What is Lowering operator: Definition and 15 Discussions

In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.

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2. I Raising the ladder operators to a power

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...
3. N

Lowering Operator Simple Harmonic Oscillator n=3

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...
4. Pauli Spin Matrices - Lowering Operator - Eigenstates

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...
5. A What are L+ and L- matrices for l=3 ?

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...
6. Eigenstates of 3 spin 1/2 particles

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...
7. Eigenvalue of lowering operator

How to prove that eigenvalue of lowering operator is zero?
8. Quantum Mechanics - Induction Method

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...
9. QM: Expectation value of raising and lowering operator

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...
10. Quantum Mechanics - Lowering Operator

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...
11. Angular momentum raising lowering operator

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...