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. T

    Quantum Mechanics - Ladder Operators

    I'm trying to show that N=a^\dagger a and K_r=\frac{a^\dagger^r a^r}{r!} commute. So basically I need to show [a^\dagger^r a^r,a^\dagger a]=0. I'm not quite sure what to do, I've tried using [a,a^\dagger] in a few places but so far haven't had much success.
  2. T

    Ladder operators for angular momentum

    This might be a basic question, but I'm having some difficulty understanding expectation values and ladder operators for angular momentum. <L+> = ? I know that L+ = Lx+iLy, but I don't know what the expectation value would be? Someone told me something that looked like this...
  3. T

    Quantum harmonic oscillator: ladder operators

    Homework Statement For a particle of mass m moving in the potential V(x) = \frac{1}{2}m\omega^2x^2 (i.e. a harmonic oscillator), it is often convenient to express the position and momentum operators in terms of the ladder operators a_{\pm}: x = \sqrt{\frac{\hbar}{2m\omega}}(a_+ + a_-) p =...
  4. N

    Understanding Ladder Operators

    Calculate [Lz,L+] By defintion ladder operators are: L+=Lx+iLy L-=Lx-iLy Important Relations: LxLy = i\hbarLz, LyLz = i\hbarLx, LzLx = i\hbarLy Lx = ypz - zpy, Ly = xpz - zpx, Lz = xpy - ypx To start solving; [Lz,L+] Lz - (Lx + iLy) = 0 Multiply through by \hbar...
  5. P

    Angular Momentum Operator in terms of ladder operators

    Homework Statement http://img716.imageshack.us/i/captur2e.png/ http://img716.imageshack.us/i/captur2e.png/ Homework Equations Stuck on the last part The Attempt at a Solution http://img689.imageshack.us/i/capturevz.png/ http://img689.imageshack.us/i/capturevz.png/
  6. G

    Ladder Operators acting upon N Ket

    I can't seem to find information regarding this anywhere. I understand why when the ladder operators act upon an energy eigenstate of energy E it produces another eigenstate of energy E \mp\hbar \omega. What I don't understand is why the following is true: \ a \left| \psi _n \right\rangle...
  7. A

    Ladder operators, a technical question

    Forgive me if I am putting this in the wrong place, but this is my first post here. The question that I have is directed to the more experienced researchers than I am, I guess. In the Hamiltonian formulation of QFTs we write everything in terms of the ladder operators, right? So in practice...
  8. ?

    What is the Missing Step to Prove the Ladder Operator Equation?

    Homework Statement The problem is to show that, \hat{a_{+}}|\alpha>=A_{\alpha}|\alpha+1> using \hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha>It's not hard to manipulate \hat{a_{+}}\hat{a_{-}}|\alpha>=\alpha|\alpha> into the form...
  9. C

    Ladder operators and harmonic oscillator

    1. Explain why any term (such as AA†A†A†)with unequal numbers of raising and lowering operators has zero expectation value in the ground state of a harmonic oscillator. Explain why any term (such as AA†A†A) with a lowering operator on the extreme right has zero expectation value in the...
  10. maverick280857

    Expressing the Klein Gordon Hamiltonian in terms of ladder operators

    Hi everyone I'm trying to express each term of the Hamiltonian H = \int d^{3}x \frac{1}{2}\left[\Pi^2 + (\nabla \Phi)^2 + m^2\Phi^2\right][/tex] in terms of the ladder operators a(p) and [itex]a^{\dagger}(p). This is what I get for the first term \int d^{3}x...
  11. B

    Ladder operators in quantum mechanics

    Homework Statement This is problem 2.11 from Griffith's QM textbook under the harmonic oscillator section. Show that the lowering operator cannot generate a state of infinite norm, ie, \int | a_{-} \psi |^2 < \infty Homework Equations This isn't so hard, except that I consistently get the...
  12. J

    Infinite Well: Ladder Operators for Simplified Expression

    Is there a simple expression for the ladder operators, in terms of x and -i\hbar\partial_x, for the infinite potential well? After some attempts, I couldn't figure out any nice operators that would map functions like this \sin\frac{\pi n x}{L} \mapsto \sin\frac{\pi(n\pm 1)x}{L}.
  13. T

    Expectation value using ladder operators

    I wonder if someone could examine my argument for the following problem. Homework Statement Using the relation \widehat{x}^{2} = \frac{\hbar}{2m\omega}(\widehat{A}^{2} + (\widehat{A}^{+})^{2} + \widehat{A}^{+}\widehat{A} + \widehat{A}\widehat{A}^{+} ) and properties of the ladder operators...
  14. M

    Angular Momentum Ladder Operators

    I thought that I had angular momentum very well understood, but something has been giving me problems recently. It is often stated in textbooks and webpages alike, that the angular momentum ladder operators defined as J_{\pm} \equiv J_x \pm i J_y Then the texts often go on to say that these...
  15. N

    Why Must Ladder Operators Have Specific Forms in Quantum Mechanics?

    Why must the ladder operators be \sqrt{\dfrac{m\omega}{2\hbar}}(x+\dfrac{ip}{m\omega}) and \sqrt{\dfrac{m\omega}{2\hbar}}(x-\dfrac{ip}{m\omega})? What is the method that obtain them from schrodinger Equation? And why we know that they are creation and anihilation operator?
  16. M

    Harmonic Oscillator, Ladder Operators, and Dirac notation

    Defining the state | \alpha > such that: | \alpha > = Ce^{\alpha {\hat{a}}^{\dagger}} | 0 >\ ,\ C \in \mathbf{R};\ \alpha \in \mathbf{C}; Now, | \alpha > is an eigenstate of the lowering operator \hat{a}, isn't it? In other words, the statement that \hat{a} | \alpha >\ =\ \alpha | \alpha >...
  17. S

    QM: Ladder Operators Explained Step-by-Step

    I am taking a QM course and we are using griffiths intro to QM text, 2nd edition. I like the text but I find it lacking when it comes to explaining ladder operators. I need to see how to use them in a very detailed step-by-step problem. Does anyone know of any good textbooks or websites that...
  18. E

    Matrix representation of ladder operators

    Homework Statement Find the matrices which represent the following ladder operators a+,a_, and a+a- All of these operators are supposed to operate on Hilbert space, and be represented by m*n matrices. Homework Equations a+=1/square root(2hmw)*(-ip+mwx) a_=1/square root(2hmw)*(ip+mwx)...
  19. quasar987

    Standard Basis and Ladder operators

    Cohen-Tanoudji defines a "standard basis" of the state space as an orthonormal basis {|k,j,m>} composed of eigenvectors common to J² and J_z such that the action of J_± on the basis vectors is given by J_{\pm}|k,j,m>=\hbar\sqrt{j(j+1)-m(m\pm 1)}|k,j,m\pm 1> But isn't is automatic that such...
  20. Gamma

    Angular Momentum and ladder operators

    Hi, I have done most of the problem in this word document (attached). I have some trouble though. In my QM class, we assumed that the z component of angular momentum Lz satisfies, Lz Ylm = m hbar Ylm and the ladder operator L+ and L- were defined as L+_ = Lx +- iLy. We were able to find the...
  21. Z

    Why is the Commutator Between Ladder Operators for a SHO Not Zero?

    Hi, Having trouble understanding something here, hoping someone can help...when dealing with a SHO, we can define two ladder operators a and a-dagger. The way I understand it is, applying a-dagger to an eigenstate of H (and that has, for instance, eigenvalue E) will give us a new eigenstate...
  22. P

    Finding Pauli matrices WITHOUT ladder operators

    Does anyone know of an alternative way of calculating the Pauli spin matrices \mbox{ \sigma_x} and \mbox{ \sigma_y} (already knowing \mbox { \sigma_z} and the (anti)-commutation relations), without using ladder operators \mbox{ \sigma_+} and \mbox{ \sigma_- }? Thanks!
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