Operator Definition and 1000 Threads

I've been reading Thomas Jordan's Linear Operators for Quantum Mechanics, and I am stalled out at the bottom of page 40. He has just defined the projection operator E(x) by E(x)(f(y)) = {f(y) if y≤x, or 0 if y>x.} Then he defines dE(x) as E(x)-E(x-ε) for ε>0 but smaller than the gap between...

Homework Statement Find the eigenfunctions of a particle in a infinite well and express the position operator in the basis of said functions.Homework Equations The Attempt at a Solution Tell me if I'm right so far (the |E> are the eigenkets) X_{ij}= \langle E_i \vert \hat{X} \vert E_j \rangle...

Hi all, I haven't been able to find an answer online but this seems like a pretty basic question to me. What are the commutator relations between the position/momentum operators and the field operator? I'm not even certain what the commutation relations between X/P and a single ladder operator...

Hi I have been looking at the solutions to a past exam question. The question gives the annihilation operator for the harmonic oscillator as a= x + ip ( I have left out the constants ). The question then asks to calculate the Hermitian conjugate a(dagger). I thought to find the Hermitian...

Hi. In 2-fold degenerate perturbation theory we can find appropiate "unperturbate" wavefunctions by looking for simultaneous eigenvectors (with different eigenvalues) of and H° and another Hermitian operator A that conmutes with H° and H'. Suppose we have the eingenvalues of H° are ##E_n =...

Hi. I have a little language problem. I'm studying in Germany, and my German is... nicht sehr gut, so I sometimes have problems understanding the exercises. The one I'm having issues right now has a part which says einen höchstens explizit zeitabhängigen Ope-rator I am Schrödingerbild. My...

Homework Statement Prove that the creation operator a_+ has no eigenvalues, for instance in the \vert n \rangle . Homework Equations Action of a_+ in a harmonic oscillator eigenket \vert n \rangle : a_+\vert n \rangle =\vert n +1\rangle The Attempt at a Solution Calling a the...

Hi. In this development (c ∇+ d A)(c ∇+dA)= c^{2} ∇^{2} + d^{2}A^{2} + cd A∇ + cd ∇A (c ∇+ d A)^{2}= c^{2} ∇^{2} + d^{2}A^{2} + cd A∇+ cd A∇+ cd (∇A) I feel like we have "two" different ∇ operators. At the end of the first line ∇ acts on A and the test function (not shown). At the...

Homework Statement Let ##n## be a unit vector in ##V## . Define a linear operator ##F_n## on ##V## such that $$F_n(u) = u-2\langle u, n \rangle n \; \mathrm{for} \; u \in V.$$ ##F_n## is called the reflection on ##V## along the direction of ##n##. Let ##S## be an orthogonal linear operator on...

Homework Statement #1 in the attachment Homework Equations The Attempt at a Solution My code is working for all of the numeric, logical, character portions. I got 7/8 points if ischar(X_input) Y_output = upper(X_input); elseif isnumeric(X_input) switch...

Hi. This is driving me mad: \hat{\vec{\nabla}}(\hat{\vec{A}})f=(\vec{\nabla}\cdot\vec{A})f + \vec{A}\cdot(\vec{\nabla}f) for an arbitrary vector operator ##\hat{\vec{A}}## So if we set ##\vec{A} = \vec{\nabla}## this should be correct...

Hi everyone. In QFT one usually defines the "number of valence quarks" of a certain particle via the operator: $$ \hat N_{val}=\sum_f |\hat Q_f|,$$ where: $$ \hat Q_f=\int d^3x \bar \psi_f\gamma_0\psi_f.$$ According to this definition I expected, for example, for the J/\psi to have...

Hi, everybody: I encountered a problem when I am reading a book. It's about the atom-photon interaction. Let the Hamiltonian for the free photons be H_0=\hbar \omega(a^{\dagger}a+\frac{1}{2}). so the commutator of the annihilation operator and the Hamiltonian is [a,H_0]=\hbar\omega a and I...

If ##A## is not projection operator. Could ##A^k## be a projection operator where ##k## is even or odd degree. Thanks for the answer.

In the operation $$U(\Lambda)|{\bf p}\rangle=|{\Lambda\bf p}\rangle,$$ if we define the state covariantly, $$|{\bf p}\rangle=\sqrt{2E_{\bf p}}a_{\bf p}^\dagger|0\rangle,$$ then does the unitary operator U(\Lambda) affect the factor of \sqrt{2E_{\bf p}}? In other words, can we write...

hello i have to proof that Px (linear momentum operator ) is hermitian or not i have added my solution in attachments please look at my solution and tell me if its correct thank you all

I am reading an intriguing article on rigged Hilbert space http://arxiv.org/abs/quant-ph/0502053 On page 8, the author describes the need for rigged Hilbert space. For that, he considers an unbounded operator A, corresponding to some observable in space of square integrable functions...

I found two "forms" of it: p=\frac{\hbar}{i}\frac{d}{dx} p=-i\hbar\frac{d}{dx} how could they be the same??

Hi. Before question, sorry about my bad english. It's not my mother tongue. My QM textbook(Schiff) adopt J x J = i(h bar)J. as the defining equations for the rotation group generators in the general case. My question is, then tensor J must have one index which has three component? (e.g...

When two vectors are dotted, the result is a scalar. But why here http://www.cobalt.chem.ucalgary.ca/ziegler/educmat/chm386/rudiment/mathbas/vectors.htm , the del-squared still maintains its unit vectors i, j, k? Isn't it this way ∇2 = (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2) and not (i∂2/∂x2 + j∂2/∂y2 +...

Homework Statement Spin Operator S has eigenvectors |R> and |L>, S|R> = |R> S|L> =-|L> eigenvectors are orthonormal Homework Equations Operator A is Hermitian if <ψ|A|Θ> = <Θ|A|ψ>* The Attempt at a Solution <ψ|S|L> = <L|S|ψ>* // Has to be true if S is Hermitian LHS...

Homework Statement We have a linear differential operator ##Ly=-y^{''}## working on all ##y## that can be derived at least twice on ##[-\pi ,\pi ]## and also note that ##y(-\pi )=y(\pi )## and ##y^{'}(-\pi )=y^{'}(\pi )##. a) Is ##0## eigenvalue for ##L##? b) Is ##L## symmetric? (I think the...

When we say expectation value of an operator like the pauli Z=[1 0; 0 -1], like when <Z> = 0.6 what does it mean? What is difference between calculating expectation value of Z and its POVM elements{E1,E2}? thanks

I'm considering a system where an electron is subjected to magnetic field which is produced by dirac monopole. Here I'm interested in looking for a translation operator. Now how can I get a translation operator in presence of field and in absence of field.?? I need both the operators. Can...

This picture from https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20 is all you need to derive the Cauchy-Riemann equations, i.e. from the picture we see i \frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} should hold so we have i \frac{\partial f}{\partial x} = i...

Homework Statement Part (a): Find eigenvalues of X, show general relation of X and show X commutes with KE. Part (b): Give conditions on V1, V2 and VI for X to commute with them. Part (c): Write symmetric and antisymmetric wavefunctions. Find energies JD and JE. Part (d): How are...

Hello, who can suggest me a book, or a PDF where i can find an introduction to Liouvillian operator in statistical mechanics? I understand that it's correlated to time evolution of density of an Hamiltonian system but i don't know anything else thank you sorry for my wrong english.. :(

Definition: ##f(x+k) = \exp(k \frac{d}{dx}) f(x)## So I thought, how take advantage this definition? Maybe it be usefull in integration like is the laplace transform. So I tried to integrate the expression ##\int f(x+k) dx = \int \exp(k \frac{d}{dx}) f(x) dx ## that is an integration by parts...

If \hat{U}(r) = e^{\hat{A}(r)}, can we say \frac{d\hat{U}}{dr} = \frac{d\hat{A}}{dr}e^{\hat{A}(r)}?

i'm trying to integrate this: $$W=\frac{ε}{2}\int{\vec{∇}\cdot\vec{E})Vdτ}$$ where ε is a constant, E= -∇V, τ is a volume element how do i end up with the following via integration by parts? $$W=\frac{ε}{2}[-\int{\vec{E}\cdot(\vec{∇}V)dτ}+\oint{V\vec{E}\cdot d\vec{a}}$$] where the vector a...

The question is to calculate the time evoution of S_{x} wrt <\Psi(t)\pm l where <\Psi\pm (t) l= ( \frac{1}{\sqrt{2}}(exp(^{+iwt})< \uparrow l , \pm exp(^{-iwt})< \downarrow l ) [1] Sx=\frac{}{2}(^{0}_{1}^{1}_{0} ) Here is my attempt: - First of all from [1] I see that l \Psi\pm (t) > = (...

Homework Statement Write out matrix representation of P. Also, do P|ψ> Homework Equations ψ=ψ0 + 2ψ1 ψ0=(1/∏)1/4 exp(- u2/2) ψ1=(1/∏)1/4 √2 exp(- u2/2) P= 1/(i*∏) d/du The Attempt at a Solution I've no clue what to do. If I had a ψm ψn I would, but what do I do...

Homework Statement Let ##n## be a positive integer and let ##V = P_n## be the space of polynomials over ##R##. Let D be the differentiation operator on ##V## . Find a basis for the null space of the transpose operator ##D^t: V^*\to V^*##. Homework Equations Let ##T:V\to W## be a linear...

In "Quantum Computation and Quantum Information" by Nielsen & Chuang, on pp. 88-89, applying basic statistical definitions to operators, one of the intermediary steps uses the expression M-<M> where M is a Hermitian operator, and <M> is the expected value = <ψ|M|ψ> for a given vector ψ...

Homework Statement Show that if T is a normal operator on a finite dimensional vector space than it has the same image as its adjoint. Homework Equations N/A The Attempt at a Solution I have been able to show that both T and T^{*} have the same kernel. Thus, by using the finite...

Hi there, Let S denote the shift operator on the Hardy space on the unit disc H^2, that is (Sf)(z)=zf(z). My question is to show the following identity (1-\lambda S^*)^{-1}S^*f (z)=\frac{f(z)-f(\lambda)}{z-\lambda}, where \lambda,z\in\mathbb{D} Thanks in advance

Homework Statement Consider a time-dependent harmonic oscillator with Hamiltonian \hat{H}(t)=\hat{H}_0+\hat{V}(t) \hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right) \hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right) (i)...

Homework Statement I am solving a problem and I arrived near the end, and can't figure out what to do here: (1/(2m)) [P^2,X]+[P^2,X] m - mass P - Momentum operator X - Position operator Homework Equations P = -iħ(∂/∂x) [A,B]=AB-BA [AB,C]=A[B,C]+B[A,C] where A, B...

Homework Statement Consider a system of two spin 1/2 particles, labeled 1 and 2. The Pauli spin matrices associated with each particle may then be written as \vec{\hat{\sigma _{1}}} , \vec{\hat{\sigma _{2}}} a)Prove that the operator \hat{A]}\equiv \vec{\hat{\sigma _{1}}}\cdot...

This is not really a homework problem but rather a homework-related question.. When I came across my homework (and my textbook: Atkin's physical chemistry 9th Ed.), they defined the momentum operator as: p_x = - ( \hbar / i ) * d/dx... but i have seen in other sources that they define it...

A Hermitian operator A is defined by A=A(dagger) which is the transpose and complex conjugate of A. In 1-D the momentum operator is -i(h bar)d/dx. How can this be Hermitian as the conjugate has the opposite sign ? Thanks

Homework Statement I have a question asking me to find the expectation value of S_{12} for a system of two nucleons in a state with total spin S = 1 and M_s = +1 , where S_{12} is the tensor operator inside the one-pion exchange nuclear potential operator, equal to S_{12} =...

Question and symbols: Consider a state|ε> that is in a quantum superposition of two particle-in-a-box energy eigenstates corresponding to n=2,3, i.e.: |ε> = ,[1/(2^.5)][|2> + |3>], or equivalently: ε(x) = [1/(2^.5)][ψ2(x) + ψ3. Compute the expectation value of momentum: <p> = <ε|\widehat{}p|ε>...

In any textbooks I have seen, vacuum states are defined as: a |0>= 0 What is the difference between |0> and 0? Again, what happens when a+ act on |0> and 0? and Number Operator a+a act on |0> and 0?

Homework Statement I want to show that <n',l',m'|\hat{z}|n,l,m> = 0 unless m=m', using the form of the spherical harmonics. Homework Equations Equations for spherical harmonics The Attempt at a Solution Not sure how to begin here since there aren't any simple eigenvalues for...

In deriving the Heisenberg uncertainty relation for 2 general Hermitian operators A and B , the uncertainty operators ΔA and ΔB are introduced defined by ΔA=A - (expectation value of A) and similarly for B. My question is this - how can you subtract(or add) an expectation value , which is just...

In wolframpage there is follows definition for shape operator in a given point by vector v: I think that this equation means: S(\vec{v})=-\frac{d\hat{n}}{d \vec{v}} correct, or not? If yes, of according with the matrix calculus...

Homework Statement Hey guys. So here's the situation: Consider the Hilbert space H_{\frac{1}{2}}, which is spanned by the orthonormal kets |j,m_{j}> with j=\frac{1}{2}, m_{j}=(\frac{1}{2},-\frac{1}{2}). Let |+> = |\frac{1}{2}, \frac{1}{2}> and |->=|\frac{1}{2},-\frac{1}{2}>. Define the...

Homework Statement So I have this rather komplex example and I am looking for help. ∇(3(r*a)r)/R5 -a/R5) r=xex+yey+zez a-constant vector R=r1/2 Homework Equations The Attempt at a Solution So the nabla " works" on every member individualy,and i have to careful here:(r*∇a),because...

Hi, I was just wondering if anyone know of an operator, which has some realistic analogue, that would perform the following action: A|N,0> = A|0,N> Where the ket's represent two joint fock states (i.e. two joint cavities) and A is the opeator I desire. I thought that the beam...