Ladder operators Definition and 72 Threads

I'm trying to show that N=a^\dagger a and K_r=\frac{a^\dagger^r a^r}{r!} commute. So basically I need to show [a^\dagger^r a^r,a^\dagger a]=0. I'm not quite sure what to do, I've tried using [a,a^\dagger] in a few places but so far haven't had much success.

This might be a basic question, but I'm having some difficulty understanding expectation values and ladder operators for angular momentum. <L+> = ? I know that L+ = Lx+iLy, but I don't know what the expectation value would be? Someone told me something that looked like this...

Homework Statement For a particle of mass m moving in the potential V(x) = \frac{1}{2}m\omega^2x^2 (i.e. a harmonic oscillator), it is often convenient to express the position and momentum operators in terms of the ladder operators a_{\pm}: x = \sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_-) p =...

Calculate [Lz,L+] By defintion ladder operators are: L+=Lx+iLy L-=Lx-iLy Important Relations: LxLy = i\hbarLz, LyLz = i\hbarLx, LzLx = i\hbarLy Lx = ypz - zpy, Ly = xpz - zpx, Lz = xpy - ypx To start solving; [Lz,L+] Lz - (Lx + iLy) = 0 Multiply through by \hbar...

Homework Statement http://img716.imageshack.us/i/captur2e.png/ http://img716.imageshack.us/i/captur2e.png/ Homework Equations Stuck on the last part The Attempt at a Solution http://img689.imageshack.us/i/capturevz.png/ http://img689.imageshack.us/i/capturevz.png/

I can't seem to find information regarding this anywhere. I understand why when the ladder operators act upon an energy eigenstate of energy E it produces another eigenstate of energy E \mp\hbar \omega. What I don't understand is why the following is true: \ a \left| \psi _n \right\rangle...

Forgive me if I am putting this in the wrong place, but this is my first post here. The question that I have is directed to the more experienced researchers than I am, I guess. In the Hamiltonian formulation of QFTs we write everything in terms of the ladder operators, right? So in practice...

Homework Statement The problem is to show that, \hat{a_{+}}|\alpha>=A_{\alpha}|\alpha+1> using \hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha>It's not hard to manipulate \hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha> into the form...

1. Explain why any term (such as AA†A†A†)with unequal numbers of raising and lowering operators has zero expectation value in the ground state of a harmonic oscillator. Explain why any term (such as AA†A†A) with a lowering operator on the extreme right has zero expectation value in the...

Hi everyone I'm trying to express each term of the Hamiltonian H = \int d^{3}x \frac{1}{2}\left[\Pi^2 + (\nabla \Phi)^2 + m^2\Phi^2\right][/tex] in terms of the ladder operators a(p) and [itex]a^{\dagger}(p). This is what I get for the first term \int d^{3}x...

Homework Statement This is problem 2.11 from Griffith's QM textbook under the harmonic oscillator section. Show that the lowering operator cannot generate a state of infinite norm, ie, \int | a_{-} \psi |^2 < \infty Homework Equations This isn't so hard, except that I consistently get the...

Is there a simple expression for the ladder operators, in terms of x and -i\hbar\partial_x, for the infinite potential well? After some attempts, I couldn't figure out any nice operators that would map functions like this \sin\frac{\pi n x}{L} \mapsto \sin\frac{\pi(n\pm 1)x}{L}.

I wonder if someone could examine my argument for the following problem. Homework Statement Using the relation \widehat{x}^{2} = \frac{\hbar}{2m\omega}(\widehat{A}^{2} + (\widehat{A}^{+})^{2} + \widehat{A}^{+}\widehat{A} + \widehat{A}\widehat{A}^{+} ) and properties of the ladder operators...

I thought that I had angular momentum very well understood, but something has been giving me problems recently. It is often stated in textbooks and webpages alike, that the angular momentum ladder operators defined as J_{\pm} \equiv J_x \pm i J_y Then the texts often go on to say that these...

Why must the ladder operators be \sqrt{\dfrac{m\omega}{2\hbar}}(x+\dfrac{ip}{m\omega}) and \sqrt{\dfrac{m\omega}{2\hbar}}(x-\dfrac{ip}{m\omega})? What is the method that obtain them from schrodinger Equation? And why we know that they are creation and anihilation operator?

Defining the state | \alpha > such that: | \alpha > = Ce^{\alpha {\hat{a}}^{\dagger}} | 0 >\ ,\ C \in \mathbf{R};\ \alpha \in \mathbf{C}; Now, | \alpha > is an eigenstate of the lowering operator \hat{a}, isn't it? In other words, the statement that \hat{a} | \alpha >\ =\ \alpha | \alpha >...

I am taking a QM course and we are using griffiths intro to QM text, 2nd edition. I like the text but I find it lacking when it comes to explaining ladder operators. I need to see how to use them in a very detailed step-by-step problem. Does anyone know of any good textbooks or websites that...

Homework Statement Find the matrices which represent the following ladder operators a[SIZE="3"]+,a_, and a[SIZE="3"]+a- All of these operators are supposed to operate on Hilbert space, and be represented by m*n matrices. Homework Equations a[SIZE="3"]+=1/square root(2hmw)*(-ip+mwx)...

Cohen-Tanoudji defines a "standard basis" of the state space as an orthonormal basis {|k,j,m>} composed of eigenvectors common to J² and J_z such that the action of J_± on the basis vectors is given by J_{\pm}|k,j,m>=\hbar\sqrt{j(j+1)-m(m\pm 1)}|k,j,m\pm 1> But isn't is automatic that such...

Hi, I have done most of the problem in this word document (attached). I have some trouble though. In my QM class, we assumed that the z component of angular momentum Lz satisfies, Lz Ylm = m hbar Ylm and the ladder operator L+ and L- were defined as L+_ = Lx +- iLy. We were able to find the...

Hi, Having trouble understanding something here, hoping someone can help...when dealing with a SHO, we can define two ladder operators a and a-dagger. The way I understand it is, applying a-dagger to an eigenstate of H (and that has, for instance, eigenvalue E) will give us a new eigenstate...

Does anyone know of an alternative way of calculating the Pauli spin matrices \mbox{ \sigma_x} and \mbox{ \sigma_y} (already knowing \mbox { \sigma_z} and the (anti)-commutation relations), without using ladder operators \mbox{ \sigma_+} and \mbox{ \sigma_- }? Thanks!