量子力学英文课件格里菲斯Charter10

In molecular physics, this technique is known as the Born-Oppenheimer （玻恩-奥本海默）approximation.
In quantum mechanics, the essential content of the adiabatic approximation can be cast in the form of a theorem.
Here we assume that the spectrum is discrete and nondegenerate throughout the transition from Hi to Hf , so there is no ambiguity（歧义） about the
ordering of the states; these conditions can be relaxed, given a suitable procedure for “tracking” （跟踪）the eigenfunctions, but we’re not going to pursue that
A case in point is our discussion of the hydrogen molecule ion.
We began by assuming that the nuclei were at rest, a fixed distance R apart, and we solved for the motion of the electron.
and they are complete, so the general solution to the time-dependent Schrödinger equation
can be expressed as a linear combination of them :
where
generalizes the “standard” phase factor to the case where En varies with time.
But if you very gently and steadily move the support (Figure 10.1), the pendulum will continue to swing in a nice, smooth way, in the same plane (or one parallel to it) with the same amplitude.
of opening angle , at constant angular velocity
(Figure 10.5):
The Hamiltonian (Eq.[4.158]) is where
The normalized eigenspinors of H(t) are
and
they represent spin up and spin down, respectively, along the instantaneous direction of B(t) (see Problem 4.31).
Outline
Imagine a perfect pendulum, with no friction or air resistance, oscillating back and forth in a vertical plane.
If you grab the support and shake it in a jerky(忽动忽停的) manner, the bob will swing around in a wild chaotic fashion.
or
Now, differentiating Eq.[10.9] with respect to time yields
and hence (again taking the inner product with ψm )
Putting this into Eq.[10.16] we conclude that
If the Hamiltonian changes with time, then the eigenfunctions and eigenvalues are themselves timedependent:
But they still constitute (at any particular instant) an orthonormal set
state is still i(x) (Figure 10.3c), which is a
complicated linear combination of eigenstates of the new Hamiltonian.
The adiabatic theorem is simple to state, and it sounds plausible, but it is not easy to prove.
Notice that we’re not talking about a small change in the Hamiltonian -- this one is a huge change. All we require is that it happens slowly.
By contrast, if the well expands suddenly, the resulting
system itself. Te , the “external” time, over which the parameters of
the system change appreciably.
An adiabatic process is one for which Te >>Ti .
The basic strategy (策略) for analyzing an adiabatic process is first to solve the problem with the external parameters held fixed, and only at the end of the calculation allow them to change with time.
This result is exact. Now comes the adiabatic approximation: Assume that the dH/dt in Eq.[10.19] is extremely small, and drop the second term, leaving
Substituting Eq.[10.12] into Eq.[10.11] we obtain
In view of Eqs.[10.9] and [10.13] the last two terms cancel, leaving
Taking the inner product with ψm, and invoking the orthonormality of the instantaneous eigenfunctions (Eq.[10.10])
This gradual change in the external conditions characterizes an adiabatic process.
Notice that there are two characteristic times involved: Ti , the “internal” time, representing the motion of the
Once we had found the ground state energy of the system as a function of R, we located the equilibrium separation and from the curvature of the graph we obtained the frequency of vibration of the nuclei.
Suppose that the Hamiltonian changes gradually from some initial form Hi to some final form Hf (Figure 10.2).
The Adiabatic Theorem states that if the particle was initially in the the nth eigenstate of Hi, it will be carried (under the Schrödinger equation) into the nth eigenstate of Hf.
with the solution :
where
In particular, if the particle starts out in the nth eigenstate (which is to say, if cn(0)=1, and cm(0)=0 for m≠n). then (Eq.[10.12])
The corresponding eigenvalues are
Suppose the electron starts out with spin up, along B(0):
The exact solution to the time-dependent Schrödinger equation is (Problem 10.3)
here.
For example, suppose we prepare a particle in the ground state of the infinite square well (Figure 10lly move the right wall out to 2a, the adiabatic theorem says that the particle win end up in the ground state of the expanded well (Figure 10.3b):
so it remains in the nth eigenstate, picking up only a couple of phase factors.
QED
Imagine an electron (charge e, mass m) at rest at the origin.
In the presence of a magnetic field whose magnitude (B0) is constant but whose direction sweeps out a cone,
For example, the classical period of a pendulum of (constant) length L is
if the length is now gradually changing, the period will presumably be.
When you stop to think about it, we actually use the adiabatic approximation (implicitly) all the time without noticing it.
If the Hamiltonian is independent of time, then a particle which starts out in the nth eigenstate, ψn ,
remains in the nth eigenstate, simply picking up a phase factor: