meopemuk said:
Please remind me what was the field theory explanation for the interference experiment
with atomic deposition? In particular, how the atomic "field" collapses to a specific position
on the substrate? Isn't it the same dreadful Copenhagen collapse?
I hope in this explanation you [Arnold] wouldn't invoke the quantum nature of the
substrate as M&W did in their Chapter 9.
I don't see anything fundamentally wrong with a similar calculation as in M&W ch9.
For the benefit of other readers, I'll quickly review sections 9.2 and 9.3 of Mandel & Wolf:
M&W work in the interaction picture, with the Schroedinger equation
<br />
\partial_t \; |\psi(t)\rangle<br />
~=~ \frac{1}{i\hbar} \, \hat{H}_I(t) \, |\psi(t)\rangle<br />
~~~~~~~~(9.2-1)<br />
where |\psi(t)\rangle is the state of an electron in the detector,
and \hat{H}_I(t) is the interaction part of the Hamiltonian.
For such a quantized detector interacting with a classical EM field, the latter is
<br />
\hat{H}_I(t) ~=~ - \frac{e}{m}\, \hat{p}_i(t) \hat{A}^i(r,t) ~.<br />
~~~~~~~~(9.2-3)<br />
where \hat{p}_i(t) is the electron momentum,
\hat{A}_i(r,t) is the (classical, c-number) EM vector potential,
and the electron is initially in a tightly bound state at position r.
By standard techniques, equation (9.2-1) is formally integrated
iteratively (as usual for Volterra-type integral equations) to obtain:
<br />
|\psi(t)\rangle ~=~ |\psi(t_0)\rangle<br />
~+~ \frac{1}{i\hbar} \int_{t_0}^t dt_1 \hat{H}_I(t_1) \, |\psi(t_0)\rangle<br />
~+~ \mbox{(higher order terms in } \hat{H}_I) \dots<br />
~~~~~~~~(9.2-8)<br />
In the situation being considered, truncation of the higher order terms
is acceptable. The transition probability from an initial state
|\psi(t_0)\rangle to some new state \Phi at time t
which is orthogonal to |\psi(t_0)\rangle is then approximately:
<br />
\left| \mbox{Transition probability} \right|<br />
~=~ \left| \langle \Phi | \psi(t)\rangle \right|^2<br />
~=~ \frac{1}{\hbar^2} \left| \int_{t_0}^t dt_1<br />
\langle \Phi | \hat{H}_I(t_1) |\psi(t_0)\rangle \right|^2<br />
~~~~~~~~(9.2-9)<br />
Since we're working in the interaction picture, and the EM field is
a c-number here, the interaction part of the Hamiltonian at t_1
can be expressed in terms of an arbitrary initial time t_0 as
<br />
\hat{H}_I(t) ~=~ - \frac{e}{m} ~<br />
e^{i H_0 \Delta t/\hbar} \, \hat{p}_i(t_0) \,<br />
e^{-i H_0 \Delta t/\hbar} ~ \hat{A}^i(r,t) <br />
where \Delta t := t_1 - t_0.
With an extra (analytic signal representation) assumption about the
incident EM field, this is sufficient information to evaluate formula
(9.2-9) above, giving the probability that a transition occurs (i.e., a
photoelectron is produced) within a small time interval \Delta t.
In summary, one calculates the probability from an initial product state
consisting of quantized bound electron and incident EM field to a final
state in which the electron has been excited into the conduction band
under the interaction \hat{H}_I(t).
The important point from the above is that similar computations
could be done for other kinds of interactions, involving some field
incident on a many-body plate, with other possibilities for the final
state(s). For atom deposition, we could consider an field of atom type
"I" incident upon a lattice of atoms of type "L", where the final state
consists of a new bound state between a "L" atom and an "I" atom.
All that matters for the purposes of this thread is that a different final
state be possible, resulting from an interaction. Though the numbers
may differ, we still get the result that there's a certain finite transition
probability for an atom deposition at any particular point in a finite time.
The only difference from the photodetection case is that we must
think of the incident atom beam as a field, not a collection of
particles.
In any case, more accurate results are given by also quantizing the incident
field (c.f. M&W ch14) -- we still get atoms being deposited at random
places according to a transition probability per unit time.
Minimal QM (without the extra baggage of a "collapse" interpretation)
thus seems quite adequate.
