because a=月 The Fourier transformation -(品)”E The probability density in position space eter-a while in wave vector space awP-V臣ae-ar From the standard definition of Gaussian distribution u-2a学 we easily identify the standard deviation in r and k spaces,respectively =Va=Az 0k= 编4k We thus have △x·△k=1/2 △x.△p=/2 More strict calculations show that △r·△p2h/2 which is Heisenberg's Uncertainty Principle.So the Gaussian wave packets seem to saturate the bound!We need in near future some constants to evaluate the experiments related quantities 1eV=1.602×10-12erg=1.602×10-19 Joule hc 1973eV.A,ao =0.53A,me =0.51MeV/c2 and finally the fine structure constant a=e2/hc=1/137. 21
because Z +∞ −∞ e −αx2 dx = r π α The Fourier transformation ψ(x) = µ 1 2πα¶1/4 e ik0x e − x 2 4α The probability density in position space |ψ(x)| 2 = 1 √ 2πα e − x 2 2α while in wave vector space |ϕ(k)| 2 = r 2α π e −2α(k−k0) 2 From the standard definition of Gaussian distribution P(x) = 1 √ 2πσ2 e − (x−X) 2 2σ2 we easily identify the standard deviation in x and k spaces, respectively σx = √ α = ∆x σk = 1 √ 4α = ∆k We thus have ∆x · ∆k = 1/2 ∆x · ∆p = ~/2 More strict calculations show that ∆x · ∆p ≥ ~/2 which is Heisenberg’s Uncertainty Principle. So the Gaussian wave packets seem to saturate the bound! We need in near future some constants to evaluate the experiments related quantities 1eV = 1.602 × 10−12erg = 1.602 × 10−19Joule ~c = 1973eV · A, a ˚ 0 = 0.53A, m ˚ e = 0.51MeV/c2 and finally the fine structure constant α = e 2/~c = 1/137. 21
IV.PRINCIPLE OF SUPERPOSITION OF STATES Superposition is a common property of all kinds of linear waves.The most striking characters,the interference and the diffraction,are results of superposition of waves.The distinction between wave motion and particle dynamics lies on whether there exists su- perposition.The wave nature of micro-particle is recognized and confirmed just through experimental observation of interference of matter waves.It is not surprised that the wave funtions(the states)obey superposition principle. A.Superposition of 2-states Let and represent two different states of one micro-particle.The principle of superposition asserts that =a1+a22 is also a possible state of the micro-particle-the superposition state. Here we show an interesting animation of the double slit experiment for bullets,water waves and electrons which is from http://www.upscale.utoronto.ca/GeneralInterest/Harrison/DoubleSlit/DoubleSlit.html We summarize the main conclusions of the experiments. 1.The probability of an event in an ideal experiment is given by the square of the absolute value of a complex numberwhich is called the probability amplitude P=probability =probability amplitude P= 2.When an event can occur in several alternative ways,the probability amplitude for the event is the sum of the probability amplitude for each way considered separately. There is interference: 妙=奶+归 P=l1+22 22
IV. PRINCIPLE OF SUPERPOSITION OF STATES Superposition is a common property of all kinds of linear waves. The most striking characters, the interference and the diffraction, are results of superposition of waves. The distinction between wave motion and particle dynamics lies on whether there exists superposition. The wave nature of micro-particle is recognized and confirmed just through experimental observation of interference of matter waves. It is not surprised that the wave funtions (the states) obey superposition principle. A. Superposition of 2-states Let ψ1 and ψ2 represent two different states of one micro-particle. The principle of superposition asserts that ψ = a1ψ1 + a2ψ2 is also a possible state of the micro-particle - the superposition state. Here we show an interesting animation of the double slit experiment for bullets, water waves and electrons which is from http://www.upscale.utoronto.ca/GeneralInterest/Harrison/DoubleSlit/DoubleSlit.html We summarize the main conclusions of the experiments. 1. The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number ψ which is called the probability amplitude P = probability ψ = probability amplitude P = |ψ| 2 2. When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitude for each way considered separately. There is interference: ψ = ψ1 + ψ2 P = |ψ1 + ψ2| 2 22
3.If an experiment is performed which is capable of determining whether one or another alternative is actually taken,the probability of the event is the sum of the probabilities for each alternative.The interference is lost: P=P+P The more precisely you know through which slit the electron passes (particle nature),the more unlikely you will see the interference pattern (wave nature).This is the manifestation of Heisenberg uncertainty principle B.Superposition of more than 2 states We have in this case =a1吻+2+…=∑a4 where the coefficients a determine the relative magnitudes and relative phases among the participating waves. A special case,yet the most important one,is the superposition of monochromatic plane waves(with one-dimensional case as example).The superposition of several monochromatic plane waves of different p generates a superposition state ()=+...... Generalization:the value p of monochromatic plane wave ep/changes continuously from -o to+oo,the coefficients now are written as a function of p and denoted by p,t) r+∞ (,)=2J-✉ p(p,t)ehd中 which is nothing but the Fourier transformation C.Measurement of state and probability amplitude 1.Measurement of state The measurement of state is a rather subtle subject.Let us analyze it as follows 23
3. If an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. The interference is lost: P = P1 + P2 The more precisely you know through which slit the electron passes (particle nature), the more unlikely you will see the interference pattern (wave nature). This is the manifestation of Heisenberg uncertainty principle. B. Superposition of more than 2 states We have in this case ψ = a1ψ1 + a2ψ2 + · · · · · · = X i aiψi where the coefficients ai determine the relative magnitudes and relative phases among the participating waves. A special case, yet the most important one, is the superposition of monochromatic plane waves (with one-dimensional case as example). The superposition of several monochromatic plane waves of different p generates a superposition state ψ(x) = c1e ip1x/~ + c2e ip2x/~ + · · · · · · Generalization: the value p of monochromatic plane wave e ipx/~ changes continuously from −∞ to +∞, the coefficients now are written as a function of p and denoted by ϕ(p, t) ψ(x, t) = 1 √ 2π~ Z +∞ −∞ ϕ(p, t)e ipx/~ dp which is nothing but the Fourier transformation. C. Measurement of state and probability amplitude 1. Measurement of state The measurement of state is a rather subtle subject. Let us analyze it as follows: 23
1.There is a microscopic system in a particular statethat is to be measured 2.Measurement means that we try to understand the state by means of a process called measurement.This measurement is certainly a macroscopic process and the measured quantity is some kind of physical quantities such as position,momentum, energy,angular momentum,etc. 3.It can be anticipated that the measurement results will be a statistical one. We will discuss here only measurement of two basic physical quantities,the coordinate and the momentum p. 2.Measurement of coordinat To simplify the discussion,we restrict our discussion to one-dimensional case.In fact, this is just the Born's statistical interpretation.The probability for the measuring result to be between z and r+dr is P(z)dr=(r)v(r)dz The generalization to 3D case is straightforward.The probability for the measuring result to be between (y,z)and (z+dr,y+dy,z+dz)is "(r)v(r)dr A new light for comprehending wave function: Intensity of matter wave described byat certain point r is the probability density of finding particle there Amplitude of the matter wave(probability wave)described by(r,t)at certain point r is the probability amplitude of finding the particle there. 3.Measurement of momentum In this case,we consider the case that the state()is a superposition of monochromatic waves of different but discrete p values (x)=eph+c2eph. The apparatus is an idealized equipment that can analyze momenta exactly and diffracted them in different directions,e.g.,an idealized Bragg diffraction element described by Bragg formula 24
1. There is a microscopic system in a particular state ψ that is to be measured 2. Measurement means that we try to understand the state ψ by means of a process called measurement. This measurement is certainly a macroscopic process and the measured quantity is some kind of physical quantities such as position, momentum, energy, angular momentum, etc. 3. It can be anticipated that the measurement results will be a statistical one. We will discuss here only measurement of two basic physical quantities, the coordinate x and the momentum p. 2. Measurement of coordinate To simplify the discussion, we restrict our discussion to one-dimensional case. In fact, this is just the Born’s statistical interpretation. The probability for the measuring result to be between x and x + dx is P(x)dx = ψ ∗ (x)ψ(x)dx The generalization to 3D case is straightforward. The probability for the measuring result to be between (x, y, z) and (x + dx, y + dy, z + dz) is ψ ∗ (r)ψ(r)d 3 r. A new light for comprehending wave function: Intensity of matter wave described by |ψ| 2 at certain point r is the probability density of finding particle there Amplitude of the matter wave (probability wave) described by ψ(r, t) at certain point r is the probability amplitude of finding the particle there. 3. Measurement of momentum In this case, we consider the case that the state ψ(x) is a superposition of monochromatic waves of different but discrete p values ψ(x) = c1e ip1x/~ + c2e ip2x/~ · · · · · · The apparatus is an idealized equipment that can analyze momenta exactly and diffracted them in different directions, e.g., an idealized Bragg diffraction element described by Bragg formula 24
Ψ(x) Q FIG.10:Bragg diffraction element:Different momentum components are diffracted into different directions. FIG.11:Scattering of waves by crystal planes. nA=2dsin6 The probability for the outcome of p is proportional to the intensity of the i-th component cic=c2.Generalizing to arbitrary ID state with continuous p values we have z利=V2赢o0ei严 where(p,t)plays the role of coefficientsc in the case of continuous p.Hence the probability of finding the particle to have its momentum between p and p+dp P(p)dp="(p,t)o(p.t)dp=(p.t)dp 25
p p p1 2 3 ψ(x) FIG. 10: Bragg diffraction element: Different momentum components are diffracted into different directions. FIG. 11: Scattering of waves by crystal planes. nλ = 2d sin θ The probability for the outcome of p is proportional to the intensity of the i−th component c ∗ i ci = |ci | 2 . Generalizing to arbitrary 1D state with continuous p values we have ψ(x, t) = 1 √ 2π~ Z ϕ(p, t)e i ~ pxdp where ϕ(p, t) plays the role of coefficients ci in the case of continuous p. Hence the probability of finding the particle to have its momentum between p and p + dp P(p)dp = ϕ ∗ (p, t)ϕ(p, t)dp = |ϕ(p, t)| 2 dp 25