What is Raising operator: Definition and 13 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. guyvsdcsniper

    Proving commutator relation between H and raising operator

    I am going through my class notes and trying to prove the middle commutator relation, I am ending up with a negative sign in my work. It comes from [a†,a] being invoked during the commutation. I obviously need [a,a†] to appear instead. Why am I getting [a†,a] instead of [a,a†]?
  2. Mayan Fung

    I How do we know that the raising operator only raises the state by one step?

    In the simple harmonic oscillator, I was told to use the raising and lowering operator to generate the excited states from the ground state. However, I am just thinking that how do we confirm that the raising operator doesn't miss some states in between. For example, I can define a raising...
  3. H

    I Matrix Representation of the Angular Momentum Raising Operator

    In calculating the matrix elements for the raising operator L(+) with l = 1 and m = -1, 0, 1 each of my elements conforms to a diagonal shifted over one column with values [(2)^1/2]hbar on that diagonal, except for the element, L(+)|0,-1>, where I have a problem. This should be value...
  4. 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...
  5. Pouyan

    Eigenvector of raising operator

    Homework Statement show that the raising operator at has no right eigenvectors Homework Equations We know at|n> = √(n+1)|n+1> The Attempt at a Solution we define a vector |Ψ> = ∑cn|n> (for n=0 to ∞) at|Ψ>=at∑cn|n>=∑cn(√n+1)|n+1> But further I give up!:cry:
  6. 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...
  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. lfqm

    Raising operator for s in |s,m> states

    Are there any known (collective spin) operators to raise or lower the quantum number s in \left|{s,m}\right> spin states? I'm trying to construct coherent states varying the quantum number s instead of the well known spin coherent states varying m. I found a coherent-like state similar to the...
  9. S

    Showing raising operator raising

    Hey, I have a question on showing how the raising operator in QM raises a particular eigenstate by 1 unit, the question is showed below: I think I know how to do this but not sure if what I'm doing is sufficient: \hat{N}a^{\dagger}|n>=([\hat{N},a^{\dagger}]+a^{\dagger}\hat{N})|n>...
  10. C

    Where Does the Double Hat in the Hamiltonian Raising Operator Come From?

    Homework Statement I'm given the line: (the coding stopped responding for the "hats") \hat{}H(\hat{}a|n>) = (doublehat a) H(hat) |n> + [Hhat,ahat]|n> I'm assuming Hhat= hbar *w ( aa* + 1/2) so I don't know what they are doing. where does the double hat come from. where does any...
  11. N

    Effect of the raising operator on an l=1 m=1 state

    Pretty basic question but I was wondering what the result of acting the raising operator on an l=1, m=1 quantum state for a hydrogen wavefunction would be. Specifically, L+|1,1> = ? I know that normally L+|l,m>=hbar(l(l+1)-m(m+1))1/2|l,m+1> but I wasn't sure if the eigenstate remained...
  12. R

    Angular Momentum Raising Operator

    Homework Statement In Problem 4.3 you showed that Y^{1}_{2}(\theta , \phi) = -\sqrt{15/8\pi} sin\theta cos\theta e^{i\phi} Apply the raising operator to find Y^{2}_{2}(\theta , \phi). Use Equation 4.121 to get the normalization. Homework Equations [Eq. 4.121] A^{m}_{l} = \hbar...
  13. H

    Raising Operator (Harmonic Oscillator)

    This is (another!) question I cannot solve The ground state wavefunction for the harmonic oscillator can be written as $\chi _0 = \left( {\frac{\alpha } {\pi }} \right)^{\frac{1} {4}} \exp \left( {\frac{{ - \alpha x^2 }} {2}} \right)$ where $\alpha = \sqrt {\frac{{mk}} {{\hbar ^2...
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