There are few simple examples of the formal equivalence of wave mechanics and matrix mechanics. The momentum matrix for a particle in an infinite square well is easy to calculate and rarely discussed in textbooks. We square this matrix to construct the energy levels and use the energy theorem of Fourier analysis to establish the wave-matrix connection. The key ingredients of the equivalence proofs of Schrödinger and von Neumann, such as the d/dx rule and the Riesz-Fischer theorem, find simple expression within the particle-in-a-box framework.

1.
E.
Schrödinger
, “
Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen
,”
Ann. Phys.
384
,
734
756
(
1926
), translated as “On the Relation between the Quantum Mechanics of Heisenberg, Born, and Jordan, and that of Schrödinger,” in Collected Papers on Wave Mechanics by E. Schrödinger (Chelsea Publishing, New York, 1982).
2.
J.
von Neumann
,
Mathematical Foundations of Quantum Mechanics
(
Princeton U.P.
,
Princeton, NJ
,
1955
), pp.
28
33
. This book is the English translation of von Neumann's famous treatise, Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932), which was based on his trilogy of papers published in 1927 (see Refs. 3 and 4). Much of the modern formalism of quantum theory, including Hilbert space, quantum statistics, the density matrix, and the measurement process, originated in this treatise.
3.
M.
Jammer
,
The Conceptual Development of Quantum Mechanics
(
McGraw-Hill
,
New York
,
1966
), pp.
271
276
and Chap. 6.
4.
A.
Duncan
and
M.
Janssen
, “
(Never) Mind your p's and q's: Von Neumann versus Jordan on the foundations of quantum theory
,”
Eur. Phys. J. H
38
,
175
259
(
2013
).
5.
C.
Eckart
, “
Operator calculus and the solutions of the equation of quantum dynamics
,”
Phys. Rev.
28
,
711
726
(
1926
).
6.
The letter from Pauli to Jordan, dated April 12, 1926, is published in B. L.
van der Waerden
, “
From matrix mechanics and wave mechanics to unified quantum mechanics
,”
Notices AMS
44
,
323
328
(
1997
).
7.
What we now call the “Schrödinger picture” and the “Heisenberg picture” are highly-formalized generalizations of the original wave (1926) and matrix (1925) views on how quantum systems evolve in time. The modern formalism behind this unified picture of quantum dynamics includes the time-evolution operator and unitary transformations. For example, see E. Merzbacher,
Quantum Mechanics
(
John Wiley & Sons
,
New York
,
1998
), Chap. 14.
8.
M.
Belloni
and
R. W.
Robinett
, “
Quantum mechanical sum rules for two model systems
,”
Am. J. Phys.
76
,
798
806
(
2008
).
9.
D. F.
Styer
, “
Quantum rivivals versus classical periodicity in the infinite square well
,”
Am. J. Phys.
69
,
56
62
(
2001
).
10.
R. W.
Robinett
, “
Visualizing the collapse and revival of wave packets in the infinite square well using expectation values
,”
Am. J. Phys.
68
,
410
420
(
2000
).
11.
W. A.
Fedak
and
J. J.
Prentis
, “
Quantum jumps and classical harmonics
,”
Am. J. Phys.
70
,
332
344
(
2002
).
12.
Shi-Hai
Dong
and
Zhong-Qi
Ma
, “
The hidden symmetry for a quantum system with an infinitely deep square-well potential
,”
Am. J. Phys.
70
,
520
521
(
2002
).
13.
D. S.
Rokhsar
, “
Ehrenfest's theorem and the particle-in-a-box
,”
Am. J. Phys.
64
,
1416
1418
(
1996
).
14.
R. P.
Feynman
,
R. B.
Leighton
, and
M.
Sands
,
The Feynman Lectures on Physics
(
Addison-Wesley
,
Reading, MA
,
1963
), Vol. 1, pp.
50
7
and 50-8.
15.
W.
Heisenberg
,
The Physical Principles of the Quantum Theory
(
Dover
,
New York
,
1949
), pp.
121
122
. Heisenberg's two rules addressed the question: How can a matrix correspond to physical reality? Rule 1 (Rule 2) provided a physical interpretation of diagonal (off-diagonal) elements based on the correspondence principle. The modern formulas for expectation value and transition probability can be traced back to Rule 1 and Rule 2, respectively.
16.
V.
Rojansky
,
Introductory Quantum Mechanics
(
Prentice-Hall
,
Englewood Cliffs, NJ
,
1938
), pp.
167
168
.
17.
H. C.
Ohanian
,
Principles of Quantum Mechanics
(
Prentice-Hall
,
Englewood Cliffs, NJ
,
1990
), pp.
74
77
.
18.

The only reference we could find that displays the analytical formula for pnk of a particle in a box is Ref. 17, pp. 76–77.

19.
Since pnk = 0 for n + k even, it follows that (P2)nm = 0 for n + m odd. Since xnk = 0 for n + k even (and nk), it also follows that (XPPX)nm = 0 for n + m odd. In Eqs. (9) and (13), the other off-diagonal elements have been set to zero due to their relatively small approximate values.
20.
J. J.
Prentis
and
W. A.
Fedak
, “
Energy conservation in quantum mechanics
,”
Am. J. Phys.
72
,
580
590
(
2004
).
21.
The two standard methods to calculate En for the infinite square well are (1) solve the Schrödinger wave equation; and (2) find the allowed de Broglie wavelengths. See
P. A.
Tipler
and
R. A.
Llewellyn
,
Modern Physics
(
W. H. Freeman
,
New York
,
1999
), pp.
246
249
. Note that the third method, “Square the P matrix,” is not completely wave free because a wave function is used to construct the matrix.
22.
Born, Heisenberg, and Jordan stated the fundamental problem of quantum mechanics as follows: Find the matrices X and P such that (1) the quantum condition, XP-PX=i1, is satisfied and (2) the Hamiltonian H becomes a diagonal matrix when X and P are substituted into the classical Hamiltonian function H(x,p). See
M.
Born
,
W.
Heisenberg
, and
P.
Jordan
, “
Zur Quantenmechanik II
,”
Z. Phys.
35
,
557
615
(
1926
).
23.

Ref. 2, p. 31.

24.

Equation (14) can be interpreted as the expectation value of the kinetic energy in the state ψn(x). For the infinite square well, the derivatives of ψn(x) are discontinuous at the walls. This poses no mathematical problem in calculating expectation values of powers of momentum if the power is less than 4 (see Ref. 16, pp. 160–162). To study the mathematical effect of unphysical walls (infinitely hard), one can analyze a finite square well of height V0 and then take the limit V0.

25.
S. T.
Thorton
and
J. B.
Marion
,
Classical Dynamics of Particles and Systems
(
Brooks/Cole
,
Belmont
, CA,
2004
),
pp
516
.
26.
M. L.
Boas
,
Mathematical Methods in the Physical Sciences
(
Wiley
,
Hoboken
, NJ,
2006
),
pp
375
. Note that the point of Parseval's theorem is not to calculate the mean-squared value of f(x) from its Fourier series. Given f(x), it easy to obtain this value directly by evaluating the integral. The point is to reduce a “whole entity” (such as total energy) into a set of “manageable parts.” The general relation |f(x)|2dx=|ck|2 is also called the “completeness relation” because it implies that a system of orthonormal functions ϕk(x) is complete, i.e., the infinite series ckϕk(x) converges to f(x) in the mean.
27.
Here is a modern version of the Riesz-Fischer theorem: If ck(k=1,2,3,) is a sequence of numbers such that |ck|2 converges and ϕk(x) is an orthonormal system, then there exists a function f(x) whose Fourier coefficients (with respect to ϕk) are equal to the numbers ck and |f|2dx=|ck|2. For the relation between Parseval's theorem, Bessel's theorem, and the Riesz-Fischer theorem, see
K.
Saxe
,
Beginning Functional Analysis
(
Springer
,
New York
,
2002
), pp.
81
84
.
28.
G. B.
Arfken
,
H. J.
Weber
, and
F. E.
Harris
,
Mathematical Methods for Physicists
(
Academic Press
,
Waltham, MA
,
2013
), p.
987
. Lord Rayleigh's name is attached to this theorem because he was the first to express the total energy of a wave as an integral over squared harmonic components in the following article:
J. W. S.
Rayleigh
, “
On the character of the complete radiation at a given temperature
,”
Philos. Mag.
27
,
460
469
(
1889
).
29.

To emphasize the role of addition, integration, and squared objects in quantum theory, von Neumann writes “it should be observed that one might have preferred to have quite generally, kfk=f(x)dx, or something similar, i.e., a complete analogy between addition on the one hand and integration on the other—but a closer examination shows that the addition k and the integration dx are employed in quantum mechanics only in expressions such as fk*gk or f*(x)g(x), respectively” (Ref. 2, pp. 30–31). This kind of observation led von Neumann to introduce “operator,” “inner product,” and “expectation value” into the mechanical theory as key mathematical operations from which experimental results are generated. His pioneering theory of measurement can be found in Ref. 2, Chs. III–VI.

30.
For example, in bra-ket language, Parseval's theorem is f|f=kf|kk|f and the momentum sum rule is n|P2̂|n=kn|P̂|kk|P̂|n.
31.
For an alternative proof, we could start by expanding the derivative function P̂ψm(/i)dψm/dx in terms of a basis of energy eigenfunctions ψk:P̂ψm=kpkmψk. The momentum element pkm is now interpreted as an expansion coefficient—the kth component of P̂ψm. This generalized Fourier series implies the relation ψn*P̂(P̂ψm)dx=kpnkpkm.
32.
D.
Bohm
,
Quantum Theory
(
Dover
,
New York
,
1989
), p.
441
.
33.

Ref. 16, pp. 350–352. In general, the correspondence principle says that the quantum jump from state n to state nτ corresponds to the τth harmonic component of the classical motion in the limit of large n (large orbits) and small τ (small jumps). Most applications of this principle focus on frequency and position. We focus on momentum and energy. For the infinite square well, we find pτ=8mE/πiτ and pn,nτ=(8mEn/πiτ)·(2n2τ)/(2nτ), where τ=±odd. For other values of the “jump” index τ, pτ and pn,nτ vanish.

34.

Ref. 1. Schrödinger first showed how the canonical matrices constructed from Eq. (B1) satisfy the commutation rule of Heisenberg for any arbitrary system un(x). He then showed how to solve Heisenberg''s algebraic equations of motion by choosing one definite system—the proper functions of his differential equation.

35.

Ref. 2, pp. 29–30. Instead of ck and f(x), von Neumann used the symbols xν and ϕ(q1,,qk), respectively. Von Neumann carried out the program of generalizing FZ (the original Hilbert space) and FΩ to a space that he called “abstract Hilbert space” (Ref. 2, Ch. II).

36.
It should be noted that neither Schrödinger nor von Neumann assigned a “theorem” status (title) to their physical assertions regarding the wave-matrix equivalence.
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