Dear Prof Smid,

I am highly thankful for your joining the discussion on an important and elusive question. I am sure, your sustained participation would finally help in reaching the right conclusion.

I am greatly benefited by your discussion in having a better understanding of the complexity of the issue. While I will address some important but secondary issues later, let me first address the main issue in the following.

The very fact that 1960 paper finds that two particles (say P1 and P2) in their ground state have k_1 = - k_2, indicates that their CM momentum K = k_1 + k_2 = 0. In other words P1 and P2 in the state have pure relative motion. Obviously, the state is represented by a standing wave,

sin(kx/2) = sin(qx) with relative momentum k = k_2 - k_1 = 2q and relative distance x = x_2 - x_1.

This conclusion is consistent with the fact that a free particle in its ground state has zero momentum, since CM motion of P1 and P2 too represents a freely moving body.

Our study too finds the same results; the difference lies with the value of q.

While 1960 paper finds q = \pi/L, our 2004 paper concludes q = 2pi/L.

I agree with you that one should take good care of the boundary conditions because only these conditions would decide which result is correct.

In this context let me underline the fact that stationany CM of P1 and P2 is the reference point of their motions in their ground state. Naturally, to apply necessary boundary conditions on sin(qx) accurately, it is important to identify the location of this reference point in the box. Keeping in mind that particles move away from (or come nearer to) the CM with equal and opposite momenta and this character stays till P1 and P2 are in their ground state, one can easily come to the conclusion that the CM of P1 and P2 should rest at the mid point of the box.

Since P1 and P2 are hard core particles and they meet or collide in their motion at their CM (i.e. mid point of the box) the state wavefunction has to have a node (zero value) at this point in addition to the points at the two boundary walls of the box which do not allow particles to move away from the CM by a distance more than +d and -d. One may find that a standing wave obtained by superposition of two waves of momenta q and -q with a node at the mid point of the box is characterised by \lambda = L which means q = 2\pi/L = \pi/d. This establishes the accuracy of the result obtained in 2004 paper. It appears that the 1960 paper, somehow missed to identify the physical reality that the point x_1 = x_2 = 0 (the CM point) is synonym with the mid point of the box.

Certainly the important question needs further discussion and this will be facilitated greatly if it is explained which boundary conditions render q = \pi/L and how.

Dear Prof. Smid,

I may surely have some misunderstanding of the intricacies of wave particle duality as I am a simple student of the subject who has been striving to learn it and I am fortunate to learn a lot from your interaction.

In my humble opinion, we researchers have common goal to find the truth to an extent we can and leave the rest to younger generation for further advancement. I also believe that we can achieve this goal only by intense participation in open discussion and I have no hesitation in accepting that your participation has directly helped me in having a better understanding of the subject and it will help many indirectly. It is in the best interest of our correct understanding of the subject that in the following I put my points in detail and I gratefully accept your acceptable points.

Coming to your point that the product-splitting of the wavefunction stated in 2004 paper cannot agree with the boundary conditions [ x = 2(+ or -)X and x = 2(+ or -)(L-X) ] and can at best be viewed as a (not very good) approximation. After having its critical analysis, I agree that the said boundary conditions are not exactly satisfied by P1 and P2 if they simultaneously have relative plus CM motion (a state where |k_1| ≠ |k_2|). However, they are valid in states of;

(i) pure relative motion (i.e. a state of K = 0 or k_1 = - k_2) and

(ii) pure CM motion (i.e. a state of k = k_2 - k_1 = 0 or k_2 = k_1).

We note that P1 and P2 in 1-D have only two degrees of freedom that are represented by (i) and (ii). The ground state of two particles by law of nature has, K = k_2+k_1 = 0, i.e., k_1 = -k_2 which represents a pure relative motion where P1 and P2 can be seen to start their journey in opposite direction with momenta q and -q from the points on two opposite walls and meet at the mid point of the box and return to travel in opposite direction to strike the opposite walls simultaneously to repeat the cycles. In this process they are always seen to travel with q and -q. However, if they start their journey with q and -q in such a way that they meet at an arbitrary point (other than the mid point of the box), they will surely not be travelling always with q and -q; over a part of their journey, they would be travelling in the same direction (this is simple to verify). Similarly, in the pure CM motion P1 and P2 must start from with equal momenta k_1 = k_2 from a common point, viz., say a point on either of the two walls to move towards facing wall.

In summary we used following points to conclude that the ground state is characterized by q = 2\pi/L and E_o = 2(h^2/8md^2 = 8(h^2/8mL^2).

1. The ground state of two particles has to have their CM momentum K = 0.

2. P1 and P2 in the ground state have pure relative motion with momenta (q, -q).

3. A state of P1 and P2 with momenta q and -q is represented by a standing wave, sin(qx).

4. The location of CM point in such a motion has to be at the mid point of the box.

5. Since P1 and P2 during this motion meet at their CM point, the standing wave sin(qx) has to have its node at the mid point of the box.

While points 1-3 are in agreement with 1960 paper, points 4 and 5 are simply obvious for hard core P1 and P2 with q and -q. Nevertheless, I would be happy to know which of these points are unacceptable in concluding, q = 2\pi/L and E_o = 2(h^2/8md^2 = 8(h^2/8mL^2).

As such in view of the point raised by Prof. Smid, minor modication or clarification required for the states where P1 and P2 have relative motion combined with CM motions, would be appropriately discussed separately in order to avoid mixing of these secondary points with the main issue. Certainly Prof Smid is right in stating that 1960 paper and similar similar studies are exceptionally important to understand the physics of many particles in 1-D box for the condition (0 ≤ x_1 ≤ x_2 ≤ x_3 ......... ≤ x_N≤ L) adopted by these studies; however, this condition is not followed rigorously by the particles when the system enters in its quantum regime (av. \lambda > av. inter-particle distance).

With best of my regards,

Yatendra S Jain.

Dear Prof. Smid,

Any scientific discussion can not achieve its objectives if its participant(s) have different picture of the physical realities of the system. To resolve their differences, they have to have common perception of the system and related aspects. I accepted the point that the wave function of a pair of particles (say P1 and P2) expressed as the product of the wave functions representing its CM motion and relative motion can not be consistent with the said boundary conditions if P1 and P2 are assumed to have the two motions simultaneously. However, if they have pure relative motion (where two particles have equal and opposite momenta q and -q) or pure CM motion (where two particles move with identically equal momenta as a single body of mass 2m), the said boundary conditions find no problem and such motions are unoubtedly independent of each other. I hope we can proceed further with the same spirit in accepting that 1960 paper ignores following realties related to wave particle duality and here lies the source of errors in 1960 paper.

(1) When P1 and P2 have a distance d < λ (= 2\pi/q representing their de Broglie wave length related to their relative momentum k=2q) they are bound to have their wave superposition as a natutal consequence of wave nature. It is well known that P1 and P2 in a state of their superposition occupy a single quantum state. Their identity as an individual particle is completely lost. In variance with this reality your argument implies that the 1960 paper assumes that two bosons in their ground state occupy single particle states of different momenta, -one of k_1 = \pi/L and other of k_2 = 2\pi/L. Further it is also clear that the complete wave funtion of two identical bosons in the state of their superposition can not be \Psi_T(x_1,x_2) = C(x_1-x_2)\cos((k_1)x_1)\cos((k_2)x_2); it has to be added with +/- C(x_1-x_2)\cos((k_2)x_1)\cos((k_1)x_2) and in such a state P1 and P2 undoubtedly have identically equal energy (half the energy of the pair). Evidently, the 1960 paper ingnores wave superposition of P1 and P2 which is bound to occur as a natural consequence of wave particle duality.

(2) Similarly, in what follows from one of the basic principles of wave mechanics, a particle behaves as a wave packet of size λ/2 and it exclusively occupies a space of size λ/2. In other words, a particle can not be placed in a box of size < λ/2. If we force a particle of size λ/2 in a box of size < λ/2, the particle would absorb energy from our action to so that λ/2 becomes at least equal to the size of the box. This is not merely a conjecture; it is rather an experimentally supported fact, - it is well known that an electron in its ground state in an electron bubble exclusively occupies a spherical cavity of size λ/2. Naturally, two particles of wave packet size λ/2 can not have a distance x < λ/2; the minimum distance between two particles is guided by their λ/2, (i.e. their q or corresponing energy) and for this reason the boundary condition 0 ≤ x_1 ≤ x_2 ≤ x_3 ....... ..... ≤ x_N ≤ L (as used in References 1-3 listed in 2004 paper) does not remain valid if the particles have finite energies/momenta or λ/2. It can be valid approximately if the average de Broglie wave length of particles λ_av < < d (av distance between two nearest neighbor); i.e. when the particles have large energies and the system largely has classical behavior. However, when λ_av > d ( i.e. when the system largely has quantum behavior), the said boundary condition does not fit with quantm mechanical reality that a quantum particle behaves as a wave packet of size λ/2 and two nearest particles are bound to have a separation x > or = λ/2. This is another serious problem of 1960 paper.

In summary I would like to know: (i) How can we ignore wave superposition of two particles arising as an obvious consequence of wave particle duality and use \Psi(1,2) = u_{k_1}(r_1) u_{k_2}(r_2) to describe the pair in place of \Psi(1,2) = [u_{k_1}(r_1) u_{k_2}(r_2) +/- u_{k_2}(r_1) u_{k_1}(r_2)] and (ii) How can we ignore the experimental fact that a quantum particle exclusively occupy a space of size λ/2 which also agrees with one of the well accepted aspect of wave particle duality that a particle behaves as a wave packet of size λ/2 and the experimentally observed existence of an electron bubble. Since the 1960 paper ignores these realities, its results are erroneous for the intrinsic problems in using single particle basis and faulty boundary conditions in spite of the accuracy of its mathematical derivations which I did not question. Clearly, nothing (including the said fermion-boson mapping or argument based on RG, etc.) can justify the accuracy of the ground state of the pair and other low energy states (where λ of particles is < d) concluded in 1960 paper. Mathematical accuracy of a theory and its results does not mean that they are physically relevant and accurate if the basic premises of the theory have intrinsic inconsistency with physical realities.

Thank you very much Prof Smid for letting me learn a lot from your discussion. If you wish we can discuss any issue related to this problem including the accuracy of mapping of hard core bosons on scalar fermions.

With best of my regards

Y S Jain.

All Interested Physicists,

Dr. Smid has made his concluding statement indicating that we (I and he) could not reach consesnsus on what is the correct ground state of two hard core particles in 1-D box with impenetrable walls even after an exchange of a set of e-mails with our arguments and counter arguments.

I heartly appreciate Dr. Smid for his coming forward to discuss the subject in details. I respect his spirit of participation in prolonged discussion. While he quotes a few papers as experimental support of 1960 paper, I would most humbly like to state that people would find the same experimental results to have even better agreement with our results (arxiv.org/pdf/cond-mat/0606409.pdf) obtained by using 2004 paper as their foundation. However, necessary analysis for this purpose requires a comprehensive discussion which may run over several pages (10 or more) of a paper; it can not be summed up by a short discussion appropriate for Research Gate. Consequently, I too close this discussion with a humble statement that if all interested physicists try to see the problem without any bias, they would surely find that 1960 paper has several sources of confusions, inconsistencies, and intrinsic difficulties with intricacies of wave particle duality and experimental realities. In order to facilitate their efforts, I invite them firstly to participate in discussion on several small questions related to the basics of wave particle duality and its intricacies which have to be followed to conclude the correct ground state of N hard core particles in 1D box as a general case for N=2.

I would come back to this question after having suitable discussion and answer for all necessary small questions.

Y. S. Jain