Operators Definition and 1000 Threads

Homework Statement Evaluate <n|p^2|n> where p is the momentum operator for the quantised harmonic oscillator. Homework Equations creation operator: a+|n>=sqrt(n+1)|n+1> annihilation operator: a|n>=sqrt(n)|n-1> The Attempt at a Solution the operator p can be defined in terms of...

Homework Statement If L((1,2)^T) = (-2,3)^T and L((1, -1)T) = (5,2)T determine L((7,5)T) Homework Equations If L is a linear transformation mapping a vector V into W, it follows: L(v1 + v2) = L(v1) +L(v2) (alpha = beta = 1) and L (alpha v) = alpha L(v) (v = v1, Beta = 0)...

Nice to be back here at PF and to physics after a year off in the software industry. Now it's time to catch up again :) I feel like I never really get the grasp of manipulating operators. In QM there's a lot of trixing and mixing going on, and I really would like to learn to do the magic...

Is it correct to express quantum field theory as "operator valued function" or "operator function" to spacetimepoints. Also, how value of field at each point act as a separate degrees of freedom.

I read somewhere that the meaning of applying an operator to a wave equation is measuring the quantity associated with that operator.And because the result is a function different than the wave function,the system is changed because of the measurment. But there is a problem here.If the above...

A,B and C are three hermitian operators such that [A,B]=0, [B,C]=0. Does A necessarily commutes with C?

Homework Statement I want to understand the proof of proposition 7.1 in Conway. The theorem says that if \{P_i|i\in I\} is a family of projection operators, and P_i is orthogonal to P_j when i\neq j, then for any x in a Hilbert space H, \sum_{i\in I}P_ix=Px where P is the projection...

Homework Statement Suppose that the commutator between two Hermitian operators â and \hat{}b is [â,\hat{}b]=λ, where λ is a complex number. Show that the real part of λ must vanish. Homework Equations Let A=â B=\hat{}b The Attempt at a Solution AΨ=aΨ BΨ=bΨ...

So, the rule for finding the matrix elements of an operator is: \langle b_i|O|b_j\rangle Where the "b's" are vector of the basis set. Does this rule work if the basis is not orthonormal? Because I was checking this with regular linear algebra (in R3) (finding matrix elements of linear...

I'm trying to show that \sum_{m=0}^\infty \frac{1}{m!} (-1)^m {a^{\dagger}}^m a^m =|0 \rangle\left\langle 0| Where a and {a^{\dagger}} denote the usual annihilation and creation operators. The questions suggests acting both sides with |n> but even if I did that and showed LHS=...=RHS then that...

I'm trying to show that N=a^\dagger a and K_r=\frac{a^\dagger^r a^r}{r!} commute. So basically I need to show [a^\dagger^r a^r,a^\dagger a]=0. I'm not quite sure what to do, I've tried using [a,a^\dagger] in a few places but so far haven't had much success.

This is probably easy. It's really annoying that I don't see how to do this... A finite rank operator (on a Hilbert space) is a bounded (linear) operator such that its range is a finite-dimensional subspace. I want to show that if T has finite rank, than so does T*. I'm thinking that the...

Homework Statement Show L(X,Y) is a vector space. Then if X,Y are n.l.s. over the same scalar field define B(X,Y) = set of all bounded linear operators for X and Y Show B(X,Y) is a vector space(actually a subspace of L(X,Y) Homework Equations The Attempt at a Solution im not sure if i have...

Homework Statement I am struggling to understand spin transformations and have used Sakurai's method of |new basis> = U |old basis> to change basis vectors and hence should have Sz' = Udagger Sz U to transform the operator. I thought this should give Sz' = Sy in the workings (see...

Hi, I'm doing question 2/II/32D at the top of page 68 here (http://www.maths.cam.ac.uk/undergrad/pastpapers/2005/Part_2/list_II.pdf ). I have done everything except for the last sentence of the question. This is what I have attempted so far: |\chi\rangle=|\uparrow\rangle=\left(...

Hi! Info: This is a rather elementary question about the creation a(+) and annihilation (a-) operators for the 1D H.O. The problem is to calculate the energy shift for a given state if the weak perturbation is proportional to x⁴. Using first order perturbation theory for the...

If a_m = \frac{1}{\sqrt{N}} \sum_k e^{-ikm}a_k where a_k is a bosonic operator fulfilling [a_k, a_{k'}^{\dagger}] = \delta_{kk'} then is the product a_m a_{m+1} = \frac{1}{N} \sum_k e^{-ikm}e^{-ik(m+1)}a_k a_{k+1} ? Because that's what I'm doing but it doesn't lead me anywhere near to...

I am reading through 'An Introduction to QFT' by Peskin & Schroeder and I am struggling to follow one of the computations. I follow writing the field \phi in Fourier space ϕ(x,t)=∫(d^3 p)/(2π)^3 e^(ip∙x)ϕ(p,t) And the writing the operators \phi(x) and pi(x) as ϕ(x)=∫(d^3 p)/(2π)^3...

This is basically more of a math question than a physics-question, but I'm sure you can answer it. My question is about linear operators. If I write an operator H as (<al and lb> being vectors): <alHlb> What is then the relationship between H action the ket and H action on the bra. Is this for...

Hi, this is actually more a math-problem than a physics-problem, but I thought I'd post my question here and see if anyone can help me. So I'm writing an assignment in which I have to define, what is understood by a hermitian operator. My teacher has told me to definere it as: <ϕm|A|ϕn> =...

Homework Statement Use the operator expansion theorem to show that Exp(A+B) = Exp(A)\astExp(B)\astExp(-1/2[A,B]) (1) when [A,B] = \lambda and \lambda is complex. Relationship (1) is a special case of the Baker-Hausdorff theorem. Homework Equations Operator expansion theorem...

This might be a basic question, but I'm having some difficulty understanding expectation values and ladder operators for angular momentum. <L+> = ? I know that L+ = Lx+iLy, but I don't know what the expectation value would be? Someone told me something that looked like this...

I decided to go over the mathematical introductions of QM again.The text I use is Shankar quantum, and I came across this theorem: "If \Omega and \Lambda are two commuting hermitain operators, there exists (at least) a basis of common eigenvectors that diagonalizes them both." in the proof...

Suppose we have time-dependent operator a(t) with the equal-time commutator [a(t),a^{\dag}(t)]=1 and in particular [a(0),a^{\dag}(0)]=1 with Hamiltonian H=\hbar \omega(a^\dag a+1/2) The Heisenberg equation of motion \frac{da}{dt}=\frac{i}{\hbar}[H,a]=-i\omega a implies...

Homework Statement Prove: (QR)*=R*Q*, where Q and R are operators. (Bij * I mean the hermitian conjugate! I didn't know how to produce that weird hermitian cross) The Attempt at a Solution I have to prove this for a quantum physics course, so I use Dirac's notation with two random functions f...

Can someone help me with this? When two linear operators commute, I know how to show that they must have at least one common eigenvector. Beyond this fact, what else can be said about commutative operators and their eigenvectors? Further, can they be diagonalized simultaneously (or actually, can...

I am trying to understand the idea of measurements on a system. Forgive me if any of my interpretations are incorrect...I'm hoping things can be cleared up. A measurement is taken on a system, represented by an operator, and this measurement changes the state of the system into a state...

Hi Please take a look at equation 8.60 in the following link...

Most quantum textbooks will tell you that converting between the Schrödinger and Heisenberg pictures involves something like the following: |\Psi(t)\rangle = e^{i\hat{H}(t-t_0)}|\Psi(t_0)\rangle This does make sense to me conceptually: we define eigenstates of the Hamiltonian |E\rangle, where...

Hi Say I have the creation/annihilation operators for fermions given by c and the exponential operator exp(-iHt), where H denotes the Hamiltonian of the (unperturbed) system. Is there any way for me to find out if exp(-iHt) and c (and its adjoint) commute?Niles.

Homework Statement How this would look in K space -\sum_{\vec{n},\vec{m}}I_{\vec{n},\vec{m}}\hat{a}^+_{\vec{n}}\hat{a}_{\vec{n}}\langle \hat{a}^+_{\vec{n}}\hat{b}^+_{\vec{m}}\rangle I need to get -\sum_{\vec{k}}\hat{a}^+_{\vec{n}}\hat{a}_{\vec{n}}\frac{1}{N}\sum_{\vec{q}}J(\vec{q})\langle...

Homework Statement please see attached Homework Equations The Attempt at a Solution Ok so I've done A and have worked out eigenvalues and vectors of H and B For H I get 4 possible eigenvectors (1,0,0) (0,1,1) (0,0,1) and (0,1,0) . The q is why does neither matrix uniquely...

Is there an easy way to prove the identities: e^{\hat{A}}e^{\hat{B}}=e^{\hat{A}+\hat{B}}e^{[\hat{A},\hat{B}]/2} and e^{\hat{A}}\hat{B}e^{-\hat{A}}=\hat{B}+[\hat{A},\hat{B}]+\frac{1}{2!}[\hat{A},[\hat{A},\hat{B}]]+\frac{1}{3!}[\hat{A},[\hat{A},[\hat{A},\hat{B}]]]+...In Zettili they give that...

So we all know about a and a^\dagger. My problem says that if f(a^\dagger) is an arbitrary polynomial in a^\dagger then af(a^\dagger)|n> = \frac{df(a^\dagger)}{da}|0> where |0> is the ground state energy. How can I go about proving this? A hint would be highly appreciated. Thanks,

Find < px >,< p > and < p2 > for the 1s electron ofa hydrogen atom. i am tried the solution but momentum operators Differential for x or y or z and the wave equation depends on the r !

In thermodynamics, two variables A and B are uncorrelated when: <AB>=<A><B> where <> are the expectation values in thermodynamics (for example calculated using Boltzmann distributions). What are the conditions in quantum mechanics for two operators to be uncorrelated, i.e...

"if Quantum Mechanics (QM) is complete (and there are no "hidden variables"), then there cannot be simultaneous reality to non-commuting operators" - Taken from http://drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm I am trying to understand this sentence but I do not fully...

Hi, I am trying to show that for two operators S and P: (S+P)^{-1}=S^{-1}-S^{-1}P(S+P)^{-1} I can't get anywhere and searching on google I am not even sure if it is possible to solve the general case but the question gives no more hints. Any help appreciated. Thanks. J.

Why do we have to use operators in QM?

This is probably falls within a problem of Mathematica as opposed to a question on here but I have a question about the following: Given some cylinder with the shape operator matrix: {{0,0},{0,-1/r}} We get eigenvalues 0 and -1/r and thus eigenvectors {0, -1/r} and {1/r, 0} by my...

1. Explain why <n|(a-a+)^3|n> must be zero 2. a and a+ (a dagger) are the raising and lowering operators (creation and annihilation operators). 3. Because it says explain, I am not sure any mathematical proof is needed. I am best answer is that because (ignoring that the bracket...

Obviously linear operators are ideal to work with. But is there a deeper reason explaining why they're ubiquitous in quantum mechanics? Or is it just because we've constructed operators to be linear to make life easier?

Homework Statement I must calculate [X,P].Homework Equations Not sure. What I've researched through the Internet suggests that [\hat A, \hat B]=\hat A \hat B - \hat B \hat A and that [\hat A, \hat B]=-[\hat B, \hat A]. Furthermore if the operators commute, then [\hat A, \hat B]=0 obviously...

Hi, I've been thinking about the following: In an infinitely deep box a particle's energy operator can be written as E = p^2/2m, and the momentum operator as p = -i hbar dx. (particle moves in x direction) I can see that the commutator of E and p is 0, so the operators commute, and should...

Hi All, When computing the commutator \left[x,p_{y}\right], I eventually arrived (as expected) at \hbar^{2}\left(\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) - \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)\right) and I realized that, as correct...

Hello community. My second stupid question. In the book by Nouredine Zettili , the uncertaintie operators are defined as \DeltaA = A - <A>, where A is an operator, and <A> = <\psi|A|\psi> , with repect to a normalized state vector |\psi> i was wondering, why is not the uncertainty...

1. a) The action of the parity operator, \Pi(hat), is defined as follows: \Pi(hat) f(x) = f(-x) i) Show that the set of all even functions, {en(x)}, are degenerate eigenfunctions of the parity operator. What is their degenerate eigenvalue? The same is true for the set of all odd functions...

Homework Statement To test my knowledge of Sakurai, I asked myself to: "Prove that an operator being unitary is independent of basis." The Attempt at a Solution I want to show the expansion coefficients’ squared magnitudes sum to unity at time “t”, given that they do at time t = t0...

Hello there. I'm having some problems with absolute values when they contain multiple "abs" operators and some other numbers outside the "abs"-es. For example: \left | x+2 \right | - \left | x \right | > 1 If i check it for the positive scenario, the result is true for all x-es. x+2...

Homework Statement H=(J1^2+J2^2)2A+J3^2/2B where J1,2,3 are the angular momentum operators and A and B are just numbers Homework Equations The Attempt at a Solution I rewrote the Hamiltonian as (J^2-Jz^2)/2A + J3^2/2B and got the eigenvalues to be (h^2L(l+1)-h^2m^2)/2A+h^2m^2/2B...