Lowering operator Definition and 15 Threads

Hi! I am working on homework and came across this problem: <n|X5|n> I know X = ((ħ/(2mω))1/2 (a + a+)) And if I raise X to the 5th, its becomes X5 = ((ħ/(2mω))5/2 (a + a+)5) What I'm wondering is, is there anyway to be able to solve this without going through all of the iterations the...

Homework Statement Show that application of the lowering Operator A- to the n=3 harmonic oscillator wavefunction leads to the result predicted by Equation (5.6.22). Homework Equations Equation (5.6.22): A-Ψn = -iΨn-1√n The Attempt at a Solution I began by saying what the answer should end...

This is not part of my coursework but a question from a past paper (that we don't have solutions to). 1. Homework Statement Construct the matrix ##\sigma_{-} = \sigma_{x} - i\sigma_{y}## and show that the states resulting from ##\sigma_{-}## acting on the eigenstates of ##\sigma_{z} ## are...

Hi everyone I need raising and lowering operators for l=3 (so it should be 7 dimensional ). is it enough to use this formula: (J±)|j, m > =sqrt(j(j + 1) - m(m ± 1))|j, m ± 1 > The main problem is about calculating lx=2 for a given wave function , I know L^2 and Lz but I need L+ and L- to solve...

Hi, I have learned about how to find the 4 spin states of 2 spin 1/2 particles, and how to find them by using the lowering operator twice on |1/2, 1/2> to find the triplet, then simply finding the orthogonal singlet state, |0, 0>. I started to attempt finding the states of 3 spin 1/2...

How to prove that eigenvalue of lowering operator is zero?

Let a be a lowering operator and a† be a raising operator. Prove that a((a†)^n) = n (a†)^(n-1) Professor suggested to use induction method with formula: ((a†)(a) + [a,a†]) (a†)^(n-1) But before start applying induction method, I would like to know where the given formula comes from. Someone...

Homework Statement Using J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle Derive that : \langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)] Homework Equations J_- = J_x - iJ_y J_+ = J_x + iJ_y The...

Homework Statement let A be a lowering operator. Homework Equations Show that A is a derivative respects to raising operator, A†, A=d/dA† The Attempt at a Solution I start by defining a function in term of A†, which is f(A†) and solve it using [A , f(A†)] but i get stuck after that. Can...

Homework Statement Derive [L_\pm , L^2]=0 Homework Equations L_{\pm}=L_x \pm iL_y The Attempt at a Solution [L_\pm , L^2]=[L_x,L_x^2] \pm i[L_y,L_y^2]=[L_x,L_x]L_x + L_x[L_x,L_x] \pm i([L_y,L_y]L_y+L_y[L_y,L_y]) Is this right so far? If so, how do I proceed from...

Homework Statement where a=lowering operator, ad=raising operator ad^(2)+a*ad+ad*a+a^(2) just need to find the coefficients Homework Equations ad|n>=sqrt(n+1)|n+1> a|n>=sqrt(n)|n-1> The Attempt at a Solution ad^(2)=?|n+2> ad*a=n|n> a*ad=n|n> a^(2)=?|n-2> just reviewing and can't get...

Hello, I am trying to understand some principles in my book. Is the lowering operator A, the conjugate of the raising operator A^{+}? Also I was reading that the lowering operator A is not Hermitian since it and its adjoint A^{+} are not equal? Does that imply that A^{+} is not Hermitian...

Hi all, I found a commutation relation of lowering operator(J-) and spherical operator in Shankar's QM (2ed, page 418, Eq 15.3.11): [J_-,T_k^q] = - \hbar \sqrt{(k+q)(k-q+1)} T_k^{q-1} I wonder how the minus sign in the beginning of the right hand side come out? I have googled some...

Homework Statement Consider lowering and rising operators that we encountered in the harmonic oscillator problem. 1. Find the eigenvalues and eigenfunctions of the lowering operator. 2. Does the rising operator have normalizable eigenfunctions?Homework Equations a-= 1/sqrt(2hmw) (ip + mwx) a+...