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Nine formulations of quantum mechanics

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Nine formulations of nonrelativistic quantum mechanics are reviewed. These are the wavefunction, matrix, path integral, phase space, density matrix, second quantization, variational, pilot wave, and Hamilton–Jacobi formulations. Also mentioned are the many-worlds and transactional interpretations. The various formulations differ dramatically in mathematical and conceptual overview, yet each one makes identical predictions for all experimental results. © 2002 American Association of Physics Teachers.
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Nine formulations of quantum mechanics
Daniel F. Styer,a) Miranda S. Balkin, Kathryn M. Becker, Matthew R. Burns,
Christopher E. Dudley, Scott T. Forth, Jeremy S. Gaumer, Mark A. Kramer,
David C. Oertel, Leonard H. Park, Marie T. Rinkoski, Clait T. Smith,
and Timothy D. Wotherspoon
Department of Physics, Oberlin College, Oberlin, Ohio 44074
Received 18 July 2001; accepted 29 November 2001
Nine formulations of nonrelativistic quantum mechanics are reviewed. These are the wavefunction,
matrix, path integral, phase space, density matrix, second quantization, variational, pilot wave, and
Hamilton–Jacobi formulations. Also mentioned are the many-worlds and transactional
interpretations. The various formulations differ dramatically in mathematical and conceptual
overview, yet each one makes identical predictions for all experimental results. © 2002 American
Association of Physics Teachers.
DOI: 10.1119/1.1445404
I. WHY CARE ABOUT VARIOUS FORMULATIONS?
A junior-level classical mechanics course devotes a lot of
time to various formulations of classical mechanics—
Newtonian, Lagrangian, Hamiltonian, least action, and so
forth see Appendix A. But not a junior-level quantum me-
chanics course! Indeed, even graduate-level courses empha-
size the wavefunction formulation almost to the exclusion of
all variants. It is easy to see why this should be so—learning
even a single formulation of quantum mechanics is difficult
enough—yet at the same time students must wonder why it
is so important to learn several formulations of classical me-
chanics but not of quantum mechanics. This article surveys
nine different formulations of quantum mechanics. It is a
project of the Spring 2001 offering of Oberlin College’s
Physics 412, ‘‘Applied Quantum Mechanics.’
Why should one care about different formulations of me-
chanics when, in the end, each provides identical predictions
for experimental results? There are at least three reasons.
First, some problems are difficult in one formulation and
easy in another. For example, the Lagrangian formulation of
classical mechanics allows generalized coordinates, so it is
often easier to use than the Newtonian formulation. Second,
different formulations provide different insights.1For ex-
ample, the Newtonian and least action principles provide
very different pictorializations of ‘‘what’s really going on’’ in
classical mechanics. Third, the various formulations are vari-
ously difficult to extend to new situations. For example, the
Lagrangian formulation extends readily from conservative
classical mechanics to conservative relativistic mechanics,
whereas the Newtonian formulation extends readily from
conservative classical mechanics to dissipative classical me-
chanics. In the words of the prolific chemist E. Bright
Wilson:2
‘‘I used to go to J. H. Van Vleckfor quantum me-
chanical advice and found him always patient and
ready to help, sometimes in a perplexing flow of mixed
wave mechanical, operator calculus, and matrix lan-
guage which often baffled this narrowly Schro
¨dinger-
equation-oriented neophyte. I had to learn to look at
things in these alternate languages and, of course, it
was indispensable that I do so.’’
Any attempt to enumerate formulations must distinguish
between ‘‘formulations’ and ‘‘interpretations’ of quantum
mechanics. Our intent here is to examine only distinct math-
ematical formulations, but the mathematics of course influ-
ences the conceptual interpretation, so this distinction is by
no means clear cut,3and we realize that others will draw
boundaries differently. Additional confusion arises because
the term ‘‘Copenhagen interpretation’ is widely used but
poorly defined: For example, of the two primary architects of
the Copenhagen interpretation, Werner Heisenberg main-
tained that4‘‘observation of the position will alter the mo-
mentum by an unknown and undeterminable amount,’’
whereas Niels Bohr5‘‘warned specifically against phrases,
often found in the physical literature, such as ‘disturbing of
phenomena by observation.’’’
II. CATALOG OF FORMULATIONS
A. The matrix formulation Heisenberg
The matrix formulation of quantum mechanics, developed
by Werner Heisenberg in June of 1925, was the first formu-
lation to be uncovered. The wavefunction formulation, which
enjoys wider currency today, was developed by Erwin Schro
¨-
dinger about six months later.
In the matrix formulation each mechanical observable
such as the position, momentum, or energyis represented
mathematically by a matrix also known as ‘‘an operator’.
For a system with Nbasis states where in most cases N
⬁兲 this will be an NNsquare Hermitian matrix. A quan-
tal state
is represented mathematically by an N1 col-
umn matrix.
Connection with experiment. Suppose the measurable
quantity Ais represented by the operator A
ˆ. Then for any
function f(x) the expectation value for the measurement of
f(A) in state
is the inner product
fA
ˆ
.1
Because the above statement refers to f(A) rather than to
Aalone, it can be used to find uncertainties related to
f(A)A2as well as expectation values. Indeed, it can even
produce the eigenvalue spectrum, as follows:6Consider a set
of real values a1,a2,a3,..., and form the non-negative func-
tion
gxxa12xa22xa32¯.2
288 288Am. J. Phys. 70 3, March 2002 http://ojps.aip.org/ajp/ © 2002 American Association of Physics Teachers
Then the set a1,a2,a3,..., constitutes the eigenvalues of A
if and only if
gA
ˆ
0 for all states
.3
The matrix formulation places great emphasis on opera-
tors, whence eigenproblems fall quite naturally into its pur-
view. This formulation finds it less natural to calculate time-
dependent quantities or to consider the requirements for
identical particles. Such problems fall more naturally into the
second quantization formulation discussed below.
Time development. The operator corresponding to the me-
chanical observable energy is called the Hamiltonian and
represented by H
ˆ. Any operator A
ˆ(t) changes in time ac-
cording to
dA
ˆt
dt ⫽⫺ i
A
ˆt,H
ˆ
A
t.4
The states do not change with time.
Applications. For many perhaps mostapplications, the
wavefunction formulation is more straightforward than the
matrix formulation. An exception is the simple harmonic os-
cillator, where most problems are more cleanly and easily
solved through the operator factorization technique with
raising and lowering operatorsthan through arcane manipu-
lations involving Hermite polynomials. Similar matrix tech-
niques are invaluable in the discussion of angular momen-
tum. More general factorization methods described in the
book by Green, belowcan solve more general problems, but
often at such a price in complexity that the wavefunction
formulation retains the advantage of economy.
Recommended references. Most contemporary treatments
of quantum mechanics present an amalgam of the wavefunc-
tion and matrix formulations, with an emphasis on the wave-
function side. For treatments that emphasize the matrix for-
mulation, we recommend
1. H. S. Green, Matrix Mechanics P. Noordhoff, Ltd., Groningen, The
Netherlands, 1965.
2. T. F. Jordan, Quantum Mechanics in Simple Matrix Form Wiley, New
York, 1986.
History. Matrix mechanics was the first formulation of
quantum mechanics to be discovered. The founding papers
are
3. W. Heisenberg, ‘‘U
¨ber die quantentheoretische Umdeutung kinematis-
cher und mechanischer Beziehungen,’’ ‘‘Quantum-theoretical re-
interpretation of kinematic and mechanical relations’’, Z. Phys. 33,
879–893 1925.
4. M. Born and P. Jordan, ‘‘Zur Quantenmechanik,’‘On quantum me-
chanics’’, Z. Phys. 34, 858–888 1925.
5. M. Born, W. Heisenberg, and P. Jordan, ‘‘Zur Quantenmechanik II,’’ Z.
Phys. 35, 557–615 1926.
These three papers and othersare translated into English in
6. B. L. van der Waerden, Sources of Quantum Mechanics North-Holland,
Amsterdam, 1967.
The uncertainty principle came two years after the formal
development of the theory
7. W. Heisenberg, ‘‘U
¨ber den anschaulichen Inhalt der quantentheoretis-
chen Kinematik und Mechanik,’’ ‘‘The physical content of quantum
kinematics and mechanics’’,Z.Phys.43, 172–198 1927兲关English
translation in J. A. Wheeler and W. H. Zurek, editors, Quantum Theory
and Measurement Princeton University Press, Princeton, NJ, 1983, pp.
62–84.
B. The wavefunction formulation Schro
¨dinger
Compared to the matrix formulation, the wavefunction
formulation of quantum mechanics shifts the focus from
‘‘measurable quantity’to ‘‘state.’ The state of a system with
two particles ignoring spinis represented mathematically
by a complex function in six-dimensional configuration
space, namely
x1,x2,t.5
Alternatively, and with equal legitimacy, one may use the
mathematical representation in six-dimensional momentum
space:
˜
p1,p2,t1
2
6
d3x1
d3x2ei(p1x1p2x2)/
x1,x2,t.6
Schro
¨dinger invented this formulation in hopes of casting
quantum mechanics into a ‘‘congenial’ and ‘‘intuitive’
form7—he was ultimately distressed when he found that his
wavefunctions were functions in configuration space and did
not actually exist out in ordinary three-dimensional space.8
The wavefunction should be regarded as a mathematical tool
for calculating the outcomes of observations, not as a physi-
cally present entity existing in space such a football, or a
nitrogen molecule, or even an electric field. See also Appen-
dix B.
Time development. The configuration-space wavefunction
changes in time according to
x1,x2,t
t⫽⫺ i
2
2m11
2
x1,x2,t
2
2m22
2
x1,x2,t
Vx1,x2
x1,x2,t
,7
where the particle masses are m1and m2, and where
V(x1,x2) is the classical potential energy function. Equiva-
lently, the momentum-space wavefunction changes in time
according to
˜
p1,p2,t
t⫽⫺ i
p1
2
2m1
˜
p1,p2,tp2
2
2m2
˜
p1,p2,t
d3p1
d3p2
V
˜
p1
,p2
˜
p1p1
,p2p2
,t
,8
where the Fourier transform of the potential energy function
is
V
˜
p1,p2
1
2
6
d3x1
d3x2ei(p1x1p2x2)/Vx1,x2.
9
289 289Am. J. Phys., Vol. 70, No. 3, March 2002 Styer et al.
After the measurement of a quantity, the wavefunction ‘‘col-
lapses’’ to an appropriate eigenfunction of the operator cor-
responding to that quantity.
Energy eigenstates. Most states do not have a definite en-
ergy. Those that do9satisfy the eigenequation
2
2m11
22
2m22
2Vx1,x2,t
nx1,x2
En
nx1,x2.10
The energy spectrum may be either discrete ‘quantized’’or
continuous, depending upon the potential energy function
V(x1,x2,t) and the energy eigenvalue En.
Identical particles. If the two particles are identical, then
the wavefunction is symmetric or antisymmetric under label
interchange,
x1,x2,t⫽⫾
x2,x1,t,11
depending upon whether the particles are bosons or fermi-
ons. Aprecisely parallel statement holds for the momentum
space wavefunction.
Recommended references. Most treatments of quantum
mechanics emphasize the wavefunction formulation. Among
the many excellent textbooks are
8. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic
Theory, translated by J. B. Sykes and J. S. Bell, 3rd ed. Pergamon, New
York, 1977.
9. A. Messiah, Quantum Mechanics North-Holland, New York, 1961.
10. D. J. Griffiths, Introduction to Quantum Mechanics Prentice–Hall,
Englewood Cliffs, New Jersey, 1995.
11. R. W. Robinett, Quantum Mechanics: Classical Results, Modern Sys-
tems, and Visualized Examples Oxford University Press, New York,
1997.
History. Schro
¨dinger first wrote down the configuration-
space form of the energy eigenequation 10in
12. E. Schro
¨dinger, ‘‘Quantisierung als Eigenwertproblem Erste Mittei-
lung,’’ ‘Quantization as a problem of paper values part I’’, Annalen
der Physik 79, 361–376 1926.
He wrote down the time-development equation 7兲共which
he called ‘‘the true wave equation’five months later in
13. E. Schro
¨dinger, ‘‘Quantisierung als Eigenwertproblem Vierte Mittei-
lung,’’ ‘Quantization as a problem of proper values part IV’’, An-
nalen der Physik 81, 109–139 1926.
English translations appear in
14. E. Schro
¨dinger, Collected Papers on Wave Mechanics Chelsea, New
York, 1978.
C. The path integral formulation Feynman
The path integral formulation also called the sum-over-
histories formulationshifts the focus yet again—from
‘‘state’ to ‘‘transition probability.’
Suppose, for example, that a single particle is located at
point xiwhen the time is ti, and we wish to find the prob-
ability that it will be located at xfwhen the time is tf. This
probability is calculated as follows:
• Enumerate all classical paths from the initial to the final
state.
Calculate the classical action S
(Lagrangian) dt for
each path.
Assign each path a ‘‘transition amplitude’ proportional
to eiS/.The proportionality constant is adjusted to assure
normalization.
Sum the amplitude over all paths. Because there is a
continuum of paths, this ‘‘sum’ is actually a particular kind
of integral called a ‘‘path integral.’
The resulting sum is the transition amplitude, and its
square magnitude is the transition probability.
For different problems—such as a particle changing from
one momentum to another, or for an initial state that has
neither a definite position nor a definite momentum—
variations on this procedure apply.
Applications. The path integral formulation is rarely the
easiest way to approach a straightforward problem in nonrel-
ativistic quantum mechanics. On the other hand, it has innu-
merable applications in other facets of physics and chemis-
try, particularly in quantum and classical field theory and in
statistical mechanics. For example, it is a powerful tool in
the Monte Carlo simulation of quantal systems:
15. M. H. Kalos and P. A. Whitlock, Monte Carlo Methods Wiley, New
York, 1986, Chap. 8.
In addition, many find this formulation appealing because the
mathematical formalism is closer to experiment: the center
stage is occupied by transition probabilities rather than by
the unobservable wavefunction. For this reason it can be ef-
fective in teaching:
16. E. F. Taylor, S. Vokos, J. M. O’Meara, and N. S. Thornber, ‘‘Teaching
Feynman’s sum over paths quantum theory,’’ Comput. Phys. 12, 190
199 1998.
Identical particles. The path integral procedure general-
izes in a straightforward way to collections of several non-
identical particles or of several identical bosons. The term
‘‘path’ now means the trajectories of the several particles
considered collectively.Thus it must not generalize in the
same straightforward way to identical fermions, because if it
did then bosons and fermions would behave in the same
way!
The proper procedure for identical fermions involves a
single additional step. When enumerating classical paths
from the initial situation at time tito the final situation at
time tfas in Fig. 1, notice that some of the paths inter-
change the particles relative to other paths. In Fig. 1, the
particles are interchanged in paths III and IV but not in paths
I and II.The assignment of amplitude to a fermion path
proceeds exactly as described above except that, in addition,
any amplitude associated with an interchanging path is mul-
tiplied by 1 before summing. This rule is the Pauli prin-
ciple: In your mind, slide the two particles at the final time tf
towards each other.As the separation vanishes, the amplitude
associated with path I approaches the amplitude associated
with path III. Similarly, each other direct path approaches an
interchange path. Because of the factor of 1, the ampli-
tudes exactly cancel upon summation. Thus two fermions
cannot move to be on top of each other.
This sign adjustment is not difficult for humans, but it
poses a significant challenge—know as ‘‘the fermion sign
problem’’—for computers. This important standing problem
in quantum Monte Carlo simulation is discussed in, for ex-
ample,
17. N. Makri, ‘‘Feynman path integration in quantum dynamics,’’ Comput.
Phys. Commun. 63, 389–414 1991.
18. S. Chandrasekharan and U.-J. Wiese, ‘‘Meron-cluster solution of fer-
mion sign problems,’’ Phys. Rev. Lett. 83, 3116–3119 1999.
Recommended references.
19. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Inte-
grals McGraw-Hill, New York, 1965.
290 290Am. J. Phys., Vol. 70, No. 3, March 2002 Styer et al.
20. D. F. Styer, ‘‘Additions and corrections to Feynman and Hibbs,’’ http://
www.oberlin.edu/physics/dstyer/TeachQM/Supplements.html.
21. L. S. Schulman, Techniques and Applications of Path Integration
Wiley, New York, 1981.
History. This formulation was developed by
22. R. P. Feynman, ‘‘Space–time approach to non-relativistic quantum me-
chanics,’’ Rev. Mod. Phys. 20, 367–387 1948.
D. Phase space formulation Wigner
For a single particle restricted to one dimension, the
Wigner phase-space distribution function is
Wx,p,t1
2
*x1
2y,t
x1
2y,teipy/dy.12
This function has a number of useful properties:
It is pure real, but may be positive or negative.
The integral over momentum gives the probability den-
sity in position:
Wx,p,tdp
x,t
2.13
• The integral over position gives the probability density in
momentum:
Wx,p,tdx
˜
p,t
2.14
• If the wavefunction
is altered by a constant phase fac-
tor, the Wigner function is unaltered.
• Given W(x,p,t), one can find the wavefunction through
a two-step process. First, find the Fourier transform
W
˜
x,y,t1
2
Wx,p,teipy/dp
1
2
*x1
2y,t
x1
2y,t.15
Second, select an arbitrary point x0where W
˜
(x0,0,t) does
not vanish, and find
x,t
2
W
˜
x0,0,tW
˜
1
2xx0,xx0,t.16
The Wigner function is not a probability density in phase
space—according to Heisenberg’s uncertainty principle, no
such entity can exist. Yet it has several of the same proper-
ties, whence the term ‘‘distribution function’ is appropriate.
Time development.
Wx,p,t
t⫽⫺ p
m
Wx,p,t
x
Kx,pWx,pp,tdp,17
where the kernel K(x,p)is
Kx,p1
2
2
Vx1
2yVx1
2ysinpy/dy.
18
Identical particles. If the wavefunction is either symmetric
or antisymmetric under interchange, then the Wigner func-
tion is symmetric:
Wx1,p1,x2,p2Wx2,p2,x1,p1.19
This does not, of course, mean that bosons and fermions
behave identically in this formulation: the wave-functions
produced through Eq. 16will exhibit the correct symmetry
under interchange. It does mean that the type of interchange
symmetry is more difficult to determine in the phase-space
formulation than it is in the wavefunction formulation.
Applications. For an N-state system where Nmay equal
⬁兲, the wavefunction is specified by Ncomplex numbers
with an overall phase ambiguity, that is, by 2N1 real num-
bers. For this same system the Wigner function requires N2
real numbers. Clearly the Wigner function is not the most
economical way to record information on the quantal state.
The Wigner function is useful when the desired information
is more easily obtained from the redundant Wigner form than
from the economical wavefunction form. For example, the
momentum density is obtained from the Wigner function
through a simple integral over position. The momentum den-
sity is obtained from the configuration-space wavefunction
through the square of a Fourier transform.
A number of problems, particularly in quantum optics, fall
into this category. See, for example, the following:
23. D. Leibfried, T. Pfau, and C. Monroe, ‘‘Shadows and mirrors: Recon-
structing quantum states of atom motion,’’ Phys. Today 51, 22–28
1998.
Fig. 1. If the two particles are identical fermions, then the amplitudes asso-
ciated with interchanging paths, such as III and IV, must be multiplied by
1 before summing.
291 291Am. J. Phys., Vol. 70, No. 3, March 2002 Styer et al.
24. Y. S. Kim and W. W. Zachary, editors, The Physics of Phase Space
Springer-Verlag, Berlin, 1987.
Recommended references.
25. Y. S. Kim and E. P. Wigner, ‘‘Canonical transformation in quantum
mechanics,’’Am. J. Phys. 58, 439–448 1990.
26. M. Hillary, R. F. O’Connell, M. O. Scully, and E. P. Wigner, ‘‘Distribu-
tion functions in physics: Fundamentals,’’ Phys. Rep. 106, 121–167
1984.
History. The phase space formulation was invented by
27. E. P. Wigner, ‘‘On the quantum correction for thermodynamic equilib-
rium,’’ Phys. Rev. 40, 749–759 1932.
E. Density matrix formulation
The density matrix corresponding to a pure state
is the
outer product
ˆ
典具
.20
Given the density matrix
ˆ, the quantal state
can be
found as follows: First select an arbitrary state
. The un-
normalizedket
is
ˆ
as long as this quantity does not
vanish.
The density matrix is more properly but less frequently
called ‘‘the density operator.’ As with any quantum me-
chanical operator, the operator is independent of basis
whereas the matrix elements
ij
i
ˆ
j
do depend on the
basis selected.
The density matrix formulation is particularly powerful in
dealing with statistical knowledge. For example, if the exact
state of a system is unknown, but it is known to be in one of
three states—state
with probability p
, state
with
probability p
, or state
with probability p
—then the
system is said to be in a ‘‘mixed state’in contrast to a ‘‘pure
state’’. A mixed state cannot be represented by something
like
c
c
c
,
because this represents another pure state that is a superpo-
sition of the three original states. Instead, the mixed state is
represented by the density matrix
p
典具
p
典具
p
典具
.21
All of the results that follow in this section apply to both
pure and mixed states.
Connection with experiment. The density matrix is always
Hermitian. If the measurable quantity Ais represented by the
operator A
ˆ, then the expectation value for the measurement
of f(A) is the trace
tr
fA
ˆ
ˆ
.22
Time development. The density matrix evolves in time ac-
cording to
d
ˆt
dt ⫽⫹ i
ˆt,H
ˆ,23
where H
ˆis the Hamiltonian operator. Note that this formula
differs in sign from the time-development formula in the
matrix formulation.
Identical particles. The density matrix, like the Wigner
phase-space distribution function, remains unchanged under
interchange of the coordinates of identical particles, whether
bosons or fermions. As with the Wigner distribution, this
does not mean that symmetric and antisymmetric wavefunc-
tions behave identically; it simply means that the different
behaviors are buried in the density matrix rather than readily
visible.
Applications. For an N-state system where Nmay equal
⬁兲, a pure-state wavefunction is specified by Ncomplex
numbers with an overall phase ambiguity, that is, by
2N1 real numbers. For this same system the density ma-
trix requires Nreal diagonal elements plus N(N1)/2 com-
plex above-diagonal elements for a total of N2real numbers.
Thus the density matrix is not the most economical way to
record information about a pure quantal state. Nevertheless,
the ready availability of this information through the trace
operation, plus the ability to treat mixed states, make the
density matrix formulation valuable in several areas of phys-
ics. In particular, the formula
tr
A
ˆeH
ˆ/kT
tr
eH
ˆ/kT
24
is something of a mantra in quantum statistical mechanics.
Recommended references.
28. U. Fano, ‘‘Description of states in quantum mechanics by density ma-
trix and operator techniques,’’ Rev. Mod. Phys. 29,74931957.
29. K. Blum, Density Matrix Theory and Applications, 2nd ed. Plenum,
New York, 1996.
History. The density matrix was introduced by the follow-
ing:
30. J. von Neumann, ‘‘Wahrscheinlichkeitstheoretischer Aufbau der Quan-
tenmechanik,’’ ‘Probability theoretical arrangement of quantum me-
chanics’’, Nachr. Ges. Wiss. Goettingen, 245–272 1927, reprinted in
Collected Works Pergamon, London, 1961, Vol. 1, pp. 208–235.
F. Second quantization formulation
This formulation features operators that create and destroy
particles. It was developed in connection with quantum field
theory, where such actions are physical effects for example,
an electron and a positron are destroyed and a photon is
created. However, the formulation has a much wider domain
of application and is particularly valuable in many-particle
theory, where systems containing a large but constantnum-
ber of identical particles must be treated in a straightforward
and reliable manner.
The unfortunate name of this formulation is due to a his-
torical accident—from the point of view of nonrelativistic
quantum mechanics, a better name would have been the ‘‘oc-
cupation number formulation.’’
The second-quantized creation operator a
‘‘creates’’ a
particle in quantum state
. A one-particle state is formed
by having a
act upon a state with no particles, the so-called
‘‘vacuum state’’ 0. Thus the following are different expres-
sions for the same one-particle state:
,
x,
˜
p,a
0
.25
Thus as far as one-particle systems are concerned, the second
quantization formulation is equivalent to the wavefunction
formulation, although somewhat more cumbersome.
What about many-particle systems? Suppose
,
, and
are orthonormal one-particle states. Then a state with two
identical particles is produced by creating two particles from
the vacuum: for example a
a
0
. If the particles are
bosons, then
a
a
0
a
a
0
,26
292 292Am. J. Phys., Vol. 70, No. 3, March 2002 Styer et al.
whereas for fermions
a
a
0
⫽⫺a
a
0
.27
This illustrates the general rule that bosonic creation opera-
tors commute:
a
,a
0, 28
whereas fermionic creation operators anticommute:
a
,a
0. 29
The anticommutation notation means
A
ˆ,B
ˆ
A
ˆB
ˆB
ˆA
ˆ.
The advantages of second-quantized notation for many-
particle systems are becoming apparent. Most physicists
would agree that of the two equivalent forms
a
a
0
and 1
&
x1
x2
x2
x1,30
it is easier to work with the one on the left. And nearly all
physicists find it easier to work with
a
a
a
0
31
than with the equivalent
1/
3!
x1
x2
x3
x1
x3
x2
x3
x1
x2
x3
x2
x1
x2
x3
x1
x2
x1
x3.32
Yet the greatest advantage of second quantization is not mere
compactness of notation. The wavefunction formulation al-
lows you—indeed, it almost invites you—to write down ex-
pressions such as
x1
x2,
expressions that are neither symmetric nor antisymmetric un-
der interchange, and hence expressions that do not corre-
spond to any quantal state for identical particles. Yet the
wavefunction formulation provides no overt warning that
this expression is an invitation to ruin. By contrast, in the
second quantized formulation it is impossible to even write
down a formula such as the one above—the symmetrization
or antisymmetrizationhappens automatically through the
commutation or anticommutationof creation operators, so
only legitimate states can be expressed in this formulation.
For this reason, the second quantization formulation is used
extensively in many-particle theory.
Recommended references.
31. H. J. Lipkin, Quantum Mechanics: New Approaches to Selected Topics
North-Holland, Amsterdam, 1986, Chap. 5.
32. V. Ambegaokar, ‘‘Second quantization,’’ in Superconductivity, edited by
R. D. Parks Marcel Dekker, New York, 1969, pp. 1359–1366.
33. W. E. Lawrence, ‘‘Algebraic identities relating first- and second-
quantized operators,’’Am. J. Phys. 68, 167–170 2000.
An extensive discussion of applications is
34. G. D. Mahan, Many-Particle Physics, 3rd ed. Kluwer Academic, New
York, 2000.
History. Second quantization was developed by Dirac for
photons, then extended by Jordan and Klein to massive
bosons, and by Jordan and Wigner to fermions:
35. P. A. M. Dirac, ‘‘The quantum theory of the emission and absorption of
radiation,’’ Proc. R. Soc. London, Ser. A 114, 243–265 1927.
36. P. Jordan and O. Klein, ‘‘Zum Mehrko
¨rperproblem der Quantentheo-
rie,’’ ‘On the many-body problem in quantum theory’’,Z.Phys.45,
751–765 1927.
37. P. Jordan and E. Wigner, ‘‘U
¨ber das Paulische A
¨quivalenzverbot,’’ ‘‘On
the Pauli valence line prohibition’’, Z. Phys. 47, 631–651 1928.
The Dirac and Jordan–Wigner papers are reprinted in
38. J. Schwinger, editor, Selected Papers on Quantum Electrodynamics
Dover, New York, 1958.
G. Variational formulation
The ‘‘variational formulation’ must not be confused with
the more-commonly-encountered ‘‘variational method,’
which provides a bound on the ground state energy. Instead
the variational formulation provides a full picture describing
any state—not just the ground state—and dictating its full
time evolution—not just its energy. It is akin to Hamilton’s
principle in classical mechanics.
The central entity in this formulation remains the wave-
function
(x1,x2,t), but the rule for time evolution is no
longer the Schro
¨dinger equation. We again consider a non-
relativistic two-particle system ignoring spin.Of all possible
normalized wavefunctions
(x1,x2,t), the correct wave-
function is the one that minimizes the ‘‘action integral’ over
time and configuration space, namely
dt
d3x1
d3x2Lx1,x2,t,33
where the ‘‘Lagrangian density’ is
Lx1,x2,tIm
*
t
2
2m1
1
*•“
1
2
2m22
*2
Vx1,x2
*
,34
and Im
z
means the imaginary part of z. It is not difficult to
show that this minimization criterion is equivalent to the
Schro
¨dinger time-development equation 7.
Applications. On the practical side, this formulation is di-
rectly connected to the invaluable variational method for es-
timating ground state energies. Apply the principle to the
class of time-independent trial wavefunctions, and the varia-
tional method tumbles right out.
On the fundamental side, we note that field variational
techniques often provide formulations of physical law that
are manifestly Lorentz invariant. This role is exploited for
electricity and magnetism in
39. J. Schwinger, L. L. DeRaad, Jr., K. A. Milton, and W. Tsai, Classical
Electrodynamics Perseus Books, Reading, MA, 1998, especially
Chaps. 8 and 9,
for general relativity ‘‘Hilbert’s formulation’in
40. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation Freeman,
San Francisco, 1973, Chap. 21,
and for quantum field theory in
41. C. Itzykson and J.-B. Zuber, Quantum Field Theory McGraw-Hill,
New York, 1980.
For this reason, such variational formulations are now the
preferred instrument of attack for extending physics to new
domains, for example to supersymmetric strings or mem-
branes:
42. E. Witten, ‘‘Reflections on the fate of spacetime,’’ Phys. Today 49,
24–30 April 1996.
293 293Am. J. Phys., Vol. 70, No. 3, March 2002 Styer et al.
43. E. Witten, ‘‘Duality, spacetime and quantum mechanics,’’ Phys. Today
50,2833May 1997.
However, these roles are not played directly by the formu-
lation discussed here which is iintrinsically nonrelativistic
and which iiinvolves integration over time and configura-
tion space rather than time and physical space.
Recommended reference.
44. P. M. Morse and H. Feshbach, Methods of Theoretical Physics
McGraw-Hill, New York, 1953, pp. 314–316 and 341–344.
Caution! This reference defines a Lagrangian density with
the opposite sign of Eq. 34, so the Morse and Feshbach
action integral is maximized, not minimized, by the correct
wavefunction.
History. This formulation originated in the following:
45. P. Jordan and O. Klein, ‘‘Zum Mehrko
¨rperproblem der Quantentheo-
rie,’’ ‘On the many-body problem in quantum theory’’, Z. Phys. 45,
751–765 1927.
The same paper that introduced second quantization for
massive bosons!
H. The pilot wave formulation de BroglieBohm
We outline the pilot wave formulation through the ex-
ample of an electron and a proton ignoring spin. In classi-
cal mechanics this system is represented mathematically by
two points tracing out trajectories in three-dimensional
physical space. In the wavefunction formulation this system
is represented mathematically by a complex-valued wave-
function evolving in six-dimensional configuration space. In
the pilot wave formulation this system is represented math-
ematically by both the two points in physical space and the
wavefunction in configuration space. The wavefunction is
called the ‘‘pilot wave’ and it along with the classical po-
tential energy functionprovides information telling the two
points how to move.
The most frequently cited version of the pilot wave for-
mulation is that of Bohm but see also the version by Du
¨rr,
Goldstein, and Zanghı
´, cited below. In Bohm’s version, the
wavefunction is written in terms of the realmagnitude and
phase functions as
x1,x2,tRx1,x2,teiS(x1,x2,t)/.35
If one defines the state-dependent ‘‘quantum potential’
Qx1,x2,t⫽⫺ 2
2m1
1
2R
R2
2m2
2
2R
R,36
then the pilot wave evolves in time according to
S
t⫽⫺
1S2
2m1
2S2
2m2
Vx1,x2Qx1,x2,t37
and
P
t1
m1
1P
1S1
m22P2S0, 38
where
Px1,x2,tR2x1,x2,t.39
The first equation resembles a Hamilton–Jacobi equation;
the second acts such as a continuity equation in which P
represents a probability density.
The two point particles move with accelerations
m1dv1
dt ⫽⫺
1V
1Qand m2dv2
dt ⫽⫺2V2Q.
40
In other words, the force is given not only by the gradient of
the classical potential, but by the gradient of the quantum
potential as well. The initial positions of the point particles
are uncertain: for an ensemble of systems the probability
density of initial positions is given by P(x1,x2,0). Thus both
the particle corresponding to the proton and the particle cor-
responding to the electron have a definite position and a
definite momentum; however the initial ensemble uncer-
tainty and the quantum potential work together to ensure that
any set of measurements on a collection of identically pre-
pared systems will satisfy xp⭓ប/2.
The quantum potential Q(x1,x2,t) changes instanta-
neously throughout configuration space whenever the wave-
function changes, and this mechanism is responsible for the
nonlocal correlations that are so characteristic of quantum
mechanics. A rather natural mechanism prevents human be-
ings from tapping into this instantaneous change for the pur-
pose of faster-than-light communications.
Applications. To use the pilot wave formulation one must
keep track of both trajectories and wavefunctions, so it is not
surprising that this formulation is computationally difficult
for most problems. For example the phenomenon of two-slit
interference, often set as a sophomore-level modern physics
problem using the wavefunction formulation, requires a
computational tour de force in the pilot wave formulation:
46. C. Philippidis, C. Dewdney, and B. J. Hiley, ‘‘Quantum interference and
the quantum potential,’’ Nuovo Cimento Soc. Ital. Fis., B 52, 15–28
1979.
In contrast, the pilot wave formulation is effective in rais-
ing questions concerning the general character of quantum
mechanics. For example, John Bell’s epoch-making theorem
concerning locality and the quantum theory was inspired
through the pilot wave formulation.10 And many astute ob-
servers find the pilot wave formulation intuitively insightful.
For example:
47. J. S. Bell, ‘‘Six possible worlds of quantum mechanics,’’ in Possible
Worlds in Humanities,Arts and Sciences: Proceedings of Nobel Sympo-
sium 65, 11–15 August 1986, edited by S. Alle
´nWalter de Gruyter,
Berlin, 1989, pp. 359–373. Reprinted in J. S. Bell, Speakable and
Unspeakable in Quantum Mechanics Cambridge University Press,
Cambridge, UK, 1987, Chap. 20, pp. 181–195.
48. H. P. Stapp, ‘‘Review of ‘The Undivided Universe’ by Bohm and Hi-
ley,’ Am. J. Phys. 62, 958–960 1994.
Recommended references.
49. D. Bohm, B. J. Hiley, and P. N. Kaloyerou, ‘‘An ontological basis for
the quantum theory,’’ Phys. Rep. 144, 321–375 1987.
50. D. Bohm and B. J. Hiley, The Undivided Universe: An Ontological
Interpretation of Quantum Theory Routledge, London, 1993.
51. D. Du
¨rr, S. Goldstein, and N. Zanghı
´, ‘‘Quantum equilibrium and the
origin of absolute uncertainty,’’ J. Stat. Phys. 67, 843–907 1992.
History. Louis de Broglie proposed the germ of this ap-
proach which was discussed at, for example, the Solvay Con-
gress of 1927. But the substantial development of these ideas
began with
52. D. Bohm, ‘‘Asuggested interpretation of the quantum theory in terms of
‘hidden’ variables, I and II,’’ Phys. Rev. 35, 166–179 and 180–193
1952.
294 294Am. J. Phys., Vol. 70, No. 3, March 2002 Styer et al.
I. The HamiltonJacobi formulation
The classical Hamilton–Jacobi formulation systematically
finds changes of variable such that the resulting equations of
motion are readily integrated. In particular, if this results in a
new set of variables of the so-called ‘‘action-angle’ form,
one can find the period of a repetitive motion without actu-
ally finding the motion itself.
Classical Hamilton–Jacobi theory provided important in-
spiration in the development of quantum mechanics. Dirac’s
‘‘transformation theory’ places a similar emphasis on strate-
gic changes of variable, and the Wilson–Sommerfeld version
of old quantum theory relies on action-angle variables.But
it was not until 1983 that Robert Leacock and Michael Pag-
dett produced a treatment extensive enough to be regarded as
a full Hamilton–Jacobi formulation of quantum mechanics.
The central entity of this formulation is ‘‘Hamilton’s princi-
pal function’’ S(x1,x2,t) such that
x1,x2,texpiSx1,x2,t/.41
Caution: This function may be complex...it is not the same S
as in the pilot-wave defining Eq. 35.Hamilton’s principal
function satisfies the quantum Hamilton–Jacobi equation,
S
ti
2m11
2S1
2m1
1S•“
1Si
2m22
2S1
2m22S
2SVx1,x2.42
Caution: The name ‘‘quantum Hamilton–Jacobi equation’’
is applied both to this equation and to the pilot wave equa-
tion 37.
If the resulting change in variables is of action-angle form,
then this formulation can find the energy eigenvalues without
needing to find the eigenfunctions.
Recommended references.
53. R. A. Leacock and M. J. Padgett, ‘‘Hamilton–Jacobi/action-angle quan-
tum mechanics,’’ Phys. Rev. D 28, 2491–2502 1983.
54. R. S. Bhalla, A. K. Kapoor, and P. K. Panigrahi, ‘‘Quantum Hamilton–
Jacobi formalism and the bound state spectra,’’Am. J. Phys. 65, 1187
1194 1997.
55. J.-H. Kim and H.-W. Lee, ‘‘Canonical transformations and the
Hamilton–Jacobi theory in quantum mechanics,’’ Can. J. Phys. 77,
411–425 1999.
J. Summary and conclusions
We have discussed nine distinct formulations of quantum
mechanics. Did we learn anything in the process? The most
profound lesson is already familiar from classical mechanics,
and indeed from everyday life: ‘‘There is no magic bullet.’
Each of these formulations can make some application easier
or some facet of the theory more lucid, but no formulation
produces a ‘‘royal road to quantum mechanics.’ Quantum
mechanics appears strange to our classical eyes, so we em-
ploy mathematics as our sure guide when intuition fails us.
The various formulations of quantum mechanics can repack-
age that strangeness, but they cannot eliminate it.
The matrix formulation, the first formulation to be discov-
ered, is useful in solving harmonic oscillator and angular
momentum problems, but for other problems it is quite dif-
ficult. The ever-popular wavefunction formulation is standard
for problem solving, but leaves the conceptual misimpres-
sion that wavefunction is a physical entity rather than a
mathematical tool. The path integral formulation is physi-
cally appealing and generalizes readily beyond the domain of
nonrelativistic quantum mechanics, but is laborious in most
standard applications. The phase space formulation is useful
in considering the classical limit. The density matrix formu-
lation can treat mixed states with ease, so it is of special
value in statistical mechanics. The same is true of second
quantization, which is particularly important when large
numbers of identical particles are present. The variational
formulation is rarely the best tool for applications, but it is
valuable in extending quantum mechanics to unexplored do-
mains. The pilot wave formulation brings certain conceptual
issues to the fore. And the HamiltonJacobi formulation
holds promise for solving otherwise-intractable bound state
problems.
We are fortunate indeed to live in a universe where nature
provides such bounty.
III. ADDITIONAL ISSUES
This section treats two interpretations of quantum mechan-
ics that might instead be considered formulations, then goes
on to briefly discusses four miscellaneous items.
A. The many-worlds interpretation Everett
The many-worlds interpretation is close to the boundary
between a ‘‘formulation’ and an ‘‘interpretation’’—indeed
its founder, Hugh Everett, called it ‘‘the relative state formu-
lation,’’while it is most widely known under Bryce DeWitt’s
name of ‘‘the many-worlds interpretation.’
In this interpretation there is no such thing as a ‘‘collapse
of the wavefunction.’’At the same time, the question changes
from ‘‘What happens in the world?’ to ‘‘What happens in a
particular story line?.’’This change in viewpoint is best dem-
onstrated through an example: Suppose a scientist cannot
make up her mind whether to marry or to break off her
engagement. Rather than flip a coin, she sends a single
circularly-polarized photon into a sheet of polaroid. If a pho-
ton emerges linearly polarizedfrom the polaroid, a photo-
detector will register and an attached bell will chime. The
scientist decides beforehand that if the bell chimes, she will
marry. Otherwise, she will remain single. In the Bohr version
of quantum mechanics, the question is ‘‘What happens?’and
the answer is that the scientist has a 50% chance of marriage
and a 50% chance of breaking her engagement. In the Ever-
ett version, this is not the right question: There is one story
line in which the photon emerges, the bell rings, and the
marriage occurs. There is another story line in which the
photon is absorbed, silence reigns, and the engagement ter-
minates. Each story line is consistent. To find out which story
line we are living in, we simply check on the marital status
of the scientist. If she is married, we are living in the story
line where a linearly polarized photon emerged and the bell
rang. Otherwise, we live in the other story line. The question
‘‘What happens?’ is ill-posed—one must ask instead ‘‘What
happens in a particular story line?’’ Just as the question
‘‘How far is Chicago?’ is ill-posed—one must ask instead
‘‘How far is Chicago from San Francisco?’
In the relative state formulation, the wavefunction never
collapses—it merely continues branching and branching.
Each branch is consistent, and no branch is better than any of
the other branches. In the many-worlds version, one speaks
of coexisting branching universes rather than of multiple
story lines.In summary, the relative state formulation places
the emphasis on correlations and avoids collapse.
Applications. The relative state formulation is mathemati-
cally equivalent to the wavefunction formulation, so there
can be no technical reason for preferring one formulation
295 295Am. J. Phys., Vol. 70, No. 3, March 2002 Styer et al.
over the other. On the other hand, some find that the concep-
tual orientation of the relative state formulation produces in-
sights in what would otherwise be fallow ground. For ex-
ample, David Deutsch’s 1985 paper, which founded the
enormously fertile field of quantum computing, expressed
his opinion that ‘‘The intuitive explanation of these proper-
ties places an intolerable strain on all interpretations of quan-
tum theory other than Everett’s.’
56. D. Deutsch, ‘‘Quantum theory, the Church-Turing principle and the uni-
versal quantum computer,’’ Proc. R. Soc. London, Ser. A 400, 97–117
1985.
Recommended references.
57. H. Everett III, ‘‘‘Relative state’ formulation of quantum mechanics,’
Rev. Mod. Phys. 29, 454–462 1957.
58. B. S. DeWitt and N. Graham, in The Many-Worlds Interpretation of
Quantum Mechanics Princeton University Press, Princeton, NJ, 1973.
59. Y. Ben-Dov, ‘‘Everett’s theory and the ‘many-worlds’ interpretation,’’
Am. J. Phys. 58, 829–832 1990.
60. B. S. DeWitt, ‘‘Quantum mechanics and reality,’’ Phys. Today 23,
30–35 September 1970.
61. L. E. Ballentine, P. Pearle, E. H. Walker, M. Sachs, T. Koga, J. Gerver,
and B. DeWitt, ‘‘Quantum mechanics debate,’’ Phys. Today 24, 36–44
April 1971.
B. The transactional interpretation Cramer
This interpretation or formulationis coherent and valu-
able, but it is difficult to describe in brief compass, so many
who have inspected it only briefly consider it simply bizarre.
If our short description here leaves you with that misimpres-
sion, we urge you to consult the recommended references.
In the transactional interpretation sources and detectors of,
say, electrons emit both retarded waves moving forward in
timeand advanced waves moving backward in time.An
electron moving from a source to a detector involves an ‘‘of-
fer wave’’ corresponding to
from the source and a ‘‘con-
formation wave’’ corresponding to
*from the detector
which interfere to produce ‘‘a handshake across
space–time.’’11 Destructive interference between these two
waves assures that the electron cannot arrive at the detector
before it leaves its source.
Applications. According to John Cramer,12 ‘‘the transac-
tional interpretation...makes no predictions that differ from
those of conventional quantum mechanics that is, the wave-
function formulation... We have found it to be more useful
as a guide for deciding which quantum-mechanical calcula-
tions to perform than as an aid in the performance of such
calculations... The main utility of the transactional interpre-
tation is asa conceptual model which provides the user
with a way of clearly visualizing complicated quantum pro-
cesses and of quickly analyzing seemingly ‘paradoxical’ situ-
ations... It also seems to have considerable value in the de-
velopment of intuitions and insights into quantum
phenomena that up to now have remained mysterious.’’
Identical particles. Discussions of the transactional inter-
pretation are usually carried out in the context of one-particle
quantum mechanics. It is not clear to us whether, in a two-
particle system, there are two ‘‘handshakes across space-
time’’ or one ‘‘handshake across configuration-space-time.’
Consequently, we cannot report on how the transactional for-
mulation differentiates between bosons and fermions.
Recommended references.
62. J. G. Cramer, ‘‘The transactional interpretation of quantum mechanics,’’
Rev. Mod. Phys. 58, 647–687 1986.
63. J. G. Cramer, ‘‘An overview of the transactional interpretation of quan-
tum mechanics,’’ Int. J. Theor. Phys. 27, 227–236 1988.
History. This interpretation originated in
64. J. G. Cramer, ‘‘Generalized absorber theory and the Einstein–
Podolsky–Rosen paradox,’’ Phys. Rev. D 22, 362–376 1980.
C. Miscellaneous items
Most physicists interested in formulations will also be in-
terested in density functional theory, in decoherence, in the
consistent histories interpretation, and in the possibility of
continuous spontaneous localization, so these matters are
briefly touched upon here.
The density functional theory of Hohenberg and Kohn is a
powerful quantum-theoretic tool, but it is not a formula-
tion...it deals only with the ground state. Admittedly, this is
the only state of interest for much of chemistry and con-
densed matter physics.
65. R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Mol-
ecules Oxford University Press, New York, 1989.
From the birth of quantum mechanics, everyone recog-
nized the importance of a correct classical limit. Results such
as Ehrenfest’s famous theorem assure that some quantal
states behave nearly classically. But this does not completely
answer the need: It is also true that other quantal states be-
have far from classically. Why do we never encounter such
states in the day-to-day world? The phenomenon of decoher-
ence attempts to explain this absence. The vast technical lit-
erature is best approached through the two reviews
66. W. H. Zurek, ‘‘Decoherence and the transition from quantum to classi-
cal,’’ Phys. Today 44, 36–44 October 1991.
67. S. Haroche, ‘‘Entanglement, decoherence and the quantum/classical
boundary,’’ Phys. Today 51, 36–42 July 1998.
Robert Griffiths’s consistent histories interpretation is not
a formulation, but provides an interesting point of view. See
68. R. B. Griffiths and R. Omne
`s, ‘‘Consistent histories and quantum mea-
surements,’’ Phys. Today 52,2631August 1999.
The idea of continuous spontaneous localization deals
with wavefunction collapse and the classical limit by modi-
fying the Schro
¨dinger equation in such a way that extended
quantal states naturally collapse, as if under their own
weight. There are several such schemes, the most prominent
of which is
69. G. C. Ghirardi, A. Rimini, and T. Weber, ‘‘Unified dynamics for micro-
scopic and macroscopic systems,’’ Phys. Rev. D 34, 470–491 1986.
Finally, everyone should be aware of the two wide-ranging
reviews
70. S. Goldstein, ‘‘Quantum theory without observers,’’ Phys. Today 51,
42–46 March 1998and ibid. 38–42 April 1998.
71. F. Laloe
¨, ‘‘Do we really understand quantum mechanics? Strange cor-
relations, paradoxes, and theorems,’’Am. J. Phys. 69, 655–701 2001.
APPENDIX A: FORMULATIONS OF CLASSICAL
MECHANICS
The formulations of classical mechanics known to us are
the following:
Newtonian
Lagrangian
Hamiltonian
Hamilton’s principle called by Feynman and Landau ‘‘the
principle of least action’’
296 296Am. J. Phys., Vol. 70, No. 3, March 2002 Styer et al.
the Maupertuis principle of least action also associated
with the names of Euler, Lagrange, and Jacobi
least constraint Gauss
least curvature Hertz
Gibbs–Appell
Poisson brackets
Lagrange brackets
Liouville
Hamilton–Jacobi
These formulations are discussed to a greater or lesser
extent in any classical mechanics textbook. The definitive
scholarly work appears to be
72. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and
Rigid Bodies, 4th ed. Cambridge University Press, Cambridge, UK,
1937.
APPENDIX B: GAUGE TRANSFORMATIONS
The wavefunction plays such a central role in most discus-
sions of quantum mechanics that one is easily trapped into
thinking of it as a physical entity rather than a mathematical
tool. Anyone falling into this trap will be dissuaded by the
following argument. Consider a single particle of charge q
moving in an electromagnetic field described by scalar po-
tential
(x,t) and vector potential A(x,t). Then the
configuration-space Schro
¨dinger equation is
x,t
t⫽⫺ i
1
2m
iq
cAx,t
2
q
x,t
x,t.B1
On the other hand, we can describe exactly the same system
using the gauge-transformed potentials:
Ax,tAx,t
x,t,B2
x,t
x,t1
c
⳵␹
x,t
t.B3
One can easily show that the wavefunction obtained using
these new potentials is related to the original wavefunction
by
x,teiq
(x,t)/c
x,t.B4
This gauge transformation has not changed the system at all,
and any experimental result calculated will be the same re-
gardless of which gauge is employed. Yet the wavefunction
has changed dramatically. Indeed, although the probability
density cannot and does not change through a gauge trans-
formation, the all-important phase—responsible for all inter-
ference effects—can be selected at will.
ACKNOWLEDGMENTS
Professor Martin Jones of the Oberlin College Philosophy
Department presented our class with a guest lecture that in-
fluenced the shape of Sec. III A. Two referees made valuable
suggestions and corrected some embarrassing sign errors in
the manuscript.
aElectronic mail: Dan.Styer@oberlin.edu
1To our classical sensibilities, the phenomena of quantum mechanics—
interference, entanglement, nonlocal correlations, and so forth—seem
weird. The various formulations package that weirdness in various ways,
but none of them can eliminate it because the weirdness comes from the
facts, not the formalism.
2E. B. Wilson, ‘‘Some personal scientific reminiscences,’’ International
Journal of Quantum Chemistry: Quantum Chemistry Symposium, Proceed-
ings of the International Symposium held at Flagler Beach, Florida, 10–20
March 1980, Vol. 14, pp. 17–29, 1980 see p. 21. Wilson co-authored one
of the very earliest quantum mechanics textbooks, namely L. Pauling and
E. B. Wilson, Introduction to Quantum Mechanics McGraw-Hill, New
York, 1935.
3C. A. Fuchs and A. Peres, ‘‘Quantum theory needs no ‘interpretation’,’
Phys. Today 53, 70–71 March 2000; D. Styer, ‘‘Quantum theory—
interpretation, formulation, inspiration letter,’’ ibid. 53,11September
2000; C. A. Fuchs and A. Peres, ‘‘Reply,’’ ibid. 53, 14,90 September
2000.
4W. Heisenberg, The Physical Principles of the Quantum Theory, translated
by Carl Eckart and F. C. Hoyt University of Chicago Press, Chicago,
1930,p.20.
5N. Bohr, ‘‘Discussion with Einstein on epistemological problems in atomic
physics,’’ in Albert Einstein, PhilosopherScientist, edited by P. A.
Schilpp Library of Living Philosophers, Evanston, IL, 1949,p.237.Re-
printed in N. Bohr, Atomic Physics and Human Knowledge Wiley, New
York, 1958, pp. 63–64.
6N. David Mermin, unpublished lectures given at Cornell University.
7Schro
¨dinger used both words in his first 1926 paper Ref. 12 in the text.
In the translation Ref. 14 in the text, ‘‘congenial’’ appears on p. 10 and
‘‘intuitive’’ on p. 9. The latter corresponds to the German anschaulich,
which has been variously translated as ‘‘intuitive,’’ ‘‘pictorial,’’ or ‘‘visu-
alizable.’’
8See, for example, A. Pais, Inward Bound Clarendon, Oxford, UK, 1986,
p. 256.
9We represent arbitrary wavefunctions by
(x)orby
(x), and energy
eigenfunctions by
(x), because the Greek letter
suggests ‘‘e’’ as in
‘‘energy eigenfunction’ and as in ‘‘eta’’. This admirable convention was
established by D. T. Gillespie, in A Quantum Mechanics Primer Interna-
tional Textbook Company, Scranton, PA, 1970.
10J. Bernstein, Quantum Profiles Princeton University Press, Princeton, NJ,
1991, pp. 72–77.
11Cramer, Ref. 63 in the text, p. 661.
12Cramer, Ref. 63 in the text, p. 663.
297 297Am. J. Phys., Vol. 70, No. 3, March 2002 Styer et al.
... There are many different formulations and interpretations of quantum mechanics, a concise overview is given in [2]. The pilot-wave description by de Broglie-Bohm and Nelson's stochastic mechanics are often subsumed under hidden variable theories, where the latter is less well-known and often presented as a stochastic variant of the former [3]. ...
... Let ∆ t be a fixed lag-time and Θ ∆t the flow map associated with the dynamical system. Given training data of the form {(x (i) , y (i) )} m i=1 , where y (i) = Θ ∆t (x (i) ), we define the data matrices X, Y ∈ R d×m by X = x (1) x (2) . . . x (m) and Y = y (1) y (2) . . . ...
... Given training data of the form {(x (i) , y (i) )} m i=1 , where y (i) = Θ ∆t (x (i) ), we define the data matrices X, Y ∈ R d×m by X = x (1) x (2) . . . x (m) and Y = y (1) y (2) . . . y (m) . ...
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Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson's stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schr\"odinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schr\"odinger equation. Our findings open up a new avenue towards solving Schr\"odinger's equation using recently developed tools from data science.
... 19 It is a settled fact that QM can be formulated in several ways (cf. Styer et al. 2002). Since the seminal work of von Neumann (1955) on the axiomatization of QM, developments and debates on QM employ, predominantly, the Hilbert space formulation. ...
... Our goal here is essentially to do the same, but with the term "interpretation". 20 For a critical summary of various formulations of QM, seeWightman (1976),Gudder (1979),Styer et al. (2002), and references therein.Content courtesy of Springer Nature, terms of use apply. Rights reserved. ...
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... This result is relevant for physics education researchers, e.g. for developing concept tests to investigate students' understandings of quantum physics but it is also of importance for physics teachers who want to evaluate their quantum physics lessons, because without further clarification it is not self-evident, what learning of quantum physics means. There are similar findings with regard to the diverse interpretations of quantum physics [23]. This uncertainty is also a chance: if there is no standard of quantum physics essentials, new curricula can put more emphasis on how well a new concept is learned rather than how well it represents a teaching tradition. ...
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... This distinguishability cannot occur with quantum entities, even those trapped by some device.7 The standard quantum formalism is developed within a mathematical structure called "Hilbert-space formalism," although there are alternatives ([34] mentions nine different ways of developing orthodox quantum mechanics). ...
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