Operator Definition and 1000 Threads

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...

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...

Let H and K be hermitian operators on vector space U. Show that operator HK is hermitian if and only if HK=KH. I tried some things but I don't know if it is ok. Can somebody please check? I got a hint on this forum that statements type "if only if" require proof in both directions, so here...

In an article I'm reading, the author defines an operator as below: \hat{U}_{CNOT}(\theta)=\exp{(-i \theta \hat{U}_{CNOT})}=\hat{1} \cos{\theta}-i \hat{U}_{CNOT} \sin{\theta} Where \hat{U}_{CNOT} is the controlled not gate(http://en.wikipedia.org/wiki/Controlled_NOT_gate). Then the...

Homework Statement Homework Equations N/A The Attempt at a Solution I am having trouble getting my head around these questions, the first part a) wasn't too tricky, I used the fact that eigenfunctions of a Hermitian operator \hat{O} are orthogonal and got my normalisation...

Here is the question: Here is a link to the question: Show a linear transformation is self adjoint? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Studying conformal field theory, I tried to derive general expression for the commutation relations of the modes of two chiral quasi-primary fields. At first, I expressed the modes \phi_{(i)m} and \phi_{(j)n} as contour integrals over each fields, and took commutation relation. I used...

I am confuse about the dimension of an operator? Why we need an operator of Dim six or greater for new physics?

is this true? (1/ηαβ∂α∂β)= ηαβ∂α∂β any help,pls!

Homework Statement Individual hydrogen atoms have been prepared in the energy state n = 2. However, nothing is known about the remaining quantum numbers. Fine structure and all corrections can be ignored. What is the micro-canonical statistical operator. Homework Equations \hat{\rho_{mc}} =...

Homework Statement What is the density operator (statistical operator) of a system about which nothing is known? Homework Equations \hat{\rho} = \sum p_{i} |i\rangle\langle i| The Attempt at a Solution If nothing is known about a system we must assume something in order to make...

Hi everyone, I'm stuck on the concept of the rotation operator in QM. From what I understand, one constructs a representation of SO(3) on a Hilbert space by mapping a rotation matrix R\in SO(3) specified by an angle \phi and a unit vector \vec{n} to D(R) = \exp[-\frac{i...

Greetings everyone! I have a set of tasks I need to solve using using operator norms, inner product... and have some problems with the task in the attachment. I would really appreciate your help and advice. This is what I have been thinking about so far: I have to calculate a non trivial upper...

Hi all! I was reviewing some basic quantum mechanics, and I was trying to 'derive' the equation -i \hbar \frac{\partial}{\partial x} \psi_{(t,x)} = <x| \hat{P} | \psi > using the commutator relation, and the form of the identity operator. OK, I know that the proper, mathematical way to prove the...

Homework Statement Show that every linear operator L:ℝ→ℝ has the form L(x) = cx for some c in ℝ. Homework Equations A linear operator in vector space V is a linear transformation whose domain and codomain are both V. The Attempt at a Solution If L is a vector space of the real...

Homework Statement If B is Hermitian, show that BN and the real, smooth function f(B) is as well. Homework Equations The operator B is Hermitian if \int { { f }^{ * }(x)Bg(x)dx= } { \left[ \int { { g }^{ * }(x)Bf(x) } \right] }^{ * } The Attempt at a Solution Below is my...

well i am a starter in QM, i have 2 big doubts ! let me first tell what i understood , there is phsi which defines a state of a system, phsi times x is a position operator and phsi 's derivative of x multiplied by i h is its momentum operator ... well then i operator these in phsi and what do...

Homework Statement consider operator defined as \hat{O_A} = \hat{A} -<\hat{A}> show that (ΔA)^2=<\hat{O_A}^2> Homework Equations (ΔA)^2=<\hat{A}^2>-<\hat{A}>^2 The Attempt at a Solution (ΔA)^2=<\hat{A}^2>-<\hat{A}>^2 = <\hat{A}^2> - (\hat{A} -\hat{O_A})^2 = <\hat{A}^2> -...

Homework Statement Using the operator identity: \hat{L}^2=\hat{L}_-\hat{L}_+ +\hat{L}_z^2 + \hbar\hat{L}_z show explicitly: \hat{L}^2 = -\hbar^2 \left[ \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2} + \frac{1}{\sin\theta} \frac{\partial}{\partial\theta}...

Homework Statement Question: Homework Equations Any relevant to complex numbers. The Attempt at a Solution Given, Arg(\frac{z}{w})= Arg(z)-Arg(w) z=x+yi z1 = -1-2i z2 = 2+3i Arg(z-z1)=Arg(z2-z1) LHS: Arg(x+yi+1+2i) Arg((x+1) + i(y+2)) tan(\theta)=\frac{y+2}{x+1}...

Homework Statement What is the effect of the sequence of ladder operators acting on the ground eigenfunction \psi_0 Homework Equations \hat{A}^\dagger\hat{A}\hat{A}\hat{A}^\dagger\psi_0The Attempt at a Solution I'm not sure if I'm right but wouldn't this sequence of opperators on the ground...

Why is it: $$ \langle A \rangle = \int dV ~ \psi^* (\hat{A} \psi) $$ As opposed to: $$ \langle A \rangle = \int dV ~ (\hat{A} \psi^*) (\hat{A} \psi) $$ The op can substantially change the wf, so it would seem to make more mathematical sense (at least from a linear algebra pov) to...

Dear all, There is a definition of current-density operator, and the form is as follows: j(r)=1/2iƩ[∇lδ(r-rl)+δ(r-rl)∇l] I cannot understand this form, because i think ∇ operator and δ operator are commutable. Another form of current-density operator can be found from this website...

Hello All, I was trying to prove that an operator T in a real vector space V has an upper block triangular matrix with each block being 1 X 1 or 2 X 2 and without using induction. The procedure which i followed was : We already know that an operator in a real vector space has either a one...

In covariant quantization of the string, say as in David Tong's http://arxiv.org/abs/0908.0333 (p28), time is an operator. Is the time operator Hermitian, and does it correspond to an observable?

I asked my professor if we could define time as an operator and he said no. I've read on the web that time isn't an observable because "you don't measure the time of a particle". Yet, to me at least, the sentence "you don't measure the time of a particle" is similar to "you don't measure the...

Hi there, This should be very simple... If I have a state <1|AB|2> where A and B are Hermitian operators, can I rewrite this as <2|BA|1> ? That would be, taking the complex conjugate of the matrix element and saying that A*=A and B*=B. Thank you!

I'm new to QM, but I've had a linear algebra course before. However I've never dealt with vector spaces having infinite dimension (as far as I remember). My QM professor said "the eigenvalues of the position operator don't exist". I've googled "eigenvalues of position operator", checked into...

In a Griffith's book (page 15-16) an author derives a momentum operator. In the derivation he states that he used a integration by parts two times. He starts with this equation which i do understand how to get to. $$ \begin{split} \frac{d \langle x \rangle}{dt} = -\frac{i\hbar}{2m}...

Homework Statement Let a be a fixed nonzero vector in R2. A mapping of the form L(x) = x + a is called a translation. Show that a translation is not a linear transformation. Illustrate geometrically the effect of a translation. My work is in the photo below, can you check and see if I'm...

Consider the displacement operator Tψ(x)=ψ(x+a). Is T Hermitian?

Homework Statement Consider the three operators defined by $$\left(S_i\right)_{jk} = -i\epsilon_{ijk}$$ in the x-y-z space and the basis vectors given in x-y-z space as $$e^{\left(1\right)} = -\frac{1}{\sqrt{2}}\left(e_x + ie_y\right), e^{\left(0\right)} = e_z, e^{\left(-1\right)} =...