Operators Definition and 1000 Threads

This is strictly a math question but I figured that since it is something which would show up in QM, the quantum folks might be already familiar with it. Suppose we have an operator valued function A(x) of a real parameter x and another function f, both of which have well defined derivatives...

Homework Statement I have to proove: [\hat{y},\hat{p}_y]=[\hat{z},\hat{p}_z]=i\hbar\hat{I} Homework Equations [\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A} The Attempt at a Solution Ok so I know that...

I have been told that if we have two operators, A and B, such that AB = BA then this is equivalent with that A and B have a common base of eigenfunctions. However, the proof given was made under the assumption that the operators had a non-degenerate spectrum. Now I understand that one rather...

Homework Statement I have some operators, and need to figure out which ones are Linear (or not). For example: 1. \hat{A} \psi(x) \equiv \psi(x+1) Homework Equations I have defined the Linear Operator: \hat{A}[p\psi_{1}+q\psi_{2}]=p\hat{A}\psi_{1}+q\hat{A}\psi_{2} The Attempt at a...

Homework Statement I have some operators, and need to figure out which ones are Hermitian (or not). For example: 1. \hat{A} \psi(x) \equiv exp(ix) \psi(x) Homework Equations I have defined the Hermitian Operator: A_{ab} \equiv A_{ba}^{*} The Attempt at a Solution I just don't know where...

Homework Statement Show that the operator x^kp_x^m is not hermitian, whereas \frac{x^kp_x^m+p_x^mx^k}{2} is, where k and m are positive integers. The Attempt at a Solution Is this valid? <x^kp_x^m>^*=\left(\int_{-\infty}^\infty\Psi^*x^k(-i\hbar)^m\frac{\partial^m\Psi}{\partial...

I'm pondering, since we've introduced formalism, all operators are either scalars or vector components, does it make sense to define operators like r, theta, phi (as in spherical coordinates) which are neither? In classical mechanics we can easily transform observables fro cartesian to...

Suppose \Omega_1 and \Omega_2 satisfy [\Omega_1,\Omega_2]=0 and \Omega = \Omega_1 + \Omega_2. If \Psi_1 and \Psi_2 are eigenvectors of \Omega_1 and \Omega_2, respectively, don't we know that the (tensor?) product \Psi = \Psi_1 \Psi_2 is an eigenvector of \Omega? Also, if the \Psi_i are...

Hi. If c and c^\dagger are fermion annihilation and creation operators, respectively, we know that cc^\dagger+c^\dagger c=1 and cc=0 and c^\dagger c^\dagger=0. I can use this to show the following [c^\dagger c,c]=c^\dagger cc-c c^\dagger c=-cc^\dagger c=-c(1-cc^\dagger)=-c But on the...

Homework Statement For a particle of mass m moving in the potential V(x) = \frac{1}{2}m\omega^2x^2 (i.e. a harmonic oscillator), it is often convenient to express the position and momentum operators in terms of the ladder operators a_{\pm}: x = \sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_-) p =...

Homework Statement Find the operator for position x if the operator for momentum p is taken to be \left(\hbar/2m\right)^{1/2}\left(A + B\right), with \left[A,B\right] = 1 and all other commutators zero. Homework Equations Canonical commutation relation \left [ \hat{ x }, \hat{ p } \right ] =...

Homework Statement I am trying to understand the allowed eigenvalues for the angular momentum operators J and L. In particular why, mj can take integer and half-integer values whereas ml can take only integer values. Homework Equations I have learned about angular momentum operators as...

I have some math questions about quantum theory that have been bugging me for a while, and I haven't found a suitable answer in my own resources. I'll start with the Trace operation. Question A) My understanding is that if we take system A and perform the partial trace over system B, we...

Homework Statement !( ((count<10) || (x<y)) && (count >=0) ) where count is equal to 0Homework Equations i don't think any equations here are necessary except maybe the precedence lawThe Attempt at a Solution they combined 'and' and 'or' which confused the heck out of me. How do i figure out...

In David Tong's QFT notes (http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf p. 43, eqn. 2.89) he shows how the commutator of a scalar field \phi(x) and \phi(y) vanishes for spacelike-separated 4-vectors x and y, establishing that the theory is causal. For equal time, x^0=y^0, the commutator is...

We use the antisymmetric Fock space ( "fermions"). We denote by c(h) a creator operator. I need to evaluate the following quantity: < \Omega , \big(c(h_1)+c(h_1)^{*}\big)\big(c(h_2)+c(h_2)^{*}\big) \ldots \big(c(h_n)+c(h_n)^*\big)\Omega> where \Omega is the unit vector called vaccum...

Hi, this isn't a homework question per se (it's the summer hols, I'm between semesters) but it's something that I never really got during the QM module I just did. I found myself blindly calculating exam & homework problems, and just feel like this is some stuff I should get cleared up...

Im reading in a quantum mechanics book and need help to show the following relationship, (please show all the steps): If A,B,C are operators: [A,BC] = B[A,C] + [A,B]C

Hi there, I'm neither a physicist or a mathematician, so I'm having a bit of trouble understanding commutative properties of operators. Here is an example question, if anyone could help show me how to solve it, it would be greatly appreciated. Show that Lz commutes with T and rationalize...

hi, I am a novice to quantum mechanics and get a lot of troubles with operators. I cannot explain why: - why QM uses operators for observables such as position, momentum, energy, ..ect, but classical physics does not? - what are physical interpretations of operators? - why are operators needed...

Hello, I hope I am asking this in the right area of the forums. My teacher wrote the following formula down at our last meeting, and I was wondering if it was true ( \mathcal{H} is the infinite dimension separable Hilbert space): \mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H}...

Hello, I hope I am asking this in the right area of the forums. I wanted to ask if the following formula is true (assuming H is infinite dimensional and separable): \mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H})\approx...

Homework Statement The operator Q obeys the commutation relation [Q, H] = EoQ, where Eo is a constant with units of energy. Show that if ψ(x) is a solution of the time-independent Schrodinger equation with energy E, then Qψ(x) is also a solution of the time-independent Schrodinger equation...

Hi guys The fermionic creating and annihiliations operators: Do they satisfy c_{i,\sigma }^\dag c_{i,\sigma }^{} = - c_{i,\sigma }^{} c_{i,\sigma }^\dag for some quantum number i and spin σ, i.e. do they commute?

Hello there! Above is a problem that has to do with Lie Theory. Here it is: The operators P_{i},J,T (i,j=1,2) satisfy the following permutation relations: [J,P_{i}]= \epsilon_{ij}P_{ij},[P_{i},P_{j}]= \epsilon_{ij}T, [J,T]=[P_{i},T]=0 Show that these operators generate a Lie algebra. Is that...

Homework Statement So we have an observable K = \begin{bmatrix} 0 & -i \\ -i & 0 \end{bmatrix} and its eigenvectors are v1 = (-i, 1)T and v2 = (i, 1)T corresponding to eigenvalues 1 and -1, respectively. Now if we take the outer products, we get these... |1><1| = (-i, 1)T*(i, 1) =...

I am led to believe (because it is in a paper I am reading) that \frac{1}{H - z} \left|\phi\rangle = \frac{1}{E - z}\left|\phi\rangle where H is the hamiltonian, \left|\phi\rangle is an energy eigenstate with energy E, and z is a complex variable. In attempting to understand this expression...

Fredkin Gates are supposed to be universal. So far I've gotten AND, OR and NOT out of them but I can't figure out XOR. Any help? I know that A XOR B = (A AND NOT B) OR (B AND NOT A), but trying to recreate that with Fredkin Gates is not very elegant... is that the only way? Edit: I guess I...

Homework Statement Hi, I'm currently studying for a quantum mechanics exam but I am stuck on a line in my notes: Ha\left|\Psi\right\rangle =\hbar\omega\left(a^{t}a a + \frac{a}{2}\right)\left|\Psi\right\rangleHa\left|\Psi\right\rangle =\hbar\omega\left(\left(a a^{t} - 1\right)a +...

Is it possible to express fermion annihilation operator as a function of position and momentum? I've seen on Wikipedia the formula for boson annihilation operator: \begin{matrix} a &=& \sqrt{m\omega \over 2\hbar} \left(x + {i \over m \omega} p \right) \\ a^{\dagger} &=& \sqrt{m \omega...

Homework Statement Hi, my problem is with part two of the question I've attached. I'm not exactly sure what they are expecting me to do, is it simply calculating the expectation value of L_z , from the wavefunction given (i.e. cos(φ)) Thanks.

Hi, Here's my problem, probably not that difficult in reality but I don't get how to approach it, and I've got an exam coming up soon... An atom with total angular momentum l=1 is prepared in an eigenstate of Lx, with an eigenvalue of \hbar. (Lx is the angular momentum operator for the...

urgent help!.. Finding eigenvalues of angular momentum operators the question is asking to find the eigenvalues of: 3/5 Lx - 4/5 Ly ... I feel that i have to connect it with the L^2 and Lz operators but i just have no idea how to start .. any suggestions will be greatly appreciated ..

Hello! I met some annoying problems on quantum field operators in QFT.They are as follows: (1)The quantum field operator( scalar field operator, for example),is often noted as φ（r,t). Can it be interpreted as like this: φ（r,t) is the coordinate represetation of a...

Homework Statement If an electron is in an eigen state with eigen vector : [1] [0] what are the expectation values of the operators S_{x}, and S_{z} Interpret answer in terms of the Stern-Gerlach experiment. The Attempt at a Solution Im not too sure how to calculate the...

I'm learning fluid mechanics, and I am confused about the following differential operators. What's the difference between each?

I am trying to find an example of a diagonal linear operator T in L(H) H is hilbert space that is bounded but not compact and also one which is compact but not Hilbert-Schmidt. any Ideas?? Where diagonal means Ten=§en where § is the eigenvalue and en is on orthonormal basis.

I always get slightly confused with the rules of differentials. now \frac{d^{2}y}{dx^{2}} is the scond derivative of the function y(x but rooting this does NOT give the first derivative dy/dx However, with the operator \frac{d^{2}}{dx^{2}}, it seems that you can root this and it DOES...

Hi, I'm reading Shankar's Principles of QM and I find it not very clear on how exactly should I change basis of operator. I know how to change basis of a vector so can I treat the columns of operator matrix as vectors and change them? Or is it something else?

Hi guys When working with operators in second quantization, I always imagine c^\dagger_ic_j as denoting the "good old" matrix element \left\langle {i} \mathrel{\left | {\vphantom {i j}} \right. \kern-\nulldelimiterspace} {j} \right\rangle . But how should I interpret an...

[SIZE="3"]Hi everyone ! Could you tell me who is the inventor of these two operators (creation/anihilation) ? Was he a mathematician ? A physicist ? or both ? Who was the first to use them in Quantum Mechanics ? It's hard to find this kind of information. Thanks Jonathan

based on what is it concluded that this is operation precedes that operation?

hi all, Simple questions.. I am dealing with the del operator (grad, div curl) in one coord system, but say I parametrise my system into another one. How then do I redefine the grad, div, and curl operators. Any links would be really helpful.

Today in class, by the existence of an operator that exchanges the states of two indistinguishable particles, we attempted to derive the existence of fermions and bosons & how this relates to the symmetries of multiparticle wave functions. The argument given in my textbook is: define an...

Definie linear operators S and T on the x-y plane as follows: S rotates each vector 90 degress counter clockwise, and T reflects each vector though the y axis. If ST = S o T and TS = T o S denote the composition of the linear operators, and I is the indentity map which of the following is true...

I've never seen an expectation value taken and would greatly appreciate seeing a step by step of how it is done. Feel free to use any wavefunction, this is the one I've been trying to do: In the case of \Psi=c1\Psi1 + c2\Psi2 + ... + cn\Psin And the operator A(hat) => A(hat)\Psi1 =...

Calculate [Lz,L+] By defintion ladder operators are: L+=Lx+iLy L-=Lx-iLy Important Relations: LxLy = i\hbarLz, LyLz = i\hbarLx, LzLx = i\hbarLy Lx = ypz - zpy, Ly = xpz - zpx, Lz = xpy - ypx To start solving; [Lz,L+] Lz - (Lx + iLy) = 0 Multiply through by \hbar...

L(A)=2A My book doesn't have any examples of how to do this with matrices so I don't know how to approach this.

So I have a couple of questions in regards to linear operators and their eigenvalues and how it relates to their matrices with respect to some basis. For example, I want to show that given a linear operator T such that T(x_1,x_2,x_3) = (3x_3, 2x_2, x_1) then T can be represented by a diagonal...

Homework Statement http://img716.imageshack.us/i/captur2e.png/ http://img716.imageshack.us/i/captur2e.png/ Homework Equations Stuck on the last part The Attempt at a Solution http://img689.imageshack.us/i/capturevz.png/ http://img689.imageshack.us/i/capturevz.png/