Operators Definition and 1000 Threads

Is it okay to define a local operator as an operator whose matrix elements in position space is a finite sum of delta functions and derivatives of delta functions with constant coefficients? Suppose your operator is M, and the matrix element between two position states is <x|M|y>=M(x,y). It...

I´m having a hard time proving the next result: Let T:V→V be a linear operator on a finite dimensional vector space V . If T is irreducible then T cyclic. My definitions are: T is an irreducible linear operator iff V and { {\vec 0} } are the only complementary invariant subspaces. T...

I've been watching Leonard Susskind's videos on quantum entanglements. Naturally, one of the things that he has been discussing is spin and its various operator Hermitian matrices and eigenvalues. Now I have two main questions about this: 1. I know that if you apply a spin operator σ (which is...

Homework Statement Let Amn be a matrix representation of some operator A in the basis |φn> and let Unj be a unitary operator that changes the basis |φn> to a new basis |ψj>. I am asked to write down the matrix representation of A in the new basis. Homework EquationsThe Attempt at a Solution...

are operators solely used to find the expectation value of something? What does it mean to use the momentum operator over wavefunction? What does it give? I am guessing it doesn't give momentum since momentum can never be a function of space. How to calculate kinetic energy, given the...

Hi all. Consider a UV cutoff regulator ##\Lambda## with an effective QED lagrangian ##\mathcal{L}_{\Lambda} = \bar{\psi}_{\Lambda}(i\not \partial - m_{\Lambda})\psi_{\Lambda} - \frac{1}{4}(F^{\mu\nu}_{\Lambda})^2 - e_{\Lambda}\bar{\psi}_{\Lambda}\not A_{\Lambda}\psi_{\Lambda}##. One can of...

Hi. This might sound like a stupid question, but is it, in general, true that ##(\hat{H} \psi)^* \psi'= \psi^* \hat{H}^*\psi'##? Here ##\hat{H}## is a hermitian operator and ## \psi## a wave function. I.e. do they switch places even when not inside an inner product? I am aware of the fact that...

Homework Statement Two Hermitian operators X and Y have a complete set of mutual eigenkets. Show that [X,Y]=0 and interpret this physically. Homework Equations [X,Y]=XY-YX If [X,Y]=0, XY=YX The Attempt at a Solution I have proved that [X,Y]=0, but I'm just falling a little short of...

I know that the electric field can be expressed in term of creation and annihilation operators; Is it the same for the magnetic field B ?

Hello! I am reading about the creation and annihilation operators and I don't get how you find the creation operator from the annihilation one. The creation one is \hat{a}=\sqrt{\frac{m \omega}{2 \hbar}}\left( \hat{x}+\frac{i \hat{p}}{m \omega}\right) and the annihilation operator is...

Homework Statement The Hermitian operators \hat{A},\hat{B},\hat{C} satisfy the commutation relation[\hat{A},\hat{B}]=c\hat{C}. Show that c is a purely imaginary number. The Attempt at a Solution I don't usually post questions without some attempt at an answer but I am at a loss here.

I'm trying to derive something which shouldn't be too complicated, but I get different results when doing things symbolically and with actual operators and wave functions. Some help would be appreciated. For the hydrogenic atom, I need to calculate ##\langle \hat{H}\hat{V} \rangle## and...

I am looking at the derivation of the Heisenberg Uncertainty Principle presented here: http://socrates.berkeley.edu/~jemoore/p137a/uncertaintynotes.pdf and am confused about line (21)... I do not understand why AB and BA are complex conjugates of each other... (I'm still in high school so I...

Hi ! I have a doubt about fermionic operators with the anticommutation relations. I know they follow anticommutation, that is, \begin{eqnarray} \lbrace c_{i}^{\dagger},c_{j}\rbrace=\delta_{i,j} \end{eqnarray} That is for fermionic operators. But, suppose I have two different kind of fermionic...

Homework Statement Determine whether or not the following pairs of operators commute...and there was one I could not solve...according to the back of the textbook, I do understand 14.c does NOT commute, but I don't understand... (14)c. A = SQR B = SQRT Homework Equations ABf(x) - BAf(x) = 0...

Here is what I understand. The generalized uncertainty principle is: \sigma^{2}_{A} \sigma^{2}_{B} \geq ( \frac{1}{2i} \langle [ \hat{A}, \hat{B} ] \rangle )^2 So if \hat{A} and \hat{B} commute, then the commutator [ \hat{A}, \hat{B} ] = 0 and the operators are compatible. What I don't...

Homework Statement I have a hermitian Operator A and a quantum state |Psi>=a|1>+b|2> (so we're an in a two-dim. Hilbert space) In generally, {|1>,|2>} is not the eigenbasis of the operator A. I shall now show that the Eigenvaluse of A are the maximal (minimal) expection values <Psi|A|Psi>.The...

Suppose we have linear operators A' and B'. We define their sum C'=A'+B' such that C'|v>=(A'+B')|v>=A'|v>+B'|v>. Now we can represent A',B',C' by matrices A,B,C respectively. I have a question about proving that if C'=A'+B', C=A+B holds. The proof is Using the above with Einstein summation...

Homework Statement Suppose that a state |Ψ> is an eigenstate of operator B, with eigenvalue bi. Homework Equations i. What is the expectation value of B? ii. What is the uncertainty of B? iii. Is |Ψi an eigenstate of B2 or not? iv. What is the uncertainty of B2? part B : Suppose, instead...

this is the given: the problem is the middle term, if the h-bar w outside the set brackets is canceled with the h-bar w of the m/2hw, then there will be a h-bar w that is left introduced from the middle term, i.e. i\frac{w}{m}XP- i\frac{w}{m}PX = i\frac{w}{m}[X,P]= i\frac{w}{m}i\hbar but...

Hello all! I have the following question with regards to quantum mechanics. If ##H## is a Hilbert space with a countably-infinite orthonormal basis ##\{ \left | n \right \rangle \}_{n \ \in \ \mathbb{N} }##, and two operators ##R## and ##L## on ##H## are defined by their action on the basis...

Homework Statement Show that if two square matrices of the same rank are related by unitary transformation \hat{A}=\hat{U}^\dagger\hat{B}\hat{U} then their traces and determinants are the same. Homework Equations Tr(\hat{A}) =\sum\limits_{k=0}^{n}a_{kk} \hat{U}^\dagger\hat{U} = 1 The Attempt...

Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces? As I'm not being able to precisely phrase my doubt, consider this example: Hilbert space of a two dimensional particle is the direct product of...

Hi. Say a, A(a) and b are well behaving functions. Then say [a,b] = 0, i.e. a and b commute. Why will this automatically mean that A=A(a) will commute with b? Can somebody give me an intuitive explanation, or link me to some proof?

Operator C = I+><-I + I-><+I Wavefunction PSI = Q I+> +V I-> C PSI = Q I-> + V I+> note the I is just a straight line (BRAKET vectors), the next step is where I get confused, p is subbed in and the ket vectors switch places... C PSI = pQ I+> + pV I-> <---- why?? therefore V = pQ and Q =...

I was wondering what the difference is between the two. Would be nice if someone could explain the difference in simple terms, because it appears to be essential to my quantum mechanics course.

I want to write a program that, given the tracked position of a cube being rotated, applies analogous operations to a single qubit. The issue I'm running into is that, although operations correspond to rotations on the Bloch sphere, the mapping isn't one-to-one. So when I try to map back to...

in the time-dependent schrodinger equation , our sir told us about energy and momentum operators . He just defined them , the equation was of the form Aexp(i(kx−ωt)) .if we take the equation of the form Aexp(i(kx+ωt)) will those operators change . if so generally for a wave how do we determine...

Hi all, I've been doing some independent study on vector spaces and have moved on to looking at linear operators, in particular those of the form T:V \rightarrow V. I know that the set of linear transformations \mathcal{L}\left( V,V\right) =\lbrace T:V \rightarrow V \vert \text{T is linear}...

Consider a one dimensional harmonic oscillator. We have: $$\hat{n} = \hat{a}^{\dagger} \hat{a} = \frac{m \omega}{2 \hbar} \hat{x}^2 + \frac{1}{2 \hbar m \omega} \hat{p}^2 - \frac{1}{2}$$ And: $$\hat{H} = \hbar \omega (\hat{n} + \frac{1}{2})$$ Let's say we want to measure the total...

Following on from a previous post of mine about linear operators, I'm trying to firm up my understanding of changing between bases for a given vector space. For a given vector space V over some scalar field \mathbb{F}, and two basis sets \mathcal{B} = \lbrace\mathbf{e}_{i}\rbrace_{i=1,\ldots ...

$$\text{Consider the following decomposition of the time series }{Y}_{t}\text{ where }{Y}_{t}={m}_{t}+{\varepsilon}_{t},\text{ where }{\varepsilon}_{t}\text{ is a sequence of i.i.d }\left(0,{\sigma}^{2}\right)\text{ process. Compute the mean and variance of the process }{\nabla}_{2}{Y}_{t}\text{...

Hi guys, my quesion is quite simple but I think I need to give some background... Let's suppose I have 3 qubits, so the basis of the space is: \left\{{\left |{000}\right>,\left |{001}\right>,\left |{010}\right>,\left |{100}\right>,\left |{011}\right>,\left |{101}\right>,\left...

Hi, I'm having a bit of difficulty with the following definition of a linear mapping between two vector spaces: Suppose we have two n-dimensional vector spaces V and W and a set of linearly independent vectors \mathcal{S} = \lbrace \mathbf{v}_{i}\rbrace_{i=1, \ldots , n} which forms a basis...

In Quantum Mechanics, why do the dynamical variables become operators? What is the justification or motivation, if any exist?

I am referring to the section The Harmonic Oscillator in Griffiths's introductino to quantum mechanics (the older edition with the black cover). I understand how it all works, however there is a part that I am not sure about. How do we know when we apply a- or a+ (the ladder operators) to a...

According to my teacher, for any two operators A and B, the commutators [f(A),B]=[A,B]df(A)/dA and [A,f(B)]=[A,B]df(B)/dB He did not give any proof. I can easily prove this for the particular cases [f(x),p]=[x,p]df(x)/dx and [x,pn]=[x,p]npn-1 But I don't see how the general formula is true. I...

hey pf! can you help me understand what an adjoint operator is? I've read lots of threads and other sites, but am having trouble. maybe you could give me an example? for example, does the operator d/dx have an adjoint? is asking this question completely stupid of me? thanks!

A paper I'm reading says "Our starting point is the SU(N) generalization of the quantum Heisenberg model: H=-J\sum_{\langle i,j \rangle}H_{ij}=\frac{J}{N}\sum_{\langle i,j \rangle}\sum_{\alpha , \beta =1}^N J_{\beta}^{\alpha}(i)J_{\alpha}^{\beta}(j) The J_{\beta}^{\alpha} are the generators of...

When performing matrix multiplication with 2 matrices A and B ;AB might exist but BA might not even exist. Hermitian operators can be thought of as matrices but in everything I have seen so far AB and BA always exist even though they can be different depending on the value of the commutator. Do...

Sometimes the concept of angular momentum is presented using the idea of total angular momentum J. In those cases, its always said that we have \vec{J}=\vec L + \vec S . But I can't understand how that's possible. Because orbital angular momentum operators are differential operators and so are...

When using Fourier's trick for determining the allowable energies for stationary states, Griffiths introduces the a+- operators. When factoring the Hamiltonian, the imaginary part is assigned to the momentum operator versus the position operator. Is there a reason for this? If : a-+ = k(ip +...

Hi. I was wondering why creation and destruction operators a+ and a- do not conmute. Of course, I can show that they don't conmute by computing the conmutator [a+, a-] = -1. But I want to know the "physical" meaning of this. Isn't destruction/creation a symmetric transformation? We "go up...

Hello, I am reading Griffiths Quantum Mechanics textbook, and am having some difficulty with a derivation on page 56. To me, there seems to be something logically wrong with his arguments, but I can not pin-point precisely what it is. To provide you with a little background, Griffiths is...

Homework Statement Using the switch-case construction, write code that take a variable named Shape containing a string and assigns to the variable numSides the number of sides of the shape named in the variable Shape. Your code should be able to return the number of sides for a triangle...

Homework Statement Hello, I am working on problems 6-14 on the attached PDF. Don't be scared off, they are just one line of code each. I got number 6 correct, and I got partial credit on 7 and 8, but I am trying to figure out why it is not right. Homework Equations The Attempt at a...

Homework Statement Find ##\alpha ## and ##p## so that ##\nabla \times \vec{A}=0## and ##\nabla \cdot \vec{A}=0##, where in ##\vec{A}=r^{-p}[\vec{n}(\vec{n}\vec{r})-\alpha n^2\vec{r}]## vector ##\vec{n}## is constant. Homework Equations The Attempt at a Solution ##\nabla \times...

I am a QM beginner so go easy on me. I have just noticed something. Let $$\hat{O}$$ be an hermitian operator. Then $$\left( \hat { O } \right) ^{ \dagger }\neq \hat { O } $$ when it is by itself. For example $$\left( \hat { p } \right) ^{ \dagger }=i\hbar \frac { \partial }{ \partial x...

Hi, there. I am a little confused about the following statement in Wikipedia. and it's about the density operators. "...As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has a large number of copies of the same system, then...

Homework Statement Find the eigenfunctions and the eigenvalues of the following Hamiltonian \hat{H} = \frac{1}{2m} \left ( \frac{ \hbar}{i} \vec{\nabla}-\frac{q}{c}(0, B_z x,0) \right ) ^2 = \frac{1}{2m} \left ( - \hbar^2 \vec{\nabla}^2-\frac{\hbar q}{ic}(\vec{\nabla}·(0, B_z\hat{x},0) +...