- #1
spaghetti3451
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Homework Statement
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?
Homework Equations
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?