Operators Definition and 1000 Threads

Greetings, How do we decide on which sign to take when using the momentum operator? The question may be very simple but I need a push in the right direction. Many thanks.

Homework Statement show that [x,f(p_x)] = i \hbar d/d(p_x) f(p_x) Homework Equations x is the position operator in the x direction, p_x is the momentum operator; i \hbar d/dx [x, p_x]=xp-px The Attempt at a Solution I'm stuck. maybe chain rule for d/dx and d/d(p_x)...? But I...

Theorem: For every Hermitian operator, there exists at least one basis consisting of its orthonormal eigen vectors. It is diagonal in this basis and has its eigenvalues as its diagonal entries. The theory is apparently making an assumption that every Hermitian operator must have eigen...

Ok, I'm having some conceptual difficulty here. When discussing beta functions and the relation how these differential equations flow, I still don't quite get the difference between relevant vs. marginally relevant and irrelevant vs. marginally irrelevant. For instance, take the β function...

Hey guys, I am wondering if the following relationships hold for all operators A, regardless of whether they are linear or non-linear. A-1A = AA-1 = I [A,B] = AB - BA A|an> = λn|an>, where n ranges from 1 to N, and N is the dimension of the vector space which has an orthogonal basis...

I have a homework problem which asks me to compute the second and third excited states of the harmonic oscillator. The function we must compute involves taking the ladder operator to the n-power. My question is this: because the ladder operator appears as so, -ip + mwx, and because I am using it...

Hello all, Homework Statement I’m trying to derive a result from a book on quantum mechanics but I have trouble with bra-ket notation and operators… Let’s say we have a photon moving along the cartesian z-axis. It is polarized and its state is Psi(theta) = cos (theta) x1 + sin(theta) x1...

Often in quantum mechanics, there appears statements of the type : Expected value of operator = a value I am told that operators are instructions and I do not understand how an instruction can have a value, expected or otherwise. Even in the case where the operator is of the form "muliply...

Homework Statement [A^{+}A]=1 A|a>=\sqrt{a}|a-1> A^{+}|a>=\sqrt{a+1}|a+1> <a'|a>=\delta_{a'}_{a} Homework Equations what is 1 <a|A|a+1> 4. <a+1|A^{+}|a> 3. <a|A^{+}A|a> 4. <a|AA^{+}|a> The Attempt at a Solution 1. <a|A|a+1> =<a|\sqrt{a+1}|a+1-1>=\sqrt{a+1}<a|a> since a=a and...

Homework Statement a) Two observables A and B are represented by operators A(hat) and B(hat), which obey the following commutation relation: [A(hat), B(hat)] = iC, where C is the real number. Obtain an expression for the product of the uncertainties ΔAΔB. b) Hadrons, such as protons...

Dear forumers, I was thinking about how the Sz operator "couples" (has non zero matrix elements) states with the same expectation values for the projection of spin on the z-axis (duh! α and β are its eigenvectors), and how Sx and Sy couple different states (once again, duh!). I was also...

How exactly does a physical scientist or mathematician go about modeling the way that a phenomena works or appears to work. For example, how do I know when it's appropriate to introduce something like --> ∂ or even the integral instead of something else? Alternatively, maybe I'm asking, at...

I'm reading about symmetries in QM in "Geometry of quantum theory" by Varadarajan. In one of the proofs, he refers to theorem 2.1, which is stated without proof. He says that the theorem is proved in "Linear algebra and projective geometry" by Baer. That isn't very helpful, since he doesn't even...

I'm taking a course that's taught out of Shankar, and I'm going to be tested on Tensor Operators, which is 15.3, p.417-421 in Shankar. I've never actually worked with tensors before (except the Maxwell stress tensor in EM), and I find that section too hard to understand. Does anyone know of a...

Homework Statement Let A be a linear transformation on the space of square summable sequences \ell2 such that (A\ell)n = \elln+1 + \elln-1 - 2\elln. Find the spectrum of A. 2. The attempt at a solution I see that A is self-adjoint, so its spectrum must be a subset of the real line. We also...

Greetings chaps, This will probably be old hat to most of you, but I'm beginning to start Quantum mech. so that I can develop a deeper understanding of its application in Chemistry ( I'm a Chemistry undergrad -gauge my level from that if you will!) i.) First of all, would I be right in...

This is a doubt straight from Peskin, eq 2.43 ∅(x,t) = eiHt∅(x)e-iHt. This had been derived in Quantum Mechanics. How does this hold in the QFT framework? We don't have the simple Eψ=Hψ structure so this shouldn't directly hold. I'm sorry if this is too trivial

"Integration" on operators Hi! I am having some difficulty in finding a definition about some kind of reverse operation (integration) of a derivative with respect to an operator which may defined as follows. Suppose we have a function of n, in general non commuting, operators H(q_1 ,..., q_n)...

So, my problem statement is: Suppose that two operators P and Q satisfy the commutation relation [Q,P] = Q . Suppose that ψ is an eigenfunction of the operator P with eigenvalue p. Show that Qψ is also an eigenfunction of P, and find its eigenvalue. This shouldn't be too difficult, but...

Is it because measurement of those quantities involves action on the system. And is the idea that as light is to be used to measure momentum which effects its position fundamental of QM or is it merely like an analog to understand.

I'm having trouble figuring out why an equation simplifies the way it does. (x and p refer to x hat and px hat, h refers to h bar, and the momentum operator is h/i dψ/dx ) I want to show that xpψ - pxψ = -h/i ψ I understand that xpψ= x χ h/i dψ/dx And pxψ= h/i x dψ/dx When you try...

I am trying to understand the following which is proving difficult: It is found that (and the proof here is clear) [P, Jj] anticommutes with Vi Where P = parity operator Jj and Vi are the j th and i th components of the angular momentum vector and an arbitrary vector respectively...

Does the definition of the vector triple-product hold for operators? I know that for regular vectors, the vector triple product can be found as \mathbf{a}\times(\mathbf{b}\times\mathbf{c})=( \mathbf{a} \cdot\mathbf{c})\mathbf{b}-(\mathbf{a}\cdot\mathbf{b})\mathbf{c}. Does this identity hold...

Homework Statement This is something I've been trying to prove for a bit today. My quantum mechanics book claims that the following two definitions about hermitian operators are completely equivalent my operator here is Q (with a hat) and we have functions f,g \langle f \mid \hat Q f...

I've been trying to work out some expressions involving commutators of vector operators, and I am hoping some of y'all might know some identities that might make my job a little easier. Specifically, what is \left[\mathbf{\hat A}\cdot\mathbf{\hat B}, \mathbf{\hat C}\right]? Are there any...

what is the physical significance of the commutation of operators?

Homework Statement an operator for a system is given by \hat{H}_0 = \frac{\hbar \omega}{2}\left[\left|1\right\rangle\left\langle1\right| - \left|0\right\rangle\left\langle0\right|\right] find the eigenvalues and eigenstates Homework Equations The Attempt at a Solution so i...

I understand how the Pauli matrices can operate on the quantum state of an electron to obtain measurements of its intrinsic spin along the x, y and z axes. I also understand that since these matrices do not commute, it is impossible to determine what all three components were before measurment...

Hi, I have just started looking at angular momentum in quantum mechanics and I am considering the question, ''Write down the Schrodinger-like equations for the orbital angular momentum operators L^2 and Lz. Would I be correct in thinking this would be; L^2|ψ=l(l+1)ħ|ψ Lz|ψ=mlħ|ψ Thanks...

Hi, I'm doing some exercise about second quantization. In a exercise about spiorial field I have to explicitly write the Hamiltonian of a Majorana-Langrangian, in terms of operators of creation and annihilation: A_{\vec{k},\lambda} that acts on Fock's space. The point is that during the...

Hi, A general question.. In analytical mechanics, we take a given hamiltonian and re-write it in term of generalzed coordinates. In a way- we recode the hamiltonian to concern only the "essence" of the problem. However, it seems to me, that in QM we do the opposite- we look for operators that...

Homework Statement Determine [T]β for linear operator T and basis β T:((x1; x2]) = [2x1 + x2; x1 - x2] β = {[2; 1], [1; 0]} Homework Equations Now that would be MY question :rolleyes: The Attempt at a Solution Well the answer is [1, 1; 3, 0], but i have no idea what I'm even...

Homework Statement In my quantum class we learned that if two operators commute, we can always find a set of simultaneous eigenvectors for both operators. I'm having trouble proving this for the case of degenerate eigenvalues.Homework Equations Commutator: [A,B]=AB-BA Eigenvalue equation:A...

consider a particle in the box in one dimension with the length a. the hamiltonian is H. then the box's walls goes far away and the box length gets b. now the hamiltonian is H1. i like to know whether these two hamiltonians commute or not?

I would just like some clarification and some assertion that I've got the right idea. Please correct everything I say! For any observable A over a finite-dimensional vector space with orthonormal basis kets \{|a_i\rangle\}_{i=1}^n we can write A = IAI = \left(\sum_{i=1}^n |a_i\rangle\langle...

Homework Statement Show that f(a†a)a† = a†f(a†a + 1) Where f is any analytic function and a and a† satisfy commutation relation [a, a†] = 1. The Attempt at a Solution I have used [a, a†] = aa†-a†a=1 to write the expression like f(a†a)a†= a†f(aa†) but I don't know what to do...

Homework Statement Find the hermitian conjugates, where A and B are operators. a.) AB-BA b.) AB+BA c.) i(AB+BA) d.) A^\dagger A Homework Equations (AB)^\dagger =B^\dagger A^\dagger The Attempt at a Solution Are they correct and can I simplify them more? a.)...

Just to check something: If A and B are operators and B|a> = 0, does this imply that <a|AB|a> = 0 ? Or can you not split up the operators like <a|A (B|a>) ? Thanks.

Hi! Could anyone please tell me the meaning of Tow operators are unitary equivalent. I tried Wiki but I did not get my goal.

Homework Statement The operators J(subscript x)-hat, J(subscript y)-hat and J(subscript z)-hat are Cartesian components of the angular momentum operator obeying the usual commutation relations ([J(subscript x)-hat, J(subscript y)-hat]=i h-bar J(subscript z) etc). Use these commutation...

How can I formally demonstrate this relations with hermitian operators?(A^{\dagger})^{\dagger}=A (AB)^{\dagger}=B^{\dagger}A^{\dagger} \langle x|A^{\dagger}y \rangle=\langle y|Ax \rangle ^* If \ A \ is \ hermitian \ and \ invertible, \ then \ A^{-1} \ is \ hermitian I've tried to prove them...

Homework Statement I need to prove that, <p'|\hat{x}p> = i\hbar\frac{d}{dp'}\delta{p-p'} i.e. find the position operator in the momentum basis p for p'... It's easy to prove that <x'|\hat{x}x> = <\hat{x}x'|x> = x'<x'|x> = x'\delta{x-x'} (position operator in position basis for x') since I...

Hello everybody, long time reader, first time poster. I've searched the forums extensively (and what seems like 60% of the entire internet) for anything relevant and haven't found anything, please point me in the right direction if you've seen this before! Homework Statement Show that even...

Can anyone help with the 2 questions below: 1) Suppose f is a mathematical quantity that can take on the "states", or "values", f1, f2,...fi,...fn., where n can be finite or infinite. So, F is the Set { f1, f2,... fi,.. fn } or F = { f1, f2, fi,... fn } = { all...

Penrose says in “Cycles of Time” that rest mass isn't exactly a Casimir operator of the de Sitter group, so a very slow decay of rest mass isn't out of the question in our universe. If rest mass is strictly conserved, should it be a Casimir operator of the de Sitter group? Decay of rest...

Homework Statement Consider a quantum system that acts on an N-dimensional space. We showed that any operator could be expressed as a polynomial of the form O=\sum_{m,n=1}^{\infty}o_{mn}U^m V^n where U and V are complementary unitary operators satisfying (U^N = V^N =1) Show that if O...

Homework Statement Please see attached pdf. Can anyone please advise on how to approach the questions or provide websites where I can read up on it? I am reading Griffiths but it doesn't seem to cover this much. And I don't know how to google on this topic because I can't type the characters...

In a right-handed cartesian coordinate system the divergence and curl operators are respectively: \nabla \cdot A= \frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z} \nabla \times \mathbf{A}= \begin{vmatrix} \widehat{x} & \widehat{y} &...

Hi, I have this question for a problem sheet: Use the unit operator to show that a Hermitian operator A can be written in terms of its orthonormal eigenstates ln> and real eigenvalues a as : A=(sum of) ln>a<nl and hence deduce by induction that A^k = (sum of) ln>a^k<nl I have no...

Homework Statement A bound quantum system has a complete set of orthonormal, no-degenerate energy eigenfunctions u(subscript n) with difference energy eigenvalues E(subscript n). The operator B-hat corresponds to some other observable and is such that: B u(subscript 1)=u(subscript 2) B...