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Equation related with operators

  1. Aug 31, 2012 #1
    Hi! I was studying Shankar's Principles of Quantum Mechanics, but I was stuck to understand a relation concerned with operators.

    1. The problem statement, all variables and given/known data

    I've learned that in order to get the equation of motion, I should apply the Schrodinger's equation to the given Hamiltonian operator. In the book, it is said that a state function [itex]\lvert\psi(t)\rangle[/itex] is represented by a product of the propagator and the initial state function [itex]\lvert\psi(0)\rangle[/itex] for a time-independent Hamiltonian operator.

    In p. 149 of the book, to get the propagator expressed as eigenstates of the Hamiltonian operator, a way is introduced to get those eigenfunctions. More specifically, the Hamiltonian operator given in p.149 is [itex]H=\frac{P^2}{2m} + \frac{1}{\cosh^2 X}[/itex] where X is the position operator. Then the eigenstates [itex]\lvert E\rangle[/itex] should satisfy [itex]H\lvert E\rangle = E \lvert E\rangle[/itex]. If I put bra ##\langle x \rvert ## to both sides of the equation, then ##\langle x \rvert H \lvert E \rangle = \langle x \rvert E \lvert E\rangle##. This implies according to the book,
    $$\left(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2}+\frac{1}{\cosh^2 x}\right)\psi_E(x) = E\psi_E(x)$$ where ##\psi_E(x) = \langle x \vert E \rangle ##.

    What I want to know is as follows. I think the above equation makes sense only if I show ##\langle x \rvert \frac{1}{\cosh^2 X} \lvert E\rangle = \frac{1}{\cosh^2 x} \langle x \vert E\rangle## (I've omitted other terms). However, I don't have an idea how to prove it. Even though I try to use completeness property, it's not easy. Could you give me a hint?

    I think this kind of relation should hold generally. What I mean is for any potential operator V(X), if I put ##\langle x \rvert## and ##\lvert E \rangle## to both sides of it, I should get ##V(x)\langle x \vert E\rangle##. Am I right? I want to check this as well. (Be careful in distinguishing the letters X, x.)

    I hope somebody else answer my question. Thank you for reading my long question.

    2. Relevant equations



    3. The attempt at a solution
     
    Last edited by a moderator: Sep 1, 2012
  2. jcsd
  3. Sep 1, 2012 #2
    Perhaps this trick can be of use to you:

    [tex]\langle x \lvert V(X) \lvert \psi\rangle = \langle \psi\lvert V(X)\lvert x\rangle^* = V(x) \langle \psi\lvert x \rangle^* = V(x) \langle x\lvert \psi \rangle[/tex]
     
    Last edited: Sep 1, 2012
  4. Sep 1, 2012 #3
    For me, the second equality is still mystery... ;;
    Could you give me more hint??
     
  5. Sep 1, 2012 #4
    All it is is the definition of the adjoint of V (which is V itself because V is self-adjoint).

    Shankar p.26 :smile:
     
  6. Sep 2, 2012 #5
    btw I find the "math notation" to be clearer than the Dirac notation for this kind of thing:

    [tex]\langle x \lvert V \lvert \psi \rangle = (x, V\psi) = (V^{\dagger} x, \psi) = (Vx, \psi) [/tex]
     
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