Operator Definition and 1000 Threads

Homework Statement Homework EquationsThe Attempt at a Solution This whole thing about angular momentum has me totally confused and stumped, but I am trying this problem given in a youtube video lecture I watched. I know of this equation ##L^{2} = L_{\pm}L_{\mp} + L_{z}^{2} \mp \hbar L_{z}##...

Homework Statement A measurement is described by the operator: |0⟩⟨1| + |1⟩⟨0| where, |0⟩ and |1⟩ represent orthonormal states. What are the possible measurement outcomes? Homework Equations [/B] Eigenvalue Equation: A|Ψ> = a|Ψ> The Attempt at a Solution Apologies for the basic...

Trying to derive two functions which are eigenfunctions of the hamiltonian of 2 identical and indistinguishable particles and also eigenfunctions of the 2-particle exchange operator P. Need some help with my workings I think.Have particle '1' and particle '2' in a hamiltonian given as...

As far as I know, the momentum operator is as follows: -iħ(∂/∂x) Now let's say that I enact this operator on the famous solution to the 1-D particle in a box example: Ψ= squrt(2/L) sin(πnx/L) If the momentum operator operates on the above wave function, it yields: -iħ * squrt(2/L) * (πn/L)...

I was introduced to the d operator to help solve constant coefficient differential equations for the particular integral without using trial solutions: http://en.wikipedia.org/wiki/Differential_operator http://en.wikipedia.org/wiki/Shift_theorem The results generally seem sensible, but there...

i have a problem concerning the norm of linear integral operator. ii found the answer in a book called unbounded linear operators theory and applications by Dover books author seymour Goldberg. the proof runs as follows ||T|| is less than max over x in [a,b] of integral (|k(x,y)|dy) Then he...

I'm trying to show that \int d^3x \,x^\mu \left(\partial_\mu \partial_0-g_{\mu 0} \partial^2 \right)\phi^2(x)=0 . This term represents an addition to a component of the energy-momentum tensor \theta_{\mu 0} of a scalar field and I want to show that this does not change the dilation operator...

Homework Statement Homework EquationsThe Attempt at a Solution a) I am having some trouble understanding the notation. I'm uncertain whether it should be $$ \langle {f} | \hat {O_{2}} | g \rangle = \int_{-\infty}^{\infty} f^{*}g \frac {dg}{dx} dx $$ or $$ \langle {f} | \hat {O_{2}} | g...

Hi, I've learned that material derivative is equal to local derivative + convective derivative, but can't seem to find out which way the convective derivative acts, like for example in velocity fields: The equation my teacher gave us was (with a and v all/both vectors): Acceleration = material...

In a quantum state, if you use the position operator, it gives you position, momentum operator, momentum, Hamiltonian, energy.. can you give an example or all experiments done where a quantum state has 3 competitive operators acting on it.. I want to see the quantum state changing in between...

what are the differences between D'Alembertian and Laplacian operator? while reading Electrodynamics by Griffiths I learned that Laplacian operator is used in non-relativistic cases and D'Alembertian operator is used in relativistic cases, But i don't what is the difference between these operators

Hello All, If I apply the Divergence Operator on the incompressible Navier-Stokes equation, I get this equation: $$\nabla ^2P = -\rho \nabla \cdot \left [ V \cdot \nabla V \right ]$$ In 2D cartesian coordinates (x and y), I am supposed to get: $$\nabla ^2P = -\rho \left[ \left( \frac...

I'm aware there have been plenty of discussions about Copenhagen interpretation vs ensemble interpretations (myself I have always been more fond of the latter) but I intend to explore new perspectives and stick as much as possible to what QM practitioners do in practice as opposed to obscure...

Hi, I'm stuck with a question from one of my examples sheets from uni. The question is as follows: If G(x,x') is a greens function for the linear operator L, then what is the corresponding greens function for the linear operator L'=f(x)L, where f(x) =/=0? So I've started by writing...

Are there any known (collective spin) operators to raise or lower the quantum number s in \left|{s,m}\right> spin states? I'm trying to construct coherent states varying the quantum number s instead of the well known spin coherent states varying m. I found a coherent-like state similar to the...

Where one can find a proof of the expectation value of operator expression. <A> = < Ψ | A | Ψ > or <A> = integral( Ψ* A Ψ dx ) Thanks.

Homework Statement Simplify the following commutator involving the creation and annihilation operators. [a^{\dagger}a,a \sqrt{a^\dagger a} ] Homework Equations I know that [a,a^\dagger] = 1. The Attempt at a Solution I think I should be trying to put the creation operators to the left...

Homework Statement I know that Unitary operators act similar to hermitean operators. I want to prove that the eigenvalues of unitary operators are complex numbers of modulus 1, and that Unitary operators produce orthogonal eigenvectors. Homework Equations U†U = I U-1=U† λ = eiΦ{/SUP]...

Homework Statement Consider a two-dimensional space spanned by two orthonormal state vectors \mid \alpha \rangle and \mid \beta \rangle . An operator is expressed in terms of these vectors as A = \mid \alpha \rangle \langle \alpha \mid + \lambda \mid \beta \rangle \langle \alpha \mid +...

I am currently reading up on some algebraic topology\differential geometry and have reached the section on de Rham theory. This is my first encounter with such notions and I am a little confused by what is meant when one applies a boundary operator to a simplex. Conceptually, I know that it...

One can represent the mean of the angular momentum operator as a vector. But what is the (mathematical) justification to represent the operator by a vector which has a direction that the operator has not. Yet worse, l(l+1) h2 is the proper value of operator L^2 and from such result it is assumed...

Homework Statement Given the series of three Stern-Gerlach devices: Represent the action of the last two SG devices as matrices ##\hat{A}## and ##\hat{B}## in the ##|+z\rangle, |-z\rangle## basis. Homework Equations ##|+n\rangle = cos(\frac{\theta}{2})|+z\rangle +...

What does it mean when it is said that the momentum operator flips the parity of the function on which it operates ?

This paper is about momentum operator in curvilinear coordinates. The author says that using \vec p=\frac{\hbar}{i} \vec \nabla is wrong and this form is only limited to Cartesian coordinates. Then he tries to find expressions for momentum operator in curvilinear coordinates. He's starting...

Anyone having an idea of how to solve problem 3a) file:///C:/Users/Administrator/Downloads/handin1%20(2).pdf ? I've been stuck for a great while but have not idea.

How do I prove that the parity operator Af(x) = f(-x) commutes with the second derivative operator. I am tempted to write: A∂^2f(x)/∂x^2 = ∂^2f(-x)/∂(-x)^2 = ∂^2f(-x)/∂x^2 = ∂^2Af(x)/∂x^2 But that looks to be abuse of notation..

hello, i am trying to learn the derivation of the momentum operator and i found 2 ways of deriving it. one is using Fourier transform and the other is taking the time derivative of the expectation value of x. i just want to know what is the physical interpretation of the time rate of change...

Homework Statement Let C be the composition operator on the Hilbert space L_{2}(\mathbb{R}) with the usual inner product. Let f\in L_{2}(\mathbb{R}), then C is defined by (Cf)(x) = f(2x-1), \hspace{9pt}x\in\mathbb{R} give a demonstration, which shows that C does not have any eigenvalues...

Homework Statement Given that the function f can be expanded in a power series of a and a^\dagger, show that: [a,f(a,a^{\dagger})]=\frac{\partial f }{\partial a^\dagger} and that [a,e^{-\alpha a^\dagger a}] = (e^{-\alpha}-1)e^{-\alpha a^{\dagger} a}aThe Attempt at a Solution I've tied using...

Let $$T: X \rightarrow Y$$ be a continuous linear operator between Banach spaces. Prove that $T$ is surjective $$\iff$$ $$T^*$$ is injective and $$im T^*$$ is closed. I've proven a "similar" statement, with $$imT^*$$ replaced with $$imT$$. There I used these facts: $\overline{imT}=...

In my physical chemistry course, we are learning about the Schrödinger Equation and were introduced to the Hamiltonian Operator recently. We started out with the simple scenario of a particle in 1D space. Our professor's slide showed the following "derivation" to arrive at the expression for the...

I have 2 variables A and B. A can be computed from B like this Input: BOOL b, OBJECT B A=(B<<5) if NOT b then A=[A OR 0xFF] Now I would like to compute B from the above code Input: BOOL b, OBJECT A It can be either B=A>>5 if NOT b then B= ? or if NOT b then B=? B=A>>5 I am not...

can anyone suggest any good reading material on operator state mapping in conformal field theory? I know only elementary field theory... So it might be helpful ifsomeone suggest a book where it is done in little detailed way..

For an electron the spin operator S_zis represented by a 2×2 matrix, with spin up and down as its bases. Consider the angular momentum operator L_z with l=1 which is a 3×3 matrix. How can we treat the L_z S_z operator directly in matrix form?

Homework Statement I would appreciate feedback on the following two problems: (1) For a given operator A with no explicit time dependence I am asked to show that d/dt(eAt)=A(eAt) (2) A free wave packet of width Δx is traveling at a constant velocity v0=p0/m. I am asked to estimate the...

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...

Homework Statement I know that any unitary operator U can be realized in terms of some Hermitian operator K (see equation in #2), and it seems to me that it should also be true that, starting from any Hermitian operator K, the operator defined from that equation exists and is unitary...

This might be trivial for some people but this has been bothering lately. If P is momentum operator and p its eigenvalue then the eigenfunction is up(x) = exp(ipx/h). where h is the reduced Planck constant (sorry can't find a way to make the proper notation). While it can also be proved that...

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...

Hi! :) I'm trying to understand the following calculation. The book Quantum Mechanics by Nouredine Zettili wants to determine the form of the momentum operator $\widehat{\vec{P}}$ in the position representation. To do so he calculates as follows: $$\begin{aligned} \langle \vec{r} |...

hi, please tell me what do we mean when we say in quantum mechanics operators are linear and also vector space is also linear ?

Homework Statement [/B] I'm solving for ##f_1## from ##B(f_{1}.1+1.f_{1})## from ## \frac{\partial}{\partial x}(\frac{\partial}{\partial t}+\frac{\partial^{3}}{\partial x^{3}})f_n=-\frac{1}{2}\sum^{n-1}_{m=1}B(f_{n-m}.f_{m}) ## where ##B=D_tD_x+D_x^4##, where ##B## is the Bilinear...

Homework Statement I am trying to calculate the following quantity: $$<0|T\{\phi^\dagger(x_1) \phi(x_2) exp[i\int{L_1(x)dx}]\}|0>$$ where: $$ L_1(x) = -ieA_{\mu}[\phi^* (\partial_\mu \phi ) - (\partial_\mu \phi^*)\phi] $$[/B] I am trying to find an expression including the propagators...

What is the explicit 3x3 matrix operator which measures the colour of a quark? Essentially what I want to know is what is the analogue of ##S^z## for the measuring of spin.

What is a difference between linear operator and linear functional? Do I understand it correctly that linear operator is any operator that when applied on a vector from a vector space, gives again a vector from this vector space. And also obeys linearity conditions. And linear functional is a...

Homework Statement I'm checking to see if the momentum operator is Hermitian. Griffiths has the solution worked out, I'm just not following the integration by parts. Homework Equations int(u dv) = uv - int(v du) The Attempt at a Solution I've attached an image of my work. It seems there...

A pointer in C++ is represented by *. Sometimes the * comes after the variable/class/whatever such as 'Pointer*'. Other times it comes before, '*Pointer'. What is the difference between the two?What is the member access operator for? (->) According to my notes, a->b is equivalent to (*a).b

Homework Statement Consider the real scalar field with the Lagrangian \mathcal{L}=\frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2. Show that after normal ordering the conserved four-momentum P^\mu = \int d^3x T^{0 \mu} takes the operator form P^\mu = \int...

Homework Statement Find the eigenvalues of the angular-momentum-squared operator (L2) for hydrogen 2s and 2px orbitals... Homework Equations Ψ2s = A (2-r/a0)e-r/(2a0) Ψ2px = B (r/a0)e-r/(2a0) The Attempt at a Solution If I am not wrong, is the use of L2 in eigenfunction L2Ψ = ħ2 l(l+1) Ψ...

Hi, friends! In order to find an orthogonal basis of eigenvectors of the Fourier transform operator ##F : L_2(\mathbb{R})\to L_2(\mathbb{R}),f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x## for Euclidean separable space ##L_2(\mathbb{R})##, so that ##F## would be represented by...