Mathematical analysis for double-slit experiments

Transcription

1 1 Mathematical analysis for double-slit experiments CHong WANG College of Information Engineering, Zhejiang A&F University, Lin an, China Using the principle of least action and functional theory in Hilbert space, the author provided a pure mathematical analysis to explain why particles exhibit wave properties in the double-slit experiments and presented a new understanding for the de Broglie equation. Then, proofs for the Schrödinger equation and for the hypothesis of the Feynman path integral were given. Analysis showed that statistical properties of particles caused the wave appearance of the particles and the wave function represented the average least action of the particles. Subject Index: Intrduction Double-slit experiments indicate that particles exhibit wave properties. 1) 4) Many interpretations have been proposed to explain the wave features appearing in doubleslit experiments. 5), 6) However, all these interpretations are only philosophical views with many hypotheses; they are not conclusions supported by strict mathematical logic. The Copenhagen interpretation believes that a particle has a dual nature of both particle and wave; it regards this wave as a probabilistic wave. 7) 9) Although experiments have proved the probabilistic wave viewpoint, there is still no clear mathematical description to show why it shows a probabilistic wave. In addition, a remarkable feature of the Copenhagen interpretation is that it denies the concept of the classical trajectory for a particle. This has raised doubts among some scientists. American physicist A. Landé believed a successful interpretation should be classical; 10) he did not agree with the Copenhagen interpretation s understanding for the uncertainty principle. Philosopher of science K. Popper regarded the uncertainty principle as a statistically discrete relationship. 11) He insisted that there was no need to do away with the concept of the classical trajectory of a particle. In all interpretations for the double-slit experiments, the ensemble interpretation has the fewest hypotheses. This interpretation states that the wave property in a double-slit experiment comes from the statistical behavior, and a single particle still obeys classical laws of physics. Einstein first provided the ensemble thought. 12) He believed that the wave function described the properties of the ensemble system. Later, D. I. Blokhintsev and L. E. Ballentine developed the ensemble thought into the ensemble interpretation. 13), 14) Today, more and more scientists have accepted the ensemble interpretation. In fact, philosopher of science M. Jammer affirmed that, in practical work, physicists have actually used the logic and terminology of the ensemble interpretation, whether or not they have accepted it. 15) Although the ensemble interpretation exhibits a promising future, it is still not a theory supported typeset using PTPTEX.cls Ver.0.9

2 2 by rigorous mathematical logic. To expose the truth for why particles show wave properties, a strict mathematical analysis based on fundamental principles of physics is a must. In this work, using the universal principle of least action, a mathematical description to expose the nature of the wave appearance of particles is provided, and a new understanding for the de Broglie equation is presented. Finally, the Schrödinger equation and the hypothesis of the Feynman path integral are proved. The analysis supports the conclusion that the wave appearance of particles is rooted in the statistical behavior of particles with the wave function providing a description of the ensemble system. 2. LEAST ACTION AND FOURIER DECOMPOSITION 2.1. Least action For a free particle, the Lagrange function L is 1 2 mv2 (= E) or p2 2m, where m and v are the particle s mass and speed, respectively; E and p are the particle s kinetic energy and momentum, in that order. Suppose the particle leaves one position at moment t s arriving at another position at t f with displacement r. The particle s action s is: s = tf t s Ldt = r 0 p 2 dr = t 0 Edt. (1) Where t = t f t s. The principle of least action says that the particle is moving along its least action trajectory. In the double-slit experiment ( (a) of FIG. 1. ), consider the right border of slit A or B as a particle s initial position. A free particle leaves slit A or B, taking time t, and arriving at the screen position x accompanying displacement r. By the principle of least action, the particle has a determinate trajectory. That is, only the particle s least action s will be used to determine the particle s screen position x. (If not stated otherwise, in this article a particle s least action refers to this s.) Taking no account for Einstein s theory of relativity, by equation (1), s is a function of pr or Et. That is: s = f(pr), (2) or 2.2. Least action with Fourier decomposition s = g(et). (3) Generally speaking, the least action function x(t) (where t is a real variable on [ a, a] and a is a positive real constant) is square-integrable, i.e. a a x(t) 2 dt <.

3 This can be denoted as Analysis for double-slit experiments 3 x(t) L 2 [ a, a]. For two functions, x(t) and y(t), in L 2 [ a, a], an inner product is defined as: < x, y >= a a x(t) y(t) dt, t [ a, a]. (4) By equation (4), L 2 [ a, a] constitutes an inner product space, and the norm derived by equation (4) is a x = ( x(t) 2 dt) 1 2. (5) a From the above, three theorems can be stated: 16) Theorem 1: L 2 [ a, a] constitutes a Hilbert space by equation (5). Theorem 2: For the least action function f(pr), when its complex Fourier coefficients are given by the sum α k = 1 2a a a f(pr) e ikπpr/a d(pr), (k = 0, ±1, ±2,...). k= α k e ikπpr/a converges to f(pr) in the L 2 [ a, a] norm, i.e. f(pr) = k= α k e ikπpr/a. (6) Theorem 3: The least action function g(et) can be expressed as g(et) = k= β k e ikπet/a. (7) where β k = 1 2a a a g(et) e ikπet/a d(et), (k = 0, ±1, ±2,...) Description 3. DOUBLE-SLIT EXPERIMENT In FIG. 1. identical particles are sent out from source S. They pass through slit A or B arriving at screen x. O is the origin of screen x, and S, O and O are on one line with O the mid point of slits A and B on board m. Board m is parallel

4 4 to x with m and x perpendicular to SO. To simplify this problem, consider that the particles direction of movement, the double-slit, and screen x are on the same plane. Experiments have shown that when only one slit is opened, a diffraction phenomena exists on screen x as shown by either the solid curve or the dotted curve in (b) of (FIG. 1.). However, when the two slits are both opened simultaneously, an interference phenomenon occurs as in (c) of (Fig. 1.). (a) m x x (b) x (c) S O B A O FIG. 1. a) Particles begin at S and pass through slit A or B arriving at screen x. b) When only slit A is open, the particle s position (vertical orientation)-density (horizontal orientation) curve is described by the solid curve. When only slit B is open, the position-density curve is described by the dotted curve (the curve formed by slit A partially overlaps the dotted curve). c) When the two slits are opened simultaneously, the position-density curve appears as an interference pattern Particle classification Equations (6) and (7) can be expressed in real form as: f(pr) = a k cos(kπpr/a) + b k sin(kπpr/a), (a) k=0 and g(et) = a k cos(kπet/a) + b k sin(kπet/a). (b) k=0 where a k, b k, a k and b k are real Fourier coefficients. For the real value function f(pr), equation (a) is equal to f(pr) = 2Re( α k e ikπpr/a ), k=0 (c) and for the real function g(et), equation (b) is equal to : g(et) = 2Re( β k e ikπet/a ). k=0 (d)

5 Analysis for double-slit experiments 5 For the Fourier series of a particle s least action, such as equations (a) and (c), particle s two classification patterns are: Pattern A: equation (a) can be approximated by one term a k cos(kπpr/a)+b k sin(kπpr/a)(k=0,1,2,...), and the other terms can be ignored, or equation (c) can be approximated by one term 2Re(α k e ikprπ/a ) and the other terms can be ignored. This particle is then classified as a Pattern A particle. Generally, to simplify the analysis, 2Re(α k e ikprπ/a ) is replaced by α k e ikprπ/a. Pattern B: If a particle does not belong to pattern A, it is classified as a Pattern B particle Propositions Proposition 1: In the double-slit experiments, a Pattern A particle only arrives in the neighborhood of screen position x which corresponds to a nonnegative integer. Proof: For identical particles, if a particle s least action can be described accurately by α k e ikprπ/a (k=0,1,2,...), then the particle s least action depends on a nonnegative integer. So, according to the principle of least action, the particle s trajectory depends on this integer. That is, the particle s screen position x depends on the integer. By definition, a Pattern A particle s least action can be approximated by α k e ikprπ/a (k=0,1,2,...). Thus, its screen position is near x or in the neighborhood of x. Proposition 2: In the double-slit experiments, if a Pattern A particle with least action α k e ikprπ/a (k = 0, 1, 2,...) arrives at screen position x, then, for the particle in the neighborhood of x, its least action can be approximated by β k e ik Etπ/a, where k is an integer, β k is given by equation (7). Proof: If a particle with least action α k e ikprπ/a (k = 0, 1, 2,...) arrives at screen position x, then by equation (1), its least action s real form is s = Et or s = 1 2pr. So, α k e ikprπ/a = α k e i2ketπ/a. Letting k = 2k, by equation (6) and (7), α k = β k. Thus: α k e ikprπ/a = β k e ik Etπ/a. So, when a Pattern A particle is in the neighborhood of x, its least action can be approximated by β k e ik Etπ/a, where k is a nonnegative integer Discussion on Pattern A particles Particle s actual least action and its approximation For a Pattern A particle with actual least action f(pr), 2Re(α k e ikprπ/a ) represents its approximation. Generally, the Pattern A particle s ideal form is α k e ikprπ/a. Letting ψ(r) = α k e ikprπ/a, and supposing a particle with least action ψ(r) arrives at

6 6 position x, then a particle with least action f(pr) will arrive in the neighborhood of x. By equation (5), f(pr) 2Re(ψ(r)) can be used to describe the relative displacement between the particle s actual position and x. The less f(pr) 2Re(ψ(r)) is, the closer the particle s actual position is to x Analysis of the variance for least actions Suppose ψ(r) = α k e ikprπ/a corresponds to position x. Then, in the neighborhood of x, every particle s least action can be approximated by 2Re(ψ(r)). So, 2Re(ψ(r)) can be regarded as the average least action in the neighborhood of x. That is, the mathematical expectation of the particles actual least action in the neighborhood of x is 2Re(ψ(r)). Also, in the neighborhood of x, every particle s actual least action f(pr) (a real value) is a random variable. So, the variance of these random variables is: D(f(pr)) = E[(f(pr)) 2 ] (2Re(ψ(r))) 2. Because (Re(ψ(r))) 2 is positively correlated with ψ(r) 2, the greater the value of ψ(r) 2, the greater the value of (Re(ψ(r))) 2 and the less the value of D(f(pr)). Variance reflects the degree of concentration for random variables gathering near their mean value. Thus, the greater ψ(r) 2 is, the less D(f(pr)) will be, meaning more random variables will be concentrated near 2Re(ψ(r)). Therefore, there are more particles concentrated near position x. So, ψ(r) 2 can describe the particle s density or probability of particles appearing in the neighborhood of x General form for particle least action If α k e ikprπ/a corresponds to screen position x, then x should be the average position of the particles in the neighborhood of x. To obtain a uniform functional format, k can be merged into r. That is ψ(r) = α k e iprπ/a. (8) Where r contains a nonnegative integer, ψ(r) in equation (8) corresponds to an integer-dependent position x. Equation (8) describes the least action of the average position in the neighborhood of x. By proposition (2), in the neighborhood of x, for a particle which is in position x ( x), its least action can be approximated by β k e ik Etπ/a. Similarly, merging k into E, the least action can be denoted as ψ(t) with: ψ(t) = β k e ietπ/a. (9) In the above equation, E contains a nonnegative integer, so E should be: 0, E, 2E,... For a Pattern A particle, if its position is x and x is in the neighborhood of x, then the least action of x and x should be considered simultaneously. Generally, the average value of position x and x should be used. Is this then the arithmetic mean or the geometric mean? By the Riemann-Lebesgue lemma, 16) for equation (c) or (d), when k, α k = β k 0. This means k has a normal distribution with

7 Analysis for double-slit experiments 7 the probability that k takes on a large value being small, and the probability of k having a small value being large. For Pattern A particles in the neighborhood of x, their least actions are random variables, and their approximation is s = α k e ikprπ/a (k = 0, 1, 2,...). Because k has a normal distribution, then ln(s) = ln α k + ikprπ/a is also normally distributed. Statistically, for a Pattern A particle in the neighborhood of x, the average of the least action of x and x should be the geometric mean of ψ(r) in equation (8) and ψ(t) in equation (9). This geometric mean is ψ(r, t) = ψ(r)ψ(t), or ψ(r, t) = ψ(t)ψ(r). Thus, we have ψ(r, t) = Ae i(pr Et)π/(2a), (10) or ψ(r, t) = Ae i(et pr)π/(2a). (10 ) where A = α k β k = β k α k. For a Pattern A particle in the neighborhood of x, equation (10) or (10 ) expresses its general form of least action, and ψ(r, t) 2 describes the particle s density in the neighborhood of x. Remarkably, in the neighborhood of x, Ae iprπ/(2a) or Ae iprπ/(2a) is a constant Mathematical properties of equation (10) In equation (10), ψ(x, t) has the following properties: (a) ψ(r, t) <. (b) ψ(r, t) is single valued and continuous. (c) From the uniqueness of least action, if ψ(r, t) represents the least action of position x, then Cψ(r, t), where C is a constant, still represents the least action of x. Thus, ψ(r, t) 2 only describes the particle s relative probability distribution. (d) By the Riemann-Lebesgue lemma, when k, α k = β k 0. That is A 0. So ψ(r, t) 2 dx is convergent. Because ψ(r, t) 2 describes the relative probability to find particles in the neighborhood of x, so we have: ψ(r, t) 2 dx = Particle density in Hilbert space For Pattern A particles that start from S ((a) in FIG. 1. ), pass through slit A or B, and arrive in the neighborhood of an integer-mapped screen position x, there are three classification types: Type (I): particles which only pass through slit A and arrive in the neighborhood of x meaning the probability of passing through slit A is 1; Type (II): particles which only pass through slit B and arrive in the neighborhood of x meaning the probability of passing through slit B is 1; Type (III): the probability of passing through slit A is c 1 (0 < c 1 < 1), and the

8 8 probability of passing through slit B is c 2 (0 < c 2 < 1). When only slit A is open, the particles in the neighborhood of x consist of Type (I) particles and only a few Type (III) particles. In this case, the average least action of particles in the neighborhood of x can be stated as ψ 1. When only slit B is open, the particles in the neighborhood of x consist of Type (II) particles and a few Type (III) particles. In this case, the average least action of particles in the neighborhood of x can be stated as ψ 2. By equation (5), for least action function ψ, ψ 2 is positively correlated with ψ 2. So ψ 2 can also be used to express particle s density. That is, ψ 1 2 expresses density in the neighborhood of x when only slit A is open; ψ 2 2 expresses density in the neighborhood of x when only slit B is open. When A and B open simultaneously, symmetry is enhanced, thereby increasing the number of Type (III) particles. For these new Type (III) particles, if they pass through slit A, their particle density dedication in the neighborhood of x is c 2 1 ψ 1 2 ; if they pass through slit B, their particle density dedication in the neighborhood of x is c 2 2 ψ 2 2. The average dedication in the neighborhood of x should be the geometric average of c 2 1 ψ 1 2 and c 2 2 ψ 2) 2, that is: c 2 1 ψ 1 2 c 2 2 ψ 2 2 = c 1 c 2 ψ 1 ψ 2. So, when A and B open simultaneously, the density of particles in the neighborhood of x is: ψ ψ c 1 c 2 ψ 1 ψ 2. When experimental conditions make c 1 c 2 ψ 1 ψ 2 = 2Re< ψ 1, ψ 2 > hold, the following equation will hold. ψ ψ Re< ψ 1, ψ 2 > = ψ 1 + ψ 2 2. This means that, under certain conditions, when slits A and B are opened simultaneously, ψ 1 + ψ 2 2 can be used to describe the particle density in the neighborhood of position x Single-slit diffraction In the double-slit experiment, when only one slit is open, large identical particles pass through and are modulated by the slit. In the statistical sense, some of these particles belong to Pattern A, and their screen positions correspond to integers. So, the screen positions of Pattern A particles have a tendency to arrive at the integer-mapped position. For Pattern B particles, the screen positions are randomly distributed; thus their position on the screen is well-proportioned. So, when the experimental conditions are suitable, Pattern A particles increase, and diffraction stripes appear on the screen. By the Riemann-Lebesgue lemma, when k, in equation (10), A 0. This means if k is small, then there is a greater probability of Pattern A particles appearing in the neighborhood of the integer-mapped position x. Thus, when k = 0,

9 Analysis for double-slit experiments 9 the maximum probability for the appearance of Pattern A particles in the neighborhood of x occurs. When k increases, A decreases; this means the probability for Pattern A particles appearing in the neighborhood of x decreases. Thus, in (b) of FIG. 1, at the position which corresponds to the slit, a maximum particle density distribution exists with density peak values decreasing as the distance from the slit increases. When doing the experiment with larger mass particles, the particles momentum increases, and A 0. This means the probability of finding particles at the origin O increases. So, larger mass particles reveal weaker wave features. Due to the symmetry of the experimental system, the particle s density distribution curve is symmetrical Double-slit interference When slit A and B are opened simultaneously, spatial symmetry is enhanced, and more Type (III) particles pass through slit A or B arriving in the same neighborhood of the integer-mapped position x. In this situation, the density of Pattern A particles in the neighborhood of x is: ψ ψ c 1 c 2 ψ 1 ψ 2. If the experimental conditions are suitable, c 1 c 2 ψ 1 ψ 2 > 0 will hold. Thus, the interference phenomenon will be observed as shown in (c) of FIG. 1. When determining which slit a particle passes through, the particle s least action changes. The number of Pattern A particles decreases, and as a result, the particle s wave appearance weakens or disappears. 4. NEW UNDERSTANDING FOR THE DE BROGLIE EQUATION Suppose a classical simple harmonic wave with wavelength λ and frequency ν is propagated in the positive direction of coordinate x, then after time t, the relative displacement of an infinitesimal quantity of the medium to the balance position x is: y(x, t) = Ae i( 2π λ )x e i( 2πν)t. If a Pattern A particle with momentum p and displacement r arrives in the neighborhood of an integer-mapped screen position x, then x can be regarded as the mean position in the neighborhood. Since a particle s position is determined by the particle s least action, equation (10) can be used to describe the relative displacement between a particle s actual screen position and the mean position x. When comparing equation (10) with the classical simple harmonic wave function above, many similar characteristics are found. These include: 1) describing the relative displacement between particles and the mean position versus describing the relative displacement between an infinitesimal quantity of the medium and the balance position, 2) having the same equation form, and 3) causing the same physical phenomena (diffraction and interference). Therefore, equation (10) has the properties of the classical wave, and can be considered a classical simple harmonic wave

10 10 function. For a Pattern A particle, e iprπ/(2a) corresponds to the mean screen position x; when considering the classical wave function, e i2π λ x corresponds to the balance position x. So, e iprπ/(2a) corresponds to e i2π λ x. Thus, by corresponding dimensions, pπ 2a corresponds to 2π λ. Additionally, the relative displacement of a Pattern A particle to the mean position x is described by e 2a t. However, for a classical wave, the relative displacement Eπ i for an infinitesimal quantity of the medium from the balance position x is described by e i2πνt Eπ i. So, e 2a t corresponds to e i2πνt Eπ. By corresponding dimensions, 2a corresponds to 2πν. In equations (8) and (9) or in equation (10), consider k and k = 1, and the above two corresponding relationships are regarded as equal. We have: or pπ 2a = 2π λ, Eπ 2a = 2πν, λ = 4a p, ν = E 4a. (11) In equation (6) or (7), the dimension for the constant a is J s which has the same dimension as Planck s constant h(j s). Since this analysis applies to all double-slit experiments with different particles, equation (11) is universal. Comparison of equation (11) with the de Broglie equation reveals that they have the same properties. Thus, 4a = h. Also, comparing equations (11) and (10), the value of E in equation (10) should be: 0, hν, 2hν,... which agrees with Planck s hypothesis of quanta. 5. DERIVATION OF THE SCHRÖDINGER EQUATION In (a) of FIG. 1, when the screen moves forward or backward a small distance along line OO, the time Pattern A particles take to arrive in the neighborhood of an integer-mapped screen position x will increase or decrease. By equation (10), the average least action of Pattern A particles in the neighborhood of x is Then, by the principle of least action: Thus, From the above equation, ψ t equation (10), ψ t and ψ r is: = Eπ ψ(r, t) = Ae i(pr Et)π/(2a). δψ = ψ r 2ai ψ and r ( ψ ψ dr + dt = 0. t ψ t = ψ dr r dt. is connected with ψ r. For Pattern A particles, using pπ r ) = ( 2a )2 ψ. So, the relationship between ψ t ψ t = ia πm r ( ψ r ).

11 Letting = h 2π = 2a π Analysis for double-slit experiments 11, the above equation can be written as: i ψ t = 2 2 ψ 2m r 2. (12) Equation (12) describes variation for the average least action of Pattern A particles with time. That is, equation (12) reflects the variation of Pattern A s particle density in the neighborhood of x when the screen moves forward or backward. Comparing equation (12) with the Schrödinger equation for free particles, the same form and properties are found. They both describe the variation of a particle s density with time. So, equation (12) is equivalent to the Schrödinger equation for free particles. 6. DERIVATION FOR THE HYPOTHESIS OF PATH INTEGRALS For the hypothesis of the Feynman path integral, 17) in the double-slit experiments, when a free particle passes through one slit (A or B) and arrives at screen position x, the probability of finding the particle at x is: C 2 e is(r)/ 2. all paths where C is a constant. Using this hypothesis, there are many possible trajectories for a particle to arrive at x. If the particle passes along a possible trajectory r arriving at x, then s(r) in the above formula represents the particle s action where all paths means all possible trajectories should be taken into account when calculating s(r). Proof: If a free particle passing through one slit (A or B) arrives at an integer-dependent screen position x, then it is a Pattern A particle. From the ensemble interpretation, the hypothesis of the Feynman path integral should be statistical. This means the hypothesis should only be suitable for Pattern A particles. Using equation (10 ) and letting 2a π = in the neighborhood of x, every particle s least action can be written as ψ(r, t) = Ae ipr/ e iet/. where Ae ipr/ is the least action of the average position in the neighborhood of x making it a constant C, that is C = Ae ipr/. Suppose the number of particles in the neighborhood of x is n. Then, if Particle j(j = 1, 2,..., n) with energy E j takes time t j to move along its least action trajectory r j, it will have the least action ψ j when it arrives in the neighborhood of x. By equation (1), the actual least action of particle j is E j t j and can be denoted by s j (r j ) = E j t j. In the neighborhood of x, the position of Particle j is near x, so E j t j Et. Thus, in the neighborhood of x, the sum of the particle s least action can be written

12 12 as: or ψ = ψ 1 + ψ ψ n = C ψ = C n e ie jt j /, j=1 n e is j(r j )/. j=1 In e iet/, Et contains a nonnegative integer, so E j t j also contains this integer, and from the uniqueness of least action, s j = E j t j still represents the least action of Particle j. The above equation can also be written as ψ = C e is(r)/. all paths Because ψ n 2 describes the particle s density in the neighborhood of x and because ψ 2 is positively correlated to ψ n 2, ψ 2 can also describe the particle s density in the neighborhood of x. That is, the probability of finding particles in the neighborhood of x is: C 2 e is(r)/ 2. all paths This is slightly different from Feynman s hypothesis, as 1) this proof holds only for statistical occasions, 2) s(r) is a particle s least action-not its action, and 3) every particle has only one least action trajectory. 7. CONCLUSION For the double-slit experiments, in a statistical sense, there exists Pattern A particles with the particle s screen position in the neighborhood of an integer-mapped screen position. From this, a mathematical understanding for the particles wave appearances can be obtained. In the statistical sense, a particle s average least action, given by equation (10), has the same properties of physics and mathematics as the wave function given in Quantum Mechanics. In addition, if equation (10) is used, then the de Broglie equation, the Schrödinger equation and the Feynman hypothesis of path integrals can be derived. Thus, in the double-slit experiments, the wave phenomena are rooted in the particles s statistical behavior with the nature of the wave function being the particle s average least action. Because the principle of least action is universal, this conclusion is also universal. References 1) C. J. Davisson, L. H. Germer, Phys. Rev. 30 (1927), 705.

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