The schroedinger equation in ladder operator form looks like$$(a_+ a_- - \frac{1}{2}\hbar \omega) \psi_n = E_n \psi_n$$So I said that$$\int (a_+ \psi_n)^* (a_+ Normalization of the wave functions Although the ladder operators can be used to create a new wave function from a given normalized wave function, the new wave Obtain the matrix representation of the ladder operators ##J_{\pm}##. Homework Equations Remark that ##J_{\pm} | jm \rangle = N_{\pm}| jm \pm 1 \rangle## The Attempt
the normalization constant Aif you like). Then, to nd the rst excited state, just apply the raising operator, also written in terms of p= id=dx, to the ground state The last important property about ladder operators its eigenvalue. While it is already clear from equations 26 and 32 that ladder operators do NOT scale QHO wave There doesn't seem to be a true convention for the ladder operators; I have chosen to use: $A_{\pm}=\frac{1}{\sqrt{2m}}\left(\hat{p}\pm im\omega x\right)$ as it We can think about :::;jm 1i;jmi;jm+ 1i;:::as rungs of a ladder. J + acts as a raising operator that allows us to climb one rung of the ladder each time we use it Creation and annihilation operators can act on states of various types of particles. For example, in quantum chemistry and many-body theory the creation and
normalization R1 1 dxj 0(x)j2 = 1 together with formula (2.119) for Gaussian functions determines the normalization constant N2 = r m! ˇ~) N= m! ˇ~ 1 4: (5.22) We Ladder operators are good for solving matrix elements Xl, ml +1 Lx l, ml\ = ß 2 Lx = ILx + i Ly M+ILx - iLy M = L-+ L+ ß Lx = L-+ L+ 2 = [l, ml +1 L-2 l, ml_+[l, ml We can write the quantum Hamiltonian in a similar way. Choosing our normalization with a bit of foresight,wedefinetwoconjugateoperators, ^a = r m! 2~ X^ + i m! P^ ^ay = The use of ladder operators with simple commutation relations simplifies the calculation of matrix elements for the Morse oscillator. A further simplification arises representation of the L x operator (use the ladder operator representation of L x). Verify that the matrix is hermitian. Find the eigenvalues and corresponding
For this reason these operators are called ladder operators; they allow us to \climb up and down in energy. It is also easy to take care of the ladder operator method. This method is similar to that used for the derivation of wave This method is similar to that used for the derivation of wave function of
operators appearing as quadratic terms. We have encountered the harmonic oscillator already in Sect. 2 where we determined, in the context of a path integral approach Raising and lowering operators for angular momentum: The set of eigenvalues a and b can be obtained by making use of a trick based on a ladder operator formalism Finding the m = l Eigenket of \\(L^2\\), \\(L_z\\). Matrix representation of angular momentum operators: So far the angular momen-tum operators L2 and L iâ s are
