EINSTEIN-PAULI-YUKAWA PARADOX-WHATIS THE PHYSICAL REALITY?Ni GuangjiongAbstract　What is the momentum spectrum of a particle moving in an infinite deep square well?Einstein,Pauli and Yukawa had adopted different point of view than that in usual text books.Thetheoretical and experimental implication of this problem is discussed.Keywords　Quantum mechanics;Wave function;State vector;Measurement;Physical reality爱因斯坦—泡利—汤川佯谬——什么是“物理实在”?倪光炯复旦大学物理系，上海市 200433　　摘　要　在无限深方势阱内一个粒子的动量谱究竟是什么?爱因斯坦、泡利和汤川曾分别采取了一种与通常教科书中不同的观点.本文讨论了与此问题有关的理论上和实验上的含义.　　关键词　量子力学；波函数；态矢量；测量；物理实在

0　Introduction　　Looking at the title,our reader may wonder if there is some confusion with the well known Einstein-Podolsky-Rosen (EPR) paradox?Our answer turns out to be‘no’first but ‘yes’afterwards.　　The problem begins from an elementaryexample in almost every textbook on quantum mechanics(QM).Consider a particle with mass m confined in an infinite deep square well in one dimensional space.The potential reads(1)

The energy eigenvalue is well known(2)

while the stationary wave function of ground state (n＝1) reads(3)

If one wishes to discuss the momentumrepresentation of this state,the usual calulation runs as(4)

as shown by Landau and Lifshitz1.　　However,there is another point of view in the literature as stressed by Tao in a shortpaper2.Although Einstein3,Pauli4 and Yukawa5 (EPY) knew the calculation from (3) to (5),they did consider independently that，in the ground state (3),there are only two sharp values of momentum (p′) each with probability 1/(6)

p has a continuous spectrum.　　On the other hand,eq.(6) is a result coming from the direct observation of eq.(3) as

with operator(8)

However,′ is not well defined to have a discrete spectum(9)

before the periodical boundary condition in 4ainterval is imposed.In this case,the relationbetween

The difference between ^P and ^P′ is shown at eqs.(7) and (8) by the definition range of x and boundary condition.It can be displayed even more clearly if they are expressed in otherrepresentation,say,^P′ can be expressde by aninfinite matrix with matrix element (9) whereas ^P cannot.Furthermore,there is a momentum conservation law for p and correspondingly an uncertainty relation reads(10)

whereas the momentum conservation law related to p′ is a discrete one with possible momentum exchange with the lattice (Umkrapp process in crystal) and there is no uncertainty relation like eq.(10).Only at large quantum number (n1) case,φ′n(p′n) approaches to φ(p).　　As mentioned above,the spectrum of palong x direction could be measured if theparticle is deconfined via the z direction.On the other hand,one can imagine the measurement of p′ spectrum ‘locally’ if the observer andapparatus are also confined in the deep potential well.How the periodical boundary conditionsinfluence on the position operator was discussed recently by Resta9.　　So in our point of view,both calculations of (5) and (6) are correct because they are talking about two different kinds of momentum.2　What is the real momentum of particle?　　Let us insist on querying that ‘which one,p or p′,is the real momentum of particle before the measurement is performed’?Then we are facing a dilemma.Logically,eqs.(5) and (6) can not all be correct since they are different.　　It seems to us that the only way out of the dilemma is not taking the question seriously.Because if taking seriously,neither p not p′ is really existing before the measurement is made.　　ln Dirac notation,a quantum state,e.g,the ground state under consideration,is denoted by an abstract state vector ｜ψ1＞.In Heisenbergpicture,there is no description either x or t in｜ψ1＞.Only after some representation is chosen.can it get some description.For instance,if we choose the eigenvector of the position ｜x,t＞ as the base vector and take the contraction(projection) of ｜ψ1＞ with ｜x,t＞,we obtain the wave function in configureation space:(11)

as shown in eq.(3) with a factor exp (－iE1t/h) omitted.　　Alternatively,we can choose the eigenvector of momentum p,｜p,t＞,as the base vector to get the wave function in momentum (p)representation:(12)

or in p′ representation:(13)

The three kinds of wave function,(11)～(13),are three different descriptions for a same quantum state ｜ψ1＞.No one among the three is more fundamental than others.　　The wave functions in QM are notobservable.But they are very useful in linking the even more abstract state vector,say ｜ψ1＞,to the potential possible outcome in experiments if the latter are really performed on the state.The variable,x in (11),or p in (12),or p′ in (13) is just to characterize what kind of experiment,for position (x) measurement,or for p(p′)measurement,etc.we are going to use.Therefore,p (or p′) in wave function is by no means the momentum of particle before themeasurement.Similarly,the variable x is also not the position of ‘point particle’ before themeasurement.(The statistical interpretation of QM claims that ｜ψ(x,t)｜2 is the probabilitydensity of finding the particle at position x in themeasurement rather than that of the appearance of the particle before the measurement).It is just because they are the outcome of ‘fictituous measurement’,so the probability amplitudes,i.e.,the wave functions,contain the i＝√-1 which is unobaervable.　　In short,a quantum state in abstract sense contains no information.Certain information is gained only after certain measurement is made.However,the power of QM lies in the fact that we can predict what will happen before certain measurement is performed if the knowledge about the environment of particle,e.g.eq.(1),is known.3　The possible relevance with EPR paradox and their experimental implications　　In 1935,based on the belief of ‘physicalreality’,Einstein,Podolsky and Rosen (EPR) raised their famous paradox.Since then,especially after the study on Bell inequality,the various EPR experiments have been achieving an important conclusion that the prediction of QM is correct whereas the existence of ‘localhidden variable’ is incompatible withexperimental results (see refs.10,11).Thesubtle entanglement in two-particle system,to our understanding,implies that before thequantum coherence of entangled state isdestroyed by measurement,no information about one individual particle can be isolated from the other.Actually,such kind of information does not really exist before the measurement is made12.　　This is no surprise.Another example is the two electron state with paralled spin described by the Slater determinant(14)

We can not say that the electron 1 is staying in either state a or state b.Only after themeasurement finds an electron,say 1,being in state a (or b),can we say that the electron 2being in state b (or a).　　Now the EPY paradox discussed in thispaper,according to our understanding,has the similar implication as EPR paradox but for one particle state.　　The discussion on EPR or EPY paradox is by no means a pure academic one.We are facing more and more quantum subtlety after theestablishment of QM in 70 years ago.The fact that the canonical momentum ^p is not anobservable becomes more evident when theparticle with charge q is coupled to the gauge field,say,the vector potential of electromagnetic field,A.Then ^p should be replaced by thekinematic momentum^∏，(15)

(B is the magnetic field).An important case with degeneracy is electron (q＝－e＜0) moving in a plane with a constant B along z direction.There are three kinds of wavefunctionsdescribing the motion of electron.　　1)Landau gauge1,Ax＝－By,Ay＝Az＝0.(17)

For fixed nth Landau level,the state degeneracy G is reflected in yp.

Lx and Ly are the width of plane.　　2)Symmetric gauge13,Ax＝－By／2,Ay＝Bx／2.(18)

z＝(x＋iy)／2awhile En＝(n＋1/2)ω as before,the quantum number L＝0，1，2，…shows that for fixed n the wavefunctions are peaked at concentric circles,with equal area between successive circles.Each ring between circles supports one ψ0,giving G as before.　　3)For n＝0 case,the L degeneracy can be transformed into wavefunctions described by(19)