Operator Definition and 1000 Threads

I cannot quite understand why expression \frac{1-\gamma_5 \slashed{s}}{2} is covariant? We defined it in the rest frame, and then said that because it is in the slashed expression, it's covariant, what does that mean? s is the direction of polarization, s \cdot s = -1

Hello, I am having trouble when solving non-homogeneous DE's how to find the annihilator to find my particular solution. For example, if you have a DE that equals 24x^2cos(x), how do I find something that will annihilate this? It seems to me no matter how many derivatives you take, you...

I just have two questions relating to what I have been studying recently. 1) I know that the total energy and momentum operators don't commute, while the kinetic energy and momentum operators do. Why is this the case? (explanation rather than mathematically). 2) One form of the HUP says that...

Homework Statement I need to express the rotation operator as follows R(uj) = cos(u/2) + 2i(\hbar) S_y sin(u/2) given the fact that R(uj)= e^(iuS_y/(\hbar)) using |+-z> as a basis, expanding R in a taylor series express S_y^2 as a matrix Homework Equations I know...

I know that the average momentum <p> is defined as m\frac{d}{dt}<x>. But why is this also equal to : \intψ*\frac{h}{(2\pi)i}\frac{\partial ψ}{\partial x}dx ? the integral goes from negative inf to inf, * indicates conjugate,ψ the wavefunction. Also, why is it in general that for any average...

Let M be the space of all 2 × 2 complex matrices, satisfying 〖(X)bar〗^t = -X (skew-hermitian). Consider M as a vector space over R. Define a bilinear form B on M by B(X,Y) = -tr(XY) (1) Show that B takes real values, is symmetric and positive definite. (2) For any A ∈ M , define the...

I should Use the fact that in general the eigenvalues of the square of the angular momentum operator is J(J + 1)h and show the spin of the electron. I have J= L+S and J2 = L2+ S2 Homework Statement But how could i find the spin of the electron

Like the title says, why are the only possible values of an operator its eigenvalues? reading shankar right now and I'm having difficulty understanding why this has to be the case, given some operator/variable Ω

Suppose T belongs to L(V,V) where L(A,W) denotes the set of linear mappings from Vector spaces A to W, is such that every subspace of V with dimension dim V - 1 is invariant under T. Prove that T is a scalar multiple of the identity operator. My attempt : Let U be one of the sub spaces of V...

Change of the "Del" operator in two particle interactions Ok,so John Taylor's Classical Mechanics has this small subtopic "energy interactions between 2 particles".And,in that,hes defined a "del1" operator as the vector differential operator with respect to particle 2 at the origin.Hence,the...

There are 2 operators such that [A,B] = 0. Does [F(A),B]=0 ? Specifically, let's say we had the Hamiltonian of a 3-D oscillator H and L^2. We know that L^2 = Lx^2+Ly^2+Lz^2, and it is known that [H,Lz] = 0. Can we say that since H and Lz commute, H and Lz^2 also commute, by symmetry H and...

Hi Often in the context of multi-atom systems, such as in cavity QED, it is customary to introduce a so-called "collective pseudospin operator". An example of this is for the inversion for some atom j, \sigma_{j, z}, which becomes \sum_{j} \sigma_{z, j} = \sigma_z To me this seems very...

I want to be on the cutting edge of nuclear engineering, but I am afraid that I might not have the genius necessary to do it. I'm in my first semester of taking NucE classes, and my Fluid Mechanics class is tearing me up! Not to mention my Fundamentals of Nuclear Science/Engineering class is...

Hi all Homework Statement Given is a Hermitian Operator H H= \begin{pmatrix} a & b \\ b & -a \end{pmatrix} where as a=rcos \phi , b=rsin \phi I shall find the Eigen values as well as the Eigenvectors. Furthermore I shall show that the normalized quantum states are: \mid +...

Homework Statement Consider a Hilbert space with a (not necessarily orthogonal) basis \{f_i\} Show that G=\sum_i |f_i\rangle\langle f_i| has strictly positive eigenvalues. Homework Equations The Attempt at a Solution I know that G=\sum_i |f_i\rangle\langle f_i| is hermitian...

I am reading a quantum mechanics book. I did not clearly understand one particular idea. When the book talks about the time-evolution operator U(t,t_0), it says that one very important property is the unitary requirement for U(t,t_0) that follows from probability conservation. My question is...

Just to clarify: The del operator's a vector and the laplacian operator is just a scalar?

Homework Statement a)For a general operator A, show that and i(A-A+) are hermitian? b) If operators A and B are hermitian, show that the operator (A+B)^n is Hermitian. Homework EquationsThe Attempt at a Solution The first part I did, (A+A+)+=(A++A)=(A+A+)...

Hi If C is an operator such that C|1> = |1> and C|2>=|2>, then C^3 |1>= |1>|1>|1> =|1> ^ 3 ? If yes, then what does this C^3 represent? :confused:

My question is about both sides of the same coin. First, does a hermitian operator always represent a measurable quantity? Meaning, (or conversely) could you cook up an operator which was hermitian but had no physical significance? Second, are all observables always represented by a...

Hello Everybody, I am working through Pathria's statistical mechanics book; on page 114 I found the following definition for the density operator: \rho_{mn}= \frac{1}{N} \sum_{k=1}^{N}\left \{ a(t)^{k}_m a(t)^{k*}_n \right \}, where N is the number of systems in the ensemble and the...

Hi, In quantum optics, when we talk about atom field interaction with a classical field and quantized atom, we say that the Hamiltonian has an interaction part of the form \hat{d}.\vec{E} where d is the dipole operator. For a two level atom, the dipole operator has only off diagonal elements...

Hi. I am trying to understand a statement from Peskin and Schroeder at page 59 they write; "The one particle states |\vec p ,s \rangle \equiv \sqrt{2E_{\vec p}}a_{\vec p}^{s \dagger} |0\rangle are defined so that their inner product \langle \vec p, r| \vec q,s\rangle = 2 \vec E_\vec{p}...

Homework Statement Let x be a fixed nonzero vector in R^3. Show that the mapping g:R^3→R^3 given by g(y)=projxy is a linear operator. Homework Equations projxy = \left(\frac{x\cdot y}{\|x\|}\right)x My book defines linear operator as: Let V be a vector space. A linear operator on V is...

I'm having trouble understanding the derivation of the the position operator eigenfunction in Griffiths' book : How is it "nothing but the Dirac delta function"?? (which is not even a function). Couldn't g_{y}(x) simply be a function like (for any constant y) g_{y}(x) = 1 | x=y...

Homework Statement Let \left|x\right\rangle and \left|p\right\rangle denote position and momentum eigenstates, respectively. Show that U^n\left|x\right\rangle is an eigenstate for x and compute the eigenvalue, for U = e^{ip}. Show that V^n\left|p\right\rangle is an eigenstate for p and...

Homework Statement (A.∇)B What does this mean and how do I go about trying to expand this (using cartesian components)?

Homework Statement Determine the standard matrix for the linear operator defined by the formula below: T(x, y, z) = (x-y, y+2z, 2x+y+z) Homework Equations The Attempt at a Solution No idea

Hi there, If the evolution operator is given as follows U(t) = \exp[-i (f(p, t) + g(x))/\hbar] where p is momentum, t is time. Can I conclude that the Hamiltonian is H(t) = f(p, t) + g(x) if no, why?

Homework Statement Consider the operator A and its Hermitian adjoint A*. If [A,A*] = 1, evaluate: [A*A,A] Homework Equations standard rules of linear algebra, operator algebra and quantum mechanics The Attempt at a Solution [A,A*] = AA* - A*A = 1 A*A = (1+AA*) [A*A,A] =...

Hi there, I am reading a book in which the unitary evolution operator is U = \exp(-i H/\hbar) where H is the given Hamiltonian. But in another book, I found that the evolution operator is general given as U = \exp(-i \int H(t) dt / \hbar) which one is correct and why there are two...

Hello: I would like to understand how to compute the shape operator (and eigenvalues etc) for a complex example like the Schwarzschild spacetime. It's easy for a submanifold in Euclidean space, but I don't know how to do it for the more advanced examples like the schwarzschild spacetime in...

Hey Guys, Does anyone know where I can find a derivation of the laplace operator in spherical and cylidrical coordinates?

Why is the position operator of a particle on the x-axis defined by x multiplied by the wave function? Is there an intuitive basis for this or is it merely something that simply works in QM?

I work with a grid-based code, this means that all of my quantities are defined on a mesh. I need to compute, for every point of the mesh the divergence of the velocity field. All I have is, for every cell of my mesh, the values of the 3-d velocity in his 26 neighbors. I call neighbors the...

I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's definition of the determinant expressed in terms of an alternating bilinear form but am having...

I'd like to show that if there exists some operator \overset {\wedge}{x} which satisfies \overset {-}{x} = <\psi|\overset {\wedge}{x}|\psi> , \overset {\wedge}{x}|x> = x|x> be correct. \overset {-}{x} = \int <\psi|x> (\int<x|\overset {\wedge}{x}|x'><x'|\psi> dx')dx = \int <\psi|x>...

http://web.mit.edu/6.013_book/www/chapter2/2.4.html I was going through the curl derivation on the above link. How does equation 3 turn out? Δy is the incremental length. But how do you decide whether it is +Δy/2 or -Δy/2. And why is the line integral taken Δz when the change is in the y...

Homework Statement "Show that if the Hamiltonian depends on time and [H(t_1),H(t_2)]=0, the time development operator is given by U(t)=\mathrm{exp}\left[-\frac{i}{\hbar}\int_0^t H(t')dt'\right]." Homework Equations i\hbar\frac{d}{dt}U=HU U(dt)=I-\frac{i}{\hbar}H(t)dt The Attempt at a...

I am trying to figure out what the matrix of this linear operator would be: T:M →AMB where A, M, B are all 2X2 matrices with respect to the standard bases of a 2x2 matrix viz. e11, e12 e21 and e22. Any ideas? Il know it should be 2X8 matrix. I am trying to teach myself Abstract algebra using...

know I'm missing something obvious. for a momentum operator p = -iħ d/dx if I square the -iħ part I get (+1)ħ2 but I believe the correct value (as in the kinetic energy of the Hamiltonian) is -ħ/2m d2/dx2. how is the value of the term -ħ/2m where the square of -i = +1? Thanks!

Homework Statement I do not understand equal signs 2 and 3 the following Angular momentum operator identity: Homework Equations \hat{J}^2 = \hat{J}_1^2+\hat{J}_2^2 +\hat{J}_3^2 = \left(\hat{J}_1 +i\hat{J}_2 \right)\left(\hat{J}_1 -i\hat{J}_2 \right) +\hat{J}_3^2 + i...

Consider two Hermitian operator A, B; Define [A,B]=iC, then operator C is also Hermitian. we calculate the expectation value with respect to |a>, one eigenstate of A with the eigenvalue a. From the left side, we have: <a|[A,B]|a>=<a|(AB-BA)|a>=(a-a)<a|B|a>=0, while on the right side...

Hi, Could someone explain how the following two definitions of the displacement operator are equal? The first is the standard one 1) e^{\alpha a^{\dagger}-\alpha*a} But what about this one? This is from a Fock state decomposition of a coherent state. 2) e^{\frac{-|\alpha^{2}|}{2}}e^{\alpha...

I have some doubts about the implications of the orbital angular operators and its eigenvectors (maybe the reason is that I have a weak knowledge on QM). If we choose the measurement of the z axis and therefore the Lz operator, the are the following spherical harmonics for l=1...

Let's say I have this relation B=∇XA I know B, now I want A. What ever do I do? Is there some tried and true method out there?

Homework Statement Prove that e^{-i \pi J_x} \mid j,m \rangle =e^{-i \pi j} \mid j,-m \rangle Homework Equations J_x \mid j,m \rangle =\frac{\hbar}{2} [\sqrt{(j-m)(j+m+1)} \mid j,m+1 \rangle + \sqrt{(j+m)(j-m+1)} \mid j,m-1 \rangle] The Attempt at a Solution Expanding e^{-i \pi J_x}...

There's a geometric interpretation of the determinant of an operator in a real vector space that I've always found intuitive. Suppose we have a n-dimensional real-valued vector space. We can plot n vectors in an n-dimensional Cartesian coordinate system, and in general we'll have an...

If A is a linear operator, and we have some ordered basis (but not necessarily orthonormal), then the element Aij of its matrix representation is just the ith component of A acting on the jth basis vector. We can also represent the action of A on a ket as the matrix product of A's matrix with...

Dirac's "Quantum Mechanics" - the definition of the time evolution operator I'm reading Dirac's "Principles Of Quantum Mechanics" to learn more about the formal side of the subject. I have a question about the way he defines the time evolution operator in the book. Either there's a mistake or...