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.

## REFERENCES

*Collected Papers on Wave Mechanics by E. Schrödinger*(Chelsea Publishing, New York, 1982).

*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.

The only reference we could find that displays the analytical formula for *p _{nk}* of a particle in a box is Ref. 17, pp. 76–77.

*p*= 0 for

_{nk}*n*+

*k*even, it follows that (

**P**

^{2})

_{nm}= 0 for

*n*+

*m*odd. Since

*x*= 0 for

_{nk}*n*+

*k*even (and $n\u2260k$), it also follows that (

**XP**–

**PX**)

_{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.

*E*for the infinite square well are (1) solve the Schrödinger wave equation; and (2) find the allowed de Broglie wavelengths. See

_{n}**P**matrix,” is not completely wave free because a wave function is used to construct the matrix.

**X**and

**P**such that (1) the quantum condition, $XP-PX=i\u210f1$, 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

Ref. 2, p. 31.

Equation (14) can be interpreted as the expectation value of the kinetic energy in the state $\psi n(x)$. For the infinite square well, the derivatives of $\psi n(x)$ 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 *V*_{0} and then take the limit $V0\u2192\u221e$.

*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 $\u222b|f(x)|2dx=\u2211|ck|2$ is also called the “completeness relation” because it implies that a system of orthonormal functions $\varphi k(x)$ is complete, i.e., the infinite series $\u2211ck\varphi k(x)$ converges to

*f*(

*x*) in the mean.

*f*(

*x*) whose Fourier coefficients (with respect to $\varphi k$) are equal to the numbers

*c*and $\u222b|f|2dx=\u2211|ck|2$. For the relation between Parseval's theorem, Bessel's theorem, and the Riesz-Fischer theorem, see

_{k}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, $\u2211kfk=\u222bf(x)dx$, 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 $\u2211k$ and the integration $\u222bdx$ 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.

*p*is now interpreted as an expansion coefficient—the

_{km}*k*component of $P\u0302\psi m$. This generalized Fourier series implies the relation $\u222b\psi n*P\u0302(P\u0302\psi m)dx=\u2211kpnkpkm$.

^{th}Ref. 16, pp. 350–352. In general, the correspondence principle says that the quantum jump from state n to state $n\u2212\tau $ 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 $p\tau =8mE/\pi i\tau $ and $pn,n\u2212\tau =(8mEn/\pi i\tau )\xb7(2n\u22122\tau )/(2n\u2212\tau )$, where $\tau =\xb1odd$. For other values of the “jump” index τ, $p\tau $ and $pn,n\u2212\tau $ vanish.

Ref. 1. Schrödinger first showed how the canonical matrices constructed from Eq. (B1) satisfy the commutation rule of Heisenberg for any arbitrary system *u _{n}*(

*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.

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