operators

1. I Condition for delta operator and total time differential to commute

While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think...
2. I Hermitian operators in QM and QFT

I have always learnt that a Hermitian operator in non-relativistic QM can be treated as an "experimental apparatus" ie unitary transformation, measurement, etc. However this makes less sense to me in QFT. A second-quantised EM field for instance, has field operators associated with each...
3. Show the formula which connects the adjoint representations

That's my attempting: first I've wrote $e$ in terms of the power series, but then I don't how to get further than this $$\sum_{n=0}^\infty (-1)^n \frac {Â^n} {n!} \hat B \sum_{n=0}^\infty \frac {Â^n} {n!} = \sum_{n=0}^\infty (-1)^n \frac {Â^2n} {\left( n! \right) ^2}$$. I've alread tried to...

7. Bosonic annihilation and creation operators commutators

1. Homework Statement After proving the relations $[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0$ and $[\hat{b}_i,\hat{b}_j]=0$, I want to prove that $[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}$, however I'm not sure where to begin. 2. The attempt at a solution I tried to apply the...
8. Write the matrix representation of the raising operators...

1. Homework Statement Hi, guys. The question is: For a 3-state system, |0⟩, |1⟩ and |2⟩, write the matrix representation of the raising operators $\hat A, \hat A^\dagger$, $\hat x$ and $\hat p$. 2. Homework Equations I know how to use all the above operators projecting them on...
9. Quantum state of system after measurement

> Operator $$\hat{A}$$ has two normalized eigenstates $$\psi_1,\psi_2$$ with > eigenvalues $$\alpha_1,\alpha_2$$. Operator $$\hat{B}$$ has also two > normalized eigenstates $$\phi_1,\phi_2$$ with eigenvalues > $$\beta_1,\beta_2$$. Eigenstates satisfy: > $$\psi_1=(\phi_1+2\phi_2)/\sqrt{5}$$ >...
10. I The Ehrenfest Theorem

This may seem rather silly, but how would I go about enunciating Ehrenfest’s theorem? Also, does anyone know what this theorem implies for the relation between classical and quantum mechanics? Any suggestions or help is greatly appreciated!
11. A On spectra

Hi, why do unbounded operators and bounded operators differ so much in terms of defining their spectra? 1. The unbounded operator requires a self-adjoint extension to define its spectrum. 2. A bounded one does not require a self-adjoint extension to define the spectral properties. 3. Still the...
12. I How to find admissible functions for a domain?

Hi, in a text provided by DrDu which I am still reading, it is given that "the momentum operator P is not self-adjoint even if its adjoint $P^{\dagger}=-\hbar D$ has the same formal expression, but it acts on a different space of functions." Regarding the two main operators, X and D, each has...

17. S

A Bounded or unbounded operator

Hi, I have an operator which does not obey the following condition for boundedness: \begin{equation*} ||H\ x|| \leqslant c||x||\ \ \ \ \ \ \ \ c \in \mathscr{D} \end{equation*} where c is a real number in the Domain D of the operator H. However, this operator is also not really unbounded...
18. A What are local and non-local operators in QM?

In Hartree-Fock method, I saw the Fock operator has two integrals: Coulomb integral and exchange integral. One can define two operator. "The exchange operator is no local operator" why? Whats de diference: local and no local operator? And why do the operators have singularities? thanks
19. I Measurement Values for z-component of Angular Momentum

Given a wave function $$\Psi(r,\theta,\phi)=f(r)\sin^2(\theta)(2\cos^2(\phi)-1-2i*\sin(\phi)\cos(\phi))$$ we are trying to find what a measurement of angular momentum of a particle in such wave function would yield. Attempts were made using the integral formula for the Expectation Value over a...

26. I Angular momentum operator commutation relation

I am reading a proof of why \left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z Given a wavefunction \psi, \hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...
27. A Deriving the Lagrangian from the Hamiltonian operator

In classical mechanics, the Hamiltonian and the Lagrangian are Legendre transforms of each other. By analogy, in quantum mechanics and quantum field theory, the relationship between the Hamiltonian and the Lagrangian seems to be preserved. Where can I find a derivation of the Lagrangian...
28. A Primaries, descendents and transformation properties in CFT

I want to clarify the relations between a few different sets of operators in a conformal field theory, namely primaries, descendants and operators that transform with an overall Jacobian factor under a conformal transformation. So let us consider the the following four sets of operators...
29. I About operators

When I learnt about operators, I learnt <x> = ∫ Ψ* x Ψ dx, <p> = ∫ Ψ* (ħ/i ∂/∂x) Ψ dx. The book then told me the kinetic energy operator T = p2/2m = -ħ2/2m (∂2/∂x2) I am just think that why isn't it -ħ2/2m (∂/∂x)2 Put in other words, why isn't it the square of the derivative, but...
30. Matrix representation of certain Operator

1. Homework Statement Vectors I1> and I2> create the orthonormal basis. Operator O is: O=a(I1><1I-I2><2I+iI1><2I-iI2><1I), where a is a real number. Find the matrix representation of the operator in the basis I1>,I2>. Find eigenvalues and eigenvectors of the operator. Check if the eigenvectors...