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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  1. M

    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...
  2. 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...
  3. C

    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...
  4. niloun

    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...
  5. tomdodd4598

    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...
  6. I

    Eigenvalue of lowering operator

    How to prove that eigenvalue of lowering operator is zero?
  7. I

    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...
  8. B

    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...
  9. I

    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...
  10. V

    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...
  11. D

    Raising and lowering operator coefficients

    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...
  12. R

    Understanding Lowering Operator A and Hermitianity

    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...
  13. E

    Commutate relation of lowering operator and sperical tensor operators

    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...
  14. E

    Eigenvalues and eigenfunctions of the lowering operator

    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+...