Operators Definition and 1000 Threads

why is only one component of angular momentum is quantised, and what determines which component is quantised?

If H is a Hermitian operator, then its eigenvalues are real. Is the converse true?

In Quantum Mechanics, we have linear operators which can act on a ket to produce a new ket. However, we also allow the same operators to act on a bra vector to produce a new bra vector. That is, if \langle\phi| is a bra and A is an operator, the action of A on \langle\phi| is to produce a new...

I'm reading about selection rules, and the book is talking about how if you have a parity switching operator in between two wave vectors of opposite (definite) parity, the result is 0. For example, we have \left\langle2,0,0 \right|\hat{X}\left|2,0,0\right\rangle = 0 because...

As far as I understand, the momentum operator is: \hat{p} = -i \hbar \frac{\partial}{\partial \hat{q}} Where I'm not sure at this point if it's mathematically correct to talk about a derivative wrt. the position operator - but the point is, as far as I understand, that this equality is true...

Hello! I´m trying to read Georgi's book on Lie algebras in particle physics but am confused about the start of chapter 4. Georgi writes that "A tensor operator is a set of operators that transforms under commutation with the generators of some Lie algebra like an irreducible representation of...

Hi there, As you know, we can represent a Linear vector operator as a matrix product, i.e., if T(u) = v, there is a matrix A that u = A.v. What about a linear operator of matrices. I have a T(X) = b where X belongs to R^n_1Xn_2 and b belongs to R^p. What is a suitable representation of...

Which is the role of hermitian and unitary operators in quantum mechanics and which operator is neither hermitian nor unitary

Could you please give me a hint on how to show that a set of operators with a property P is closed under addition? In other words, how one could prove that a sum of any two operators from the set still possesses this property P. The set is assumed to be infinite. Any references, comments...

Spin 1/2--Raising and Lowering operators question Hi, Quick question regarding raising and lowering operators. Sakurai (on pg 23 of Modern QM), gives the spin 1/2 raising and lowering operators S_{+}=\hbar \left|+\right\rangle \left\langle-\right| and S_{-}=\hbar \left|-\right\rangle...

Hi, There are, for example, lists of spherical tensor operators for l=\text{integer} steps, e.g. l=0,1,2,.... T_{k}^{q}(J)\rightarrow T_{0}^{0}=1, \quad T_{1}^{\pm 1}=\mp \sqrt{\frac{1}{2}}J_{\pm},\quad T_{1}^0=J_z and this continues forever. I was wondering if there are operators...

I'm having trouble seeing how an operator can be written in matrix representation. In Sakurai, for an operator X, we have: X = \sum \sum |a''> <a''| X |a'> <a'| since of course \sum |a> <a| is equal to one. Somehow, this all gets multiplied out and you get a square matrix with the...

I'm confused about index calculation in eq. 8.25, Mandl QFT textbook. Can anyone give me a detailed explanation showing the equality below? X=\frac{1}{2}A_{\delta \alpha}^+(\bf{p'})\Gamma_{\alpha \beta}(\bf{p'})A_{\beta \gamma}^+(\bf{p})\widetilde{\Gamma}_{\gamma\delta}...

I have only heard about the use of ladder operators in connection with the harmonic oscillator and spin states. However, I would expect them to be useful in other systems as well. For example, can we find ladder operators for the discrete states of the hydrogen atom, or any other system with...

Hello. I have this problem at hand: Homework Statement A quantum mechanical system has a hamilton operator \hat{H} and another, time independent operator \hat{A}_{0}. Construct a time dependent operator \hat{A}(t) so that: <ψ(t)|\hat{A}_{0}|ψ(t)> = <ψ(0)|\hat{A}(t)|ψ(0)> for all states...

Hello, I'm trying to figure out what hypothesis I need to swap the Re{} (or Im) operator and the integral sign, but I can't find anything on the matter. I guess either it's a trivial question or a rare one. Can someone help me? Thanks in advance.

So I'm starting to learn how to program and I just learned how these operators work (difference between pre and post, etc.). However, I have a question, what real application can these operators have in actual software development? I mean, if for instance I have a variable with an X value...

I am interested in knowing that in QM what vector/tensor operators are? In fact how do they differ from scalar operators?

I have read in different places something like the following: Hermitian operators have real eigenvalues Hermitian operators/their eigenvalues are the observables in Quantum Mechanics e.g energy I am not sure what this means physically. Let us say I have a Hermitian operator operating on a...

Hi guys, My question is regarding defining spin operators in the zero field splitting principal axis system. I am currently working on a S = 2 spin system, and know how to define the Sx, Sy, and Sz spin matrices. My question is, how do I rotate them to the zfs-PAS? Some papers I came across...

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...