Operator Definition and 1000 Threads

I've dealt with both the Hamiltonian function for Hamiltonian mechanics, and the Hamiltonian operator for quantum mechanics. I have a kind of qualitative understanding of how they're similar, especially when the Hamiltonian function is just the total energy of the system, but I was wondering if...

Homework Statement sup guys! I think I've solved this set of problems, but I was just wondering if I've done it right - I have no way to tell. I'll put all the questions and answers here - plus the stuff I used. So could you please tell me if there's any mistakes? Here it is - using Word...

"It is left as a problem for the reader to show that if [S,T] commutes with S and T, then [e^{tT}, S] = -t[S,T]e^{tT} I'm not sure if I'm missing something here, but i don't even see how it is possible to arrive at this answer. I get: [e^{tT}, S] = e^{tT}S - Se^{tT} Then using the fact...

Homework Statement I have fallen behind on my Numerical Methods course and am starting to fail it. I need to know how to make a differential approximation and I'm reading through the materials but I have too little time and it doesn't even explain what a differential operator is. At first it is...

What is the exact technical difference between a linear operator and linear function?

So, I understand that the derivative operator, D=\frac{d}{dx} has translational invariance, that is: x \rightarrow x - x_0, and its eigenfunctions are e^{\lambda t}. Analogously, the theta operator \theta=x\frac{d}{dx} is invariant under scalings, that is x \rightarrow \alpha x, and its...

if 2 hermitian operator A, B is commute, then AB=BA, the expectation value <.|AB|.>=<.|BA|.>. how about if A and B is non commute operator? so we can not calculate the exp value <.|AB|.> or <.|BA|.>?

Prove the equation A\left|\psi\right\rangle = \left\langle A\right\rangle\left|\psi\right\rangle + \Delta A\left|\psi\bot\right\rangle where A is a Hermitian operator and \left\langle\psi |\psi\bot\right\rangle = 0 \left\langle A\right\rangle = The expectation value of A. \Delta A...

Homework Statement Hey everyone. Imma type this up in Word as usual: http://imageshack.com/a/img577/3654/q9ey.jpg Homework Equations http://imageshack.com/a/img22/3185/pfre.jpg The Attempt at a Solution http://imageshack.com/a/img703/8571/xogb.jpg

If we consider a spin 1/2 particle, then, the rotation of the spinor for each direction is given by a rotation matrix of half the angle let say theta Rspin=\left(\begin{array}{cc} cos(\theta/2) & -sin(\theta/2)\\sin(\theta/2) & cos(\theta/2)\end{array}\right) and the new component of the spin...

Homework Statement Hey everyone! The question is this: Consider a two-state system with normalized energy eigenstates \psi_{1}(x) and \psi_{2}(x), and corresponding energy eigenvalues E_{1} and E_{2} = E_{1}+\Delta E; \Delta E>0 (a) There is another linear operator \hat{S} that acts by...

Why do we use the coordinates of r in terms of x,y,z?Why don't we express coordinates of A in x,y,z?

For commutator, HQ-QH = 0 . But for this case as shown below, complex ψQHψ - HcomplexψQψ= 0? If the operator Q is in term of (∂/∂t) and (∂/∂x) ,then the HQ-QH may not be zero. Is there any restriction for Q operator?

I'm trying to get my head around quantum mechanics with the help of Sakurai "Modern Quantum Physics". It's been good so far, but I came across a formula I don't really understand. When discussing uncertainty relation (in 1.4) the author begins with defining an "operator": \Delta A \equiv A -...

I'm reading about planning algorithms and I'm having some difficulty understanding a bit of notation here. The pdf I'm reading is "planning.cs.uiuc.edu/ch2.pdf" and the equation in question is on page 11. I'm not sure I understand what the min operator with all the subscripts actually means...

The S_{z} operator for a spin-1 particle is S_{z}=\frac{h}{2\pi}[1 0 0//0 0 0//0 0 -1] I'm given the particle state |\phi>=[1 // i // -2] What are the probabilities of getting each one of the possible results? Now... we can say the possible measure results will be 1,0,-1 and the...

Suppose that x\in H, where H is a Hilbert space. Then x has an orthogonal decomposition x = \sum_{i=0}^\infty x_i. I have a linear operator P (more specifically a projection operator), and I want to write: P(x) = \sum_{i=0}^\infty P(x_i). How can I justify taking the operator inside the...

Hey! Is it true that when you dot the del-operator on another vector, the differentiation has priority over the dot-product? That's why you get all those weird formulas for the divergence in circular and cylindrical coordinates (which are very different to the Cartesian ones)? So in the case of...

Homework Statement Find the spectrum of the Momentum operator in the Hilbert Space defined by L^2([-L,L]), consisting of all square integrable functions ψ(x) in the range -L, to L Homework Equations We can get the resolvent set containting all λ in ℂ such that you can always find a...

Consider the following commutator for the product of the creation/annihilation operators; [A*,A] = (2m(h/2∏)ω)^1 [mωx - ip, mωx + ip] = (2m(h/2∏)ω)^1 {m^2ω^2 [x,x] + imω ([x,p] - [p,x]) + [p,p]} Since we have the identity; [x,p] = -[p,x] can one assume that.. [x,p] - [p,x] =...

Hey guys! Basically, I was wondering how to prove the following statement. I've seen it in the Hamermesh textbook without proof, so I wanted to know how you go about doing it. Let's say you have a group element g_{1}, which has a corresponding inverse g_{1}^{-1}. Let's also define a linear...

Homework Statement Given an inner product space V, a unitary operator U and a set \left\{\epsilon_i\right\}_{i=1,2,\dots} which is an orthonormal basis of V, show that the image of \left\{\epsilon_i\right\} under U is also an orthonormal basis of V Homework Equations The Attempt at a...

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...

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...