Ladder operators Definition and 72 Threads

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. H

    B Tensor product of operators and ladder operators

    Hi Pfs i have 2 matrix representations of SU(2) . each of them uses a up> and down basis (d> and u> If i take their tensor product i will get 4*4 matrices with this basis: d>d>,d>u>,u>d>,u>u> these representation is the sum equal to the sum of the 0-representation , a singlet represertation with...
  2. P

    Commutation relations between Ladder operators and Spherical Harmonics

    I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?
  3. Hamiltonian

    I Solving Schrodinger's eqn using ladder operators for potential V

    The Schrodinger equation: $$-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + \hat V\psi = E\psi$$ $$\frac{1}{2m}[\hat p^2 + 2m\hat V ]\psi = E\psi$$ The ladder operators: $$\hat a_\pm = \frac{1}{\sqrt{2m}}[\hat p \pm i\sqrt{2m\hat V}]$$ $$\hat a_\pm \hat a_\mp = \frac{1}{2m}[\hat p^2 + (2m\hat V) \mp...
  4. J

    I Zero-point energy of the harmonic oscillator

    First time posting in this part of the website, I apologize in advance if my formatting is off. This isn't quite a homework question so much as me trying to reason through the work in a way that quickly makes sense in my head. I am posting in hopes that someone can tell me if my reasoning is...
  5. chocopanda

    Harmonic oscillator with ladder operators - proof using the Sum Rule

    I'm trying verify the proof of the sum rule for the one-dimensional harmonic oscillator: $$\sum_l^\infty (E_l-E_n)\ | \langle l \ |p| \ n \rangle |^2 = \frac {mh^2w^2}{2} $$ The exercise explicitly says to use laddle operators and to express $p$ with $$b=\sqrt{\frac {mw}{2 \hbar}}-\frac...
  6. E

    Normalisation constants with ladder operators

    The previous part was to show that ##a_+ \psi_n = i\sqrt{(n+1)\hbar \omega} \psi_{n+1}##, which I just did by looking at$$\int |a_+ \psi_n|^2 dx = \int \psi_n^* (a_{-} a_+ \psi_n) dx = E+\frac{1}{2}\hbar \omega = \hbar \omega(n+1)$$so the constant of proportionality between ##a_+ \psi_n## and...
  7. Q

    A Algebraic form of Klein Gordon ##\phi^4## vacuum and ladder operators

    In theory, does an algebraic expression exist for the ground state of the Klein Gordon equation with \phi^4 interactions in the same way an algebraic expression exists for the simple harmonic oscillator ground state wavefunction in Q.M.? Is it just that it hasn't been found yet or is it...
  8. G

    Harmonic Oscillator Ladder Operators - What is (ahat_+)^+?

    I know that ahat_+ = 1/sqrt((2*m*h_bar*w)) * (mw(xhat)+i(phat)) and ahat_- = 1/sqrt((2*m*h_bar*w)) * (mw(xhat)-i(phat)). But I'm not sure what (ahat_+)^+ could be.
  9. Garlic

    Landau levels: Hamiltonian with ladder operators

    Dear PF, I hope I've formulated my question understandable enough. Thank you for your time, Garli
  10. Gbox

    Ladder Operators: Commutation Relation & Beyond

    a. ##L_{+}^{\dagger}=(L_x+iL_y)^{\dagger}=L_x-iL_y=L_{-}## b.##[L_{+},L_{-}]=[L_x+iL_y,L_x-iL_y]=(L_x+iL_y)(L_x-iL_y)-(L_x-iL_y)(L_x+iL_y)=## ##=L_x^2-iL_xL_y+iL_yL_x+L_y^2-(L_x^2+iL_xL_y-iL_yL_x-L_y^2)## ##=L_x^2-iL_xL_y+iL_yL_x+L_y^2-L_x^2-iL_xL_y+iL_yL_x+L_y^2##...
  11. D

    I Vacuum projection operator and normal ordering

    I've been reading this book, in which the author expresses the vacuum projection operator ##\vert 0\rangle\langle 0\vert## in terms of the number operator ##\hat{N}=\hat{a}^{\dagger}\hat{a}##, where ##\hat{a}^{\dagger}## and ##\hat{a}## are the usual creation and annihilation operators...
  12. 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...
  13. N

    Isospin Doublet Derivation Using Clebsch-Gordan Coefficients

    Homework Statement I am trying to improve my understanding of the Clebsch-Gordan coefficients. I am looking at page 5 of the following document https://courses.physics.illinois.edu/phys570/fa2013/chapter3.pdf Homework Equations I have derived the result for the I = 3/2 quadruplet but am...
  14. K

    I Ladder operators and SU(2) representation

    Hello! I read in many places the derivation of the representation for SU(2) using ladder operators and in all of the places they say that, due to the fact that we are looking for a finite dimensional representation, the ladder must end at a point, hence why we have an eigenvector of ##L_3##...
  15. Rabindranath

    Angular momentum operator for 2-D harmonic oscillator

    1. The problem statement I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian. The Attempt at a Solution I get...
  16. S

    Solving Spherical Harmonics Homework

    Homework Statement The spherical harmonic, Ym,l(θ,φ) is given by: Y2,3(θ,φ) = √((105/32π))*sin2θcosθe2iφ 1) Use the ladder operator, L+ = +ħeiφ(∂/∂θ+icotθ∂/∂φ) to evaluate L+Y2,3(θ,φ) 2) Use the result in 1) to calculate Y3,3(θ,φ) Homework Equations L+Ym,l(θ,φ)=Am,lYm+1,l(θ,φ)...
  17. D

    To find the energy eigenvalues in the 3D Hilbert space

    A fictitious system having three degenerate angular momentum states with ##\ell=1## is described by the Hamiltonian \hat H=\alpha (\hat L^2_++\hat L^2_-) where ##\alpha## is some positive constant. How to find the energy eigenvalues of ##\hat H##?
  18. binbagsss

    Complex scalar field -- Quantum Field Theory -- Ladder operators

    Homework Statement STATEMENT ##\hat{H}=\int \frac{d^3k}{(2\pi)^2}w_k(\hat{a^+(k)}\hat{a(k)} + \hat{b^{+}(k)}\hat{b(k)})## where ##w_k=\sqrt{{k}.{k}+m^2}## The only non vanishing commutation relations of the creation and annihilation operators are: ## [\alpha(k),\alpha^{+}(p)] =(2\pi)^3...
  19. L

    Hamiltonian in terms of creation/annihilation operators

    Homework Statement Consider the free real scalar field \phi(x) satisfying the Klein-Gordon equation, write the Hamiltonian in terms of the creation/annihilation operators. Homework Equations Possibly the definition of the free real scalar field in terms of creation/annihilation operators...
  20. L

    I Understanding the scalar field quantization

    I am getting started with QFT and I'm having a hard time to understand the quantization procedure for the simples field: the scalar, massless and real Klein-Gordon field. The approach I'm currently studying is that by Matthew Schwartz. In his QFT book he first solves the classical KG equation...
  21. DOTDO

    Ladder operators in electron field and electron's charge

    S. Weinberg says in his book, "The Quantum Theory of Fields Volume I", that Since electrons carry a charge, we would not like to mix annihilation and creation operators, so we might try to write the field as $$\psi(x)=\sum_{k}u_k (x)e^{-i\omega_k t}a_k$$ where ##u_k (x)e^{-i\omega_k t}## are a...
  22. S

    I Harmonic oscillator ladder operators

    The ladder operators of a simple harmonic oscillator which obey $$[H,a^{\dagger}]=\hbar\omega\ a^{\dagger}$$. --- I would like to see a proof of the relation $$\exp(-iHt)\exp(a^{\dagger})\exp(iHt)|0\rangle=\exp(a^{\dagger}e^{-i\omega t})|0\rangle\exp(i\omega t/2).$$ Thoughts?
  23. P

    Normalization of the Angular Momentum Ladder Operator

    Homework Statement 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 at a Solution [/B] The textbook states ##|N_{\pm}|^2=\langle jm | J_{\pm}^\dagger J_{\pm} | jm \rangle##...
  24. Xico Sim

    I Quarks and isospin ladder operators

    Hi, guys. This is actually a question about quantum mechanics, but since the context in which it appeared is particle physics, I'll post it here. On Thompson's book (page 227, equation (9.32)), we have $$T_+ |d\bar{u}\rangle = |u\bar{u}\rangle - |d\bar{d}\rangle$$ But I thought...
  25. S

    A Commutation relations - field operators to ladder operators

    I would like to show that the commutation relations ##[a_{\vec{p}},a_{\vec{q}}]=[a_{\vec{p}}^{\dagger},a_{\vec{q}}^{\dagger}]=0## and ##[a_{\vec{p}},a_{\vec{q}}^{\dagger}]=(2\pi)^{3}\delta^{(3)}(\vec{p}-\vec{q})## imply the commutation relations...
  26. B

    Do ladder operators give integer multiples of ћ?

    Say I apply a raising operator to the spin state |2,-1>, then by using the the equation S+|s,ms> = ћ*sqrt(s(s+1) - ms(ms+1))|s,ms+1> I get, S+|2,-1> = sqrt(6)ћ|2,0> Does this correspond to a physical eigenvalue or should I disregard it and only take states with integer multiples of ћ as...
  27. Raptor112

    Matrix Representation for Combined Ladder Operators

    Due to the definition of spin-up (in my project ), \begin{eqnarray} \sigma_+ = \begin{bmatrix} 0 & 2 \\ 0 & 0 \\ \end{bmatrix} \end{eqnarray} as opposed to \begin{eqnarray} \sigma_+ = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} \end{eqnarray} and the annihilation operator is...
  28. M

    Harmonic Oscillator and Ladder Operators

    Homework Statement Consider a linear harmonic oscillator with the solution defined by the ladder operators a and a†. Use the number basis |n⟩ to do the following. a) Construct a linear combination of |0⟩ and |1⟩ to form a state |ψ⟩ such that ⟨ψ|X|ψ⟩ is as large as possible. b) Suppose that...
  29. S

    Ladder operators in Klein -Gordon canonical quantisation

    The quantum Klein-Gordon field ##\phi({\bf{x}})## and its momentum density ##\pi({\bf{x}})## are given in Fourier space by ##\phi({\bf{x}}) = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2 \omega_{{\bf{p}}}}} \big( a_{{\bf{p}}} e^{i{\bf{p}} \cdot {\bf{x}}} + a^{\dagger}_{{\bf{p}}}...
  30. Activeuser

    Ladder operators and matrix elements...

    Please I need your help in such problems.. in terms of ladder operators to simplify the calculation of matrix elements... calculate those i) <u+2|P2|u> ii) <u+1| X3|u> If u is different in both sides, then the value is 0? is it right it is 0 fir both i and ii? when exactly equals 0, please...
  31. Logan Rudd

    Ladder operators to prove eigenstates of total angular momen

    Homework Statement Consider the following state constructed out of products of eigenstates of two individual angular momenta with ##j_1 = \frac{3}{2}## and ##j_2 = 1##: $$ \begin{equation*} \sqrt{\frac{3}{5}}|{\tiny\frac{3}{2}, -\frac{1}{2}}\rangle |{\tiny 1,-1}\rangle +...
  32. P

    Ladder operators for real scalar field

    Puting a minus in front of the momentum in the field expansion gives ##\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde p} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} + a_{\bf{p}}^ + {e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right){\rm{ }}\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde...
  33. 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...
  34. gfd43tg

    Ladder operators to find Hamiltonian of harmonic oscillator

    Hello, I was just watching a youtube video deriving the equation for the Hamiltonian for the harmonic oscillator, and I am also following Griffiths explanation. I just got stuck at a part here, and was wondering if I could get some help understanding the next step (both the video and book...
  35. M

    Stationary States in Griffiths Intro to QM

    I am referring to the section The Harmonic Oscillator in Griffiths's introductino to quantum mechanics (the older edition with the black cover). I understand how it all works, however there is a part that I am not sure about. How do we know when we apply a- or a+ (the ladder operators) to a...
  36. K

    Ladder operators and the momentum and position commutator

    When using Fourier's trick for determining the allowable energies for stationary states, Griffiths introduces the a+- operators. When factoring the Hamiltonian, the imaginary part is assigned to the momentum operator versus the position operator. Is there a reason for this? If : a-+ = k(ip +...
  37. B

    Finding A Solution Using the Ladder Operators

    Hello, I am reading Griffiths Quantum Mechanics textbook, and am having some difficulty with a derivation on page 56. To me, there seems to be something logically wrong with his arguments, but I can not pin-point precisely what it is. To provide you with a little background, Griffiths is...
  38. M

    How do ladder operators generate energy values in a SHO?

    Hello, I am currently studying ladder operator for a simple harmonic operator as a method for generating the energy values. This seem like a simple algebra question I am asking so I do apologize but I just can't figure it out. Here are my operator definitions...
  39. H

    SHO ladder operators & some hamiltonian commutator relations

    Homework Statement For the SHO, find these commutators to their simplest form: [a_{-}, a_{-}a_{+}] [a_{+},a_{-}a_{+}] [x,H] [p,H] Homework Equations The Attempt at a Solution I though this would be an easy problem but I am stuck on the first two parts. Here's what I did at first...
  40. C

    How to apply ladder operators?

    The total energy of a particle in a harmonic oscillator is found to be 5/2 ~!. To change the energy, if i applied the lowering operator 4 times and then the raising operator 1 times successively. What will be the new total energy? i want the calculation please
  41. P

    Quick question about raising and lowering operators (ladder operators)

    Reading through my QM text, I came across this short piece on ladder operators that is giving me trouble (see picture). What I am struggling with is how to get to equations 2 and 3 from equation 1. Can someone point me in the right direction? Where does the i infront of the x go?
  42. T

    Ladder Operators and Dirac as the source.

    Hello, I've read that Dirac introduced the idea of the creation and annihilation operators in the solution to the quantum harmonic oscillator problem, but can anyone tell me where he did this? In a paper, or maybe in a book? I've had a little search online, but I've yet to discover...
  43. JeremyEbert

    Angular momentum ladder operators and state transitions

    What is the significance of the ladder operators eigenvalues as they act on the different magnetic quantum numbers, ml and ms to raise or lower their values? How do their eigenvalues relate to the actual magnetic transitions from one state to the next?
  44. L

    Proving HO Eigenvalues Using Ladder Operators

    This has already been adressed here: https://www.physicsforums.com/showthread.php?t=173896 , but I still didn't get the answer. The Harmonic Oscillator is fully described (according to my favourite QM book) by the HO Hamiltonian, and the commutation relations between the position and momentum...
  45. G

    How Do Ladder Operators Relate to the 1-D Quantum Harmonic Oscillator?

    Homework Statement Homework Equations The Attempt at a Solution I solved part a) correctly, I believe, giving me ψ = e^{-(√(km)/\hbar)x^{2}} and a normalization constant A = ((π\hbar)/(km))^{-1/4} I'm having difficulty with part b. I'm not exactly sure how I create a...
  46. E

    Existance of ladder operators for a system

    I have only heard about the use of ladder operators in connection with the harmonic oscillator and spin states. However, I would expect them to be useful in other systems as well. For example, can we find ladder operators for the discrete states of the hydrogen atom, or any other system with...
  47. K

    Ladder Operators for Harmonic Oscillator Excited States

    I have a homework problem which asks me to compute the second and third excited states of the harmonic oscillator. The function we must compute involves taking the ladder operator to the n-power. My question is this: because the ladder operator appears as so, -ip + mwx, and because I am using it...
  48. S

    Solving Ladder Operator Problem w/ 4 Terms

    Homework Statement I have been given the following problem - the expectation value of px4 in the ground state of a harmonic oscillator can be expressed as <px4> = h4/4a4 {integral(-infin to +infin w0*(x) (AAA+A+ + AA+AA+ + A+AAA+) w0 dx} I think I know how to proceed on other...
  49. T

    QM - Deriving the Ladder Operators' Eigenbasis

    I'm am trying to derive the relations: a|n\rangle=\sqrt{n}|n-1\rangle a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle using just the facts that [a,a+]=1 and N|n>=|n> where N=a^{\dagger}a (which implies \langle n|N|n\rangle=n\geq 0). This is what I've done so far: [a,a^{\dagger}]=1 \Rightarrow...
  50. T

    Quantum Mechanics - Ladder Operators

    I'm trying to show that \sum_{m=0}^\infty \frac{1}{m!} (-1)^m {a^{\dagger}}^m a^m =|0 \rangle\left\langle 0| Where a and {a^{\dagger}} denote the usual annihilation and creation operators. The questions suggests acting both sides with |n> but even if I did that and showed LHS=...=RHS then that...
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