1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Validity of the sudden approximation

  1. Nov 17, 2016 #1
    1. The problem statement, all variables and given/known data

    The Schrodinger equation is given by

    $$i\hbar\ \frac{\partial}{\partial t}\ \mathcal{U}(t,t_{0})=H\ \mathcal{U}(t,t_{0}),$$

    where ##\mathcal{U}(t,t_{0})## is the time evolution operator for evolution of some physical state ##|\psi\rangle## from ##t_0## to ##t##.


    Rewriting time ##t## as ##t=s\ T##, where ##s## is a dimensionless parameter and ##T## is a time scale, the Schrodinger equation becomes as

    $$i\ \frac{\partial}{\partial s}\ \mathcal{U}(t,t_{0})=\frac{H}{\hbar/T}\ \mathcal{U}(t,t_{0})=\frac{H}{\hbar\ \Omega}\ \mathcal{U}(t,t_{0}),$$

    where ##\Omega \equiv 1/T##.

    In the sudden approximation, ##T \rightarrow 0##, which means that ##\hbar\ \Omega \gg H##.


    1. Are we allowed to redefine ##H## by adding or subtracting an arbitrary constant?
    2. How does this introduce some overall phase factor in the state vectors?
    3. Why does this imply that ##\mathcal{U}(t,t_{0})\rightarrow 1## as ##t\rightarrow 0##?
    4. How does this prove the validity of the sudden approximation?

    2. Relevant equations

    3. The attempt at a solution

    1. I think that we are allowed to redfine ##H## by adding or subtracting an arbitrary constant, because ##H=T-V## and the potential ##V## can be redefined by adding or subtracting an arbitrary constant without changing the physical system.

    What do you think?
     
  2. jcsd
  3. Nov 18, 2016 #2

    DrDu

    User Avatar
    Science Advisor

    Yes, you are allowed to redefine the Hamiltonian by addition of a constant. Note however that H=T+V, unless you are using an unusual definition of V.
    You also have to take in mind that ##\hbar \Omega \gg H## is meaningless as you are comparing an operator with a number. As H usually is unbound, the sudden approximation never holds for all states and convergence is non-uniform.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted