Transcript slides

The Hidden Worlds of Quantum Mechanics
Craig Callender
Philosophy, UCSD
[email protected]
Double Slit Experiment
What you would expect is…
But what actually happens is…
• 1961, Jönsson, Zeitschrift für Physik 161 454
• 1974, P. Merli, G. Missiroli and G. Pozzi in Bologna in 1974
• Hitachi (A Tonomura et al). 1989 Demonstration of singleelectron buildup of an interference pattern Am. J. Phys. 57
The Double Slit Experiment
Actual images
Hitachi 1989
Bologna 1974
Structure of Physical Theories
• Stuff: Newtonian
corpuscles
• State: (xi, pi)
• Dynamical law:
Hamilton’s equations
• Stuff: ?
• State: wavefunction
or vector
|Ψ>
• Dynamical law:
Schrodinger’s
equation
Wave function (quantum state)
Ψ(x)
x
Ψ2(x) gives the
probability of finding the
particle at position x
Most probable
location of particle
Ψ2(x)
x
 ( x)   cnun ( x)
n
where the observable is assumed to have a discrete spectrum of
eigenvalues, u are the (normalized) eigenfunctions, and the
coefficient cn of the nth term gives the probability of the nth
eigenvalue via |cn|2.
Schrodinger evolution
Ψi(x)
Schrodinger
evolution
Deterministic
Unitary
Linear
Ψf(x)
Three Ingredients for Trouble
• Linear dynamical
evolution
• Eigenstateeigenvalue rule
• Determinate
outcomes
Linearity
If the evolution takes |A> →|B>..
And takes |C> → |D>…
Then it takes the state
|A> + |C> → |B> + |D>
Eigenstate-eigenvalue link
• A system in the quantum state |Ψ> has
the value a for the observable  if and
only if |Ψ> assigns the probability 1 to
a and the probability 0 to all other
possible values of Â.
Ψ(x)
• Â |Ψ> = a|Ψ>
x
1
2
Measurement
1 |ready>M|↑>S  |up>M|↑>S
2 |ready>M|↓>S  |down>M|↓>S
3
|ready>M (a1|↑>S+a2|↓>S)=
a1|ready>M|↑>S+a2|ready>M|↓>S
 (a1|up>M|↑>S + a2|down>M|↓>S)
Schrödinger’s cat
1 |cat ready>|ready>M|↑>S → |cat dead>|up>M|↑>S
2 |cat ready>|ready>M|↓>S → |cat alive>|down>M|↓>S
3 |cat ready>|ready>M (a1|↑>S +a2|↓>S)=
a1|cat ready>|ready>M|↑>S +a2|cat
ready>|ready>M|↓>S
 (a1|dead>|up>M|↑>S + a2|alive>|down>M|↓>S
Measurement Problem
1. The quantum state is representationally complete,
i.e., the eigenstate-eigenvalue link holds
2. The quantum state always evolves according to a
linear laws of evolution, e.g., Schrodinger equation.
3. Measurements yield definite values
Contradiction!
Ψi(x)
Schrodinger
evolution
Ψf(x)
Final state is a probability
distribution; but in the real
world something actually
happens!
Ψi(x)
non-Schrodinger evolution;
miracle; collapse
Ψf(x)
MP is Here to Stay
• Quantum field theory employs
superpositions among distinct
macroscopic states
• Theories on the horizon do too, e.g.,
superstring theory, loop theory, etc.
• So MP has to be solved
The “Standard” Solution
•
Copenhagen (Bohr, Heisenberg, Dirac, von
Neumann, …)
Two types of evolution:
I.
II.
Unmeasured evolution
Measurement evolution
“Quantum philosophy” of realism about
macroscopic entities but anti-realism about
microscopic ones
Unique?
• Rosenfeld: “quantum theory eminently
possess this character of uniqueness;
every feature of it has been forced upon
us as the only way to avoid the
ambiguities which would essentially
affect any attempt at an analysis in
classical terms of typical quantum
phenomena”
Criticism
•
“It would seem that the theory is
exclusively concerned about "results
of measurement", and has nothing
to say about anything else. What
exactly qualifies some physical
systems to play the role of
"measurer"? Was the wavefunction
of the world waiting to jump for
thousands of millions of years until a
single-celled living creature
appeared? Or did it have to wait a
little longer, for some better qualified
system ... with a Ph.D.? If the theory
is to apply to anything but highly
idealized laboratory operations, are
we not obliged to admit that more
or less "measurement-like"
processes are going on more or less
all the time, more or less
everywhere.”
Solution: Deny one of the premises
1. The quantum state is
representationally complete, i.e., the
eigenstate-eigenvalue link holds
2. The quantum state always evolves
according to a linear laws of evolution,
e.g., Schrodinger equation.
3. Measurements yield definite values
Deny 1
• Bohm-like “hidden variable” theories
– De Broglie 1927
– Bohm 1952
– Bohm and Vigier
– Nelson’s stochastic mechanics
– Bell 1987
– Goldstein, Durr and Zanghi 1991
Deny 2
• Physically-specifiable Collapse theories
– Pearle 1989
– Pearle and Squires 1994
– Pearle 1996
– Ghiradhi, Rimini and Weber 1986
– Bell 1987
– Penrose
Deny 3
• “Many-world” type theories
– Everett 1957
– deWitt
– “Many minds”
– Barbour
– Rovelli’s ‘relational’ qm
Bohmian Mechanics
• De Broglie 1927; David
Bohm 1952
• The de Broglie-Bohm
“idea seems …so natural
and simple, to resolve the
wave-particle dilemma in
such a clear and ordinary
way, that it is a great
mystery… that it was so
generally ignored.” Bell,
1987.
Bohmian Mechanics
• Basic idea: Suppose
that there are some
particles and that
their velocities are
determined by Ψ…
In other words, Ψ is
not the whole story;
there are also
particles.
Bohmian mechanics

i  ( x, t )  Hˆ  ( x, t )
t
Schrodinger
equation
dQk
1 Im(  *  k )

Q
dt
mk
*
Velocity
equation
If
 (q, t0 ) |  (q, t0 ) |2
at time t0
then
 (q, t ) | (q, t ) |
at time t
2
GRW
•
The world is described by two equations, the Schrodinger equation and
the velocity equation. The latter is arguably the simplest first order
equation for the positions of particles compatible with the Galilean and
time reversal invariance of the former.
•
Given any probability distribution for the initial configuration, Bohmian
mechanics defines a probability distribution for the full trajectory.
Notice that the velocity equation is simply v= J/p, where J is the
quantum probability current and p is the quantum probability density.
It follows from the quantum continuity equation that if the distribution
of the configuration Q is given by psi at some time (say the initial time)
this will be true at all times.
•
This deterministic theory of particles completely accounts for all the
phenomena of nonrelativistic quantum mechanics, from interference
effects to spectral lines. Thus Bohmian mechanics provides us with
probabilities for complete configurational histories that are consistent
with the quantum mechanical probabilities for configurations, including
the positions of measuring devices.
Many Bohmian Theories
• Deterministic alternatives to the velocity equation
that are empirically adequate
• Indeterministic versions of velocity equation
• Spin as fundamental with position (Bohm, Schiller
and Tiomno 1955, Dewdney 1992, Holland and Vigier
1988, Bohm and Hiley 1993)
• Bell-Bub-Vink dynamics: discrete, indeterministic
• Bub: modal interpretation
Ghirardi, Rimini & Weber 1986
•
Basic Idea: The Schrodinger
equation is not quite right. The
wavefunction (r1,r2, ..., t) usually
evolves according to the
Schrodinger equation, but every
now and then, at random,  is
multiplied (‘hit’) by a gaussian
function (and then normalized).
How often the state is likely to be
‘hit’ by a gaussian is proportional to
how many particles there are in the
system.
•
The effect of this multiplication is to
collapse the state to a more
localized one. Thus, systems with
large N turn out to be
overwhelmingly likely to collapse,
and systems with small N turn out
to be unlikely to collapse.
GRW
Ψ(x)
x
Cat
Alive
Cat
Dead
1/2[(C(r1,r2, ...rN...rM, t)S(r1)) + [(C(r1,r2, ...rN...rM, t) S(r1))]
Then the Gaussian hits
Ψ(x)
j(x) = K exp (-[x - ri]2/2a2)
* = j(x, ri)(...,t)/ Ri(x).
x
Cat
Alive
a=10-5cm, jump time T=1016s
For N = 1023, collapse happens around 10-7s,
compared to observation time 10-2s
Many GRW’s
•
Ghirardi emphasizes the importance of specifying what he calls ``the physical
reality of what exists out there.''
•
Mass density interpretation: for the simple GRW theory described here can be
identified with the mass weighted sum, over all particles, of the one-particle
densities arising from integrating over the coordinates of all but one of the
particles.
•
‘Hit’ interpretation: Bell [p 205,] “that the space-time points (x,t) at which the
hits are centered (which are determined by the wave function trajectory) should
themselves serve as the ``local beables of the theory. These are the
mathematical counterparts in the theory to real events at definite places and
times in the real world (as distinct from the many purely mathematical
constructions that occur in the working out of physical theories, as distinct from
things which may be real but not localized, and as distinct from the
`observables' of other formulations of quantum mechanics, for which we have
no use here.) A piece of matter then is a galaxy of such events.'‘
•
SL v CSL
Bohm
GRW
Everett
Deterministic?
Yes*
No
Yes*
Time reversal
invariant?
Yes
No
Yes*
Monistic?
No
Yes*
Yes*
Particles?
Yes
No
No
Is spin ‘real’?
No*
Yes
Yes
Preferred
foliation?
Yes
Yes
Maybe not
•
“There is something wrong with all of these
post-Copenhagen interpretations…they
don’t offer new predictions”
• But:
1. Why should when the theory is developed
be important?
2. De Broglie 1927
3. GRW will differ from Copenhagen; Bohm
would if Copenhagen were clear; plus, one
never knows…
Underdetermination of Theory by Evidence
THEORY1
THEORY2
Observable
evidence
THEORY3
THEORY4
• Duhem:
“Shall we ever dare to assert that no
other hypothesis is imaginable? Light
may be a swarm of projectiles, or it
may be a vibratory motion whose waves
are propagated in a medium; is it
forbidden to be anything else at all?”
(1914)
Experimenta Crucis?
• Bohm v Copenhagen
– Times of arrival, etc.
– Conroversial
• Bohm v Everett
– “in principle” underdetermination
– Laudan and Leplin 1991
• GRW v Bohm (or Copenhagen…)
– Localizations  greater KE  heating
– Resistance of a superconductor different (Gallis & Flemming;
Rae & Rimini)
– One mole of H: one atom excited/sec
– “hope of reaching a crucial test…extremely dim”
Theoria crucis?
• Solving the measurement problem seems to
involve adding some new physics…
• The new physics may or may not be
experimentally detectable in the future
• But it might be crucial to new theories, e.g.,
Bohmian quantum gravity, GRW’s energy
contribution, and so on.
Philosophers: are any ‘real theories’ underdetermined?
• Our evidence is equally
compatible with T (our
best physical theory) and
T* (we and our
apparently T-governed
world is a computer
simulation)
• But T* is just skepticism,
not a ‘real’ physical
theory
Philosophers: underdetermination too local
to be interesting?
The evidence equally
supports T (newtonian
mechanics plus
gravitational theory plus
universe is at rest in
absolute space) and T*
(same, but universe
moving 5mph wrt absolute
space)
5mph
How should we react in QM case?
Lesson of Quantum Mechanics: Bad News
• GRW, Bohm, Everett, etc. show that there is
underdetermination by genuine scientific
theories, contrary to what some philosophers
suggest.
• Furthermore, it’s hardly too local to care
about…it involves the central terms of our
most fundamental theory
• Real life is stranger than fiction or philosophy
Good news
• Further testing may narrow down the available
possibilities
• Further theorizing may narrow down the possibilities
• The UT doesn’t seem to be of the kind that would
challenge realist interpretations of most science.
– First, one might appeal to non-observable facts, e.g.,
simplicity
– Second, the underdetermination is not entirely general
– Third, there are still levels that are not under-determined
wrt the evidence, e.g., energy nuclear levels (Cordero 2002),
scattering, etc.
• Nevertheless, we’re stuck with a bewildering amount
of underdetermination, like it or not—so it’s best to
learn to like it.