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arXiv:0806.4932 · American Journal of Physics

Splitting the wavefunctions of two particles in two boxes

S. J. van Enk

Published Jan 12, 2009• 1 citation
arXivHTMLPDF

Abstract

I consider two identical quantum particles in two boxes. We can split each box and thereby the wavefunction of each particle into two parts. When two half boxes are interchanged and combined with the other halves, where are the two particles? I solve this problem for two identical bosons and for two identical fermions. The solution can be used to define a measurement that yields some information about the relative phase between the two parts of a split wavefunction.

DOI
10.1119/1.2990669

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I consider two identical quantum particles in two boxes. We can split each box and thereby the wavefunction of each particle into two parts. When two half boxes are interchanged and combined with the other halves, where are the two particles? I solve this problem for two identical bosons and for two identical fermions. The solution can be used to define a measurement that yields some information about the relative phase between the two parts of a split wavefunction.
Interference is an important observed feature in quantum mechanics. It is often seen in terms of peaks and troughs in two slit interference or in the profile of bound state wavefunction (e.g. harmonic oscillator). In this note, we argue that interference also plays a central role in determining the two types of quantum particles which exist i.e. fermions and bosons. Whereas a wavefunction is subject to a global constraint (its integral over x is 1), a local spatial probability constraint is imposed (at each x) in the case of at least two particles built from a separable wavefunction Wa(x1)Wb(x2). Here there are two indistinguishable particles, two allowable energy levels a and b and one wishes to know the probability of finding a particle at x1 and one at x2 regardless of the energy level. Let us call this P(x1,x2). The modulus squared of the wavefunction yields probability P, so a constructed wavefunction may have the property that the interchange of x1 and x2 yields a phase factor. This argument is presented often although (1) suggests it should be modified, yet the conclusion that there are only fermions and bosons remains. In this note, we consider the existence of two types of interference as the physical basis of this result. From the Schrodinger equation or from P(p/x)= a(p)exp(ipx) / W one may see that W→-W and a(p)-->-a(p) is still a solution. In quantum mechanics not only do exp(ipx) terms interfere, but eigenfunctions Wn may also interfere. Given that a bound eigenfunction may be represented by Wn or -Wn, there exists two kinds of interference. Even in the case in which there is only one particle, one may already create two possibilities: Wa(x)+Wb(x) and Wa(x)-Wb(x). If one sets a=b, the second becomes 0 at each x. This we argue already points to the idea of fermions even though only one particle is considered. This result is due strictly to alternate phase (1, -1) interference). The idea of alternate phase interference Wa-Wb also seems to be linked to a difference in wavefunction i.e. a “kind” of wavefunction representing the change in moving from energy level a to b. Thus we try to argue that there is interference beyond the usual exp(ipx) namely an alternate phase interference which applies to eigenfunctions Wn(x) and leads to the existence of fermions. Bosons occur with no alternate phase interference. These two types of particles emerge from the constraint that P(x1,x2)=P(x2,x1) i.e a local spatial probability constraint.
We discuss two different approaches for splitting the wave function of a single-particle box (SPB) into two equal parts. Adiabatic insertion of a barrier in the center of a SPB in order to make two compartments which each have probability 1/2 of finding the particle in it is one of the key steps for a Szilard engine. However, any asymmetry between the volume of the compartments due to an off-center insertion of the barrier results in a particle that is fully localized in the larger compartment, in the adiabatic limit. We show that rather than exactly splitting the eigenfunctions in half by a symmetric barrier, one can use a nonadiabatic insertion of an asymmetric barrier to induce excitations to only the first excited state of the full box. As the barrier strength goes to infinity the excited state of the full box becomes the ground state of one of the new boxes. Thus, we can achieve close to exact splitting of the probability between the two compartments using the more realistic nonadiabatic, not perfectly centered barrier, rather than the idealized adiabatic and central barrier normally assumed.
Two particle product type wavefunctions may be created so as to be symmetric (bosons) or antisymmetric (fermions) on then interchange of the identical particles. As a result, spatial density at a point does not change upon interchange. This approach, however, appears to be mathematical in nature. In Part I of this note, we argued that exp(ipx) free particle momentum wavefunctions within a bound wavefunction W(x) interfere and that full wavefunctions Wn(x) in a linear combination also do the same. As a result, probability interference is ultimately linked to the physical mechanism associated with the periodic exp(ipx) wave behind the Pauli rule that fermions may not occupy the same quantum state. In this note, we try to extend this idea and argue that one does not simply exchange identical particles in the wavefunction as a mathematical procedure, but that actual exchange may occur in the probabilistic quantum mechanical case. We note that a time-independent treatment of scattering from a target already includes the idea of “negative” or out of phase interference within a wavefunction W(x)= exp(ikz) + f(theta, phi) exp(ikr)/r so as to remove some incoming flux as outgoing scattered flux now exists. We suggest that the “minus” sign used in the antisymmetric fermion wavefunction behaves in a similar manner. In (1), we argued that W(x)W(x) in a bound state includes all possible a(p1)exp(i p1 x) a(p2)exp(i p2 x) combinations and represented a product probability. In other words, a particle at x with p1 P(p1/x) may be knocked into p2 at x. Similarly, we argue that cross terms in a two particle fermion wavefunction represent flux W*b(x1)Wa(x1)W*a(x2)Wb(x2) flowing from out of states as an interchange occurs. Main terms such as W*a(x1)Wa(x1)W*b(x2)Wb(x2) represent the presence of particles in the a, b states. If a=b, the presence and flow probabilities cancel locally and there is a vanishing wavefunction i.e. the two fermions may not occupy the same state. We argue that one should be able to describe the Pauli exclusion rule in this manner and note that Feynman (2) has used probability arguments in boson scattering to argue the boson enhancement factor. In other words, the mathematical argument of a symmetric wavefunction is not used, but a more physical argument is given.
Consider a pair of particles that may be either two fermions or two bosons. e Schrödinger equation for the system is written in terms of the Hamiltonian operator for the system as H( , ) ( , ) ( , ),1 2 1 2 1 2c c= E (12.1) where 1 represents the coordinates of particle 1 and 2 the coordinates of particle 2. e Hamiltonian (1, 2) must be symmetric under particle interchange because of the indistinguishability of the particles, giving H H( , ) ( , ).1 2 2 1= (12.2) e pair wave function χ(1, 2) may be either symmetric or antisymmetric. is can be seen by introducing the permutation operator P12, which permutes the coordinates of the particles.