is confusing me is Thanks in advance!

Homework Statement Let B be the linear operator (1-x^{2}) \frac{d^2}{dx^2}-x\frac{d}{dx} Show that T_{4}(x) = 8x^{4} - 8x^{2} + 1 is an eigenvector of B, and find the corresponding eigenvalue. Attempt Righto, I find these rather difficult so a step by step solution would be nice but...

I'm trying to fit together my understanding of quantum mechanics, quantum field theory, given my lacking maths education. In quantum mechanics we have a time displacement operator and a space displacement operator, which are respectively: \hat{T}(t) = e^{-i\hat{H}t} \hat{D}(\underline{x}) =...

If my understanding is correct, the equations of QFT (Dirac, Klein-Gordon) govern the behavior of operator fields (assigning operator to each point in space). Does it mean there are no equations governing the behavior of fields (assigning a number / vector/ spinor to each point in space)? Is QFT...

If the ladder operator ##a=\sqrt {\frac{m\omega}{2\hbar}}x+\frac{ip}{\sqrt{2m\hbar \omega}}## and ##a^\dagger=\sqrt {\frac{m\omega}{2\hbar}}x-\frac{ip}{\sqrt{2m\hbar \omega}}## then I get that the number operator N, defined as ##a^\dagger a## is worth ##\frac{m \omega...

While dealing with a circling particle in an spherical symetric potential our professor said that we can replace an operator of ##z## component of angular momentum ##\hat{L}_z## with the expectation value - he denoted it just ##L_z## - of the angular momentum if ##L_z## is constant. Why is that...

While reading the article:Law of the unconscious statistician, I came across a line and then a few lines after that, the expected value of a a function g(x) is said to be given by: ∫f(x)g(x)dx. However, if g(x) is not explicitly known, how does one calculate the integtral?

I have a question about statistical operator. In statistical physics you deal with temperature. So for example ##\hat{\rho}=\frac{1}{Z}e^{-\beta \hat{H}}## where ##\beta=\frac{1}{k_BT}##. In definition there is temperature. And also equivalent definition is ##\hat{\rho}=\sum_i w_i|\psi_i\rangle...

Hi everyone, I'm stuck on proving that a certain operator is an irreducible spherical tensor operator. These are tensor operators T^{k}_{q} with -k \leq q\leq k satisfying \mathscr{D} T^{k}_{q} \mathscr{D}^{\dagger} = \sum_{q'} \mathscr{D}^{k}_{q' q} T^{k}_{q'} where...

grad, curl , div operator got any meaning?? ∇x F (t) same as the first derivative of F with respect to t , or will get the gradient of F which is normal to the F ? ∇ dot F (t) , will get the scalar value of what?? lets say F is force , then can anyone please give me the meaning of those...

representation of linear operator using "series"? I was looking into the progression of quantum states with respect to time. From what I understood the progression of a state ## \left|\psi(t)\right> ## is given by: $$ \left|\psi(t)\right> = U(t)\left|\psi(0)\right> $$ I'm not sure if that's...

I am wondering about acceleration in quantum mechanics. Let's consider spherically symmetric potential V(r). From the Heisenberg equation of motion, one finds the time derivative of the momentum operator \dot{\hat{p}}=\frac{i}{\hbar}\left[\hat{H},\hat{p}\right] = -\nabla V, from which we can...

Homework Statement Find the eigenvalues and eigenstates of the spin operator S of an electron in the direction of a unit vector n; assume that n lies in the xz plane. Homework Equations S|m>= h m|m> The Attempt at a Solution This question is from Zettili QM and they have...

Self-adjoint operator (Ben's question at Yahoo! Answers) Here is the question: Here is a link to the question: Self-adjoint and properties? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

I'm trying to work out the total momentum operator on page 22 of Peskin/Schroeder for myself, and I'm a little confused about the last few steps. Assuming I went through the first few steps correctly, I've arrive at this expression: $${\bf P}=\frac12\int\frac{d^3p}{(2\pi)^3}{\bf...

I am kind of new to this eigenvalue, eigenfunction and operator things, but i have come across this quote many times: First i need some explanation on how do we know this? All i know about operator ##\hat{H}## so far is this equation where ##\langle W \rangle## is an energy expected value...

Hi all; I'm trying to learn about classes and objects, here is a program that demonstrate operator overloading, i cannot understand how it works, when i tried this: //classes #include <iostream> using namespace std; class CVector { public: int x,y; CVector(){}...

Homework Statement What is the value of x = 5 + (9 * 5) * (3 ^ 3/2 - 20).Homework Equations Operator precedence.The Attempt at a Solution x = 5 + (45) * (3 ^ 3/2 - 20). Is the "^" here the exclusive or operator? In that case wouldn't x have 2 possible values?

Hi guys, I'm sure I'm being very stupid here but I'm reading through notes which contain various actions for fields, most of which are very similar, however there is some discrepancies with the way differential operators are shown acting on the fields and I can't for the life of me work out...

I'm trying to understand the spectrum and resolvent of a linear operator on a Banach space in as much generality as I possibly can. It seems that the furthest the concept can be "pulled back" is to a linear operator T: D(T) \to X, where X is a Banach space and D(T)\subseteq X. But here are a...

Not really a homework problem, just doing some self-studying. Homework Statement Let ##| a \rangle## by any vector in an ##N##-dimensional vector space ##\mathcal{V}##, and ##\mathbf{A}## a linear operator on ##\mathcal{V}##. The vectors $$ | a \rangle, \mathbf{A} | a \rangle...