6∫R√2m(E-V(ds=0←→6∫n(ds=0 Presently unknown ☐凸2+学=0 Presumed to be of the form 720+An20=0 Here A is an unknown constant,and the expression v2mT plays the role of"index of refraction"for the propagation of matter waves.The unknown equation now can be written as 72地+A2m(E-V(川中=0 Substitute the known free particle solution 沙=err B= V(⊙=0 into the above equation.We find the unknown constant A equals to 1/h2,therefore we have 2+01B-v(=0 for the general case.It is often written in a form -en+vnen and bears the name"Stationary Schrodinger Equation' C.Conclusion By a way of comparison,we obtained the equation of motion for a particle with definite energy moving in an external potential field V()-the Stationary Schrodinger Equation. From the procedure we stated above,here we stressed on the"wave propagation"side of the motion of micro-particle.v2mT=V2m(E-V()is treated as"refraction index"of matter waves It may happen for many cases that in some spatial districts E is less than V(),i.e.. E<V ()What will happen in such cases?
δ R B A p 2m(E − V (~r))ds = 0 ⇐⇒ δ R B A n(~r)ds = 0 ⇓ ⇓ Presently unknown ? ⇐⇒ ∇2ψ + n 2ω 2 c 2 ψ = 0 ⇓ Presumed to be of the form ∇2ψ + An2ψ = 0 Here A is an unknown constant, and the expression √ 2mT plays the role of ”index of refraction” for the propagation of matter waves. The unknown equation now can be written as ∇2ψ + A [2m(E − V (~r))] ψ = 0 Substitute the known free particle solution ψ = e i ~ ~p·~r E = 1 2m ~p 2 V (~r) = 0 into the above equation. We find the unknown constant A equals to 1/~ 2 , therefore we have ∇2ψ + 2m ~ 2 [E − V (~r)] ψ = 0 for the general case. It is often written in a form − ~ 2 2m ∇2ψ (~r) + V (~r) ψ (~r) = Eψ (~r) and bears the name ”Stationary Schr¨odinger Equation” C. Conclusion By a way of comparison, we obtained the equation of motion for a particle with definite energy moving in an external potential field V (~r) - the Stationary Schr¨odinger Equation. From the procedure we stated above, here we stressed on the ”wave propagation” side of the motion of micro-particle. √ 2mT = p 2m(E − V (~r)) is treated as ”refraction index” of matter waves. It may happen for many cases that in some spatial districts E is less than V (~r), i.e., E < V (~r). What will happen in such cases? 6
Double Slit FIG.2:Double slit experiments. .Classical mechanics:particles with total energy E can not arrive at places with E< v(). .Matter waves (Q.M.):matter wave can propagate into districts E<V(),but in that cases,the refraction index becomes imaginary. About Imaginary refraction inder:For light propagation,imaginary refraction index means dissipation,light wave will be attenuated in its course of propagation.For mat- ter waves,no meaning of dissipation,but matter wave will be attenuated in its course of propagation. II.STATISTICAL INTERPRETATION OF WAVE MECHANICS People tried hard to confirm the wave nature of micro-particles,and electron waves were first demonstrated by measuring diffraction from crystals in an experiment by Davison and Germer in 1925.They scattered electrons off a Nickel crystal which is the first experiment to show matter waves 3 years after de Broglie made his hypothesis.Series of other experiments provided more evidences,such as the double slit experiments using different particle beams: photons,electrons,neutrons,etc.and X-ray (a type of electromagnetic radiation with wavelengths of around 10-10 meters).Diffraction off polycrystalline material gives concentric rings instead of spots when scattered off single crystal
FIG. 2: Double slit experiments. • Classical mechanics: particles with total energy E can not arrive at places with E < V (~r). • Matter waves (Q.M.): matter wave can propagate into districts E < V (~r), but in that cases, the refraction index becomes imaginary. About Imaginary refraction index : For light propagation, imaginary refraction index means dissipation, light wave will be attenuated in its course of propagation. For matter waves, no meaning of dissipation, but matter wave will be attenuated in its course of propagation. II. STATISTICAL INTERPRETATION OF WAVE MECHANICS People tried hard to confirm the wave nature of micro-particles, and electron waves were first demonstrated by measuring diffraction from crystals in an experiment by Davison and Germer in 1925. They scattered electrons off a Nickel crystal which is the first experiment to show matter waves 3 years after de Broglie made his hypothesis. Series of other experiments provided more evidences, such as the double slit experiments using different particle beams: photons, electrons, neutrons, etc. and X-ray (a type of electromagnetic radiation with wavelengths of around 10−10 meters). Diffraction off polycrystalline material gives concentric rings instead of spots when scattered off single crystal. 7
Wave function is a complex function of its variables (红，t)=Aetr-E到 b(r,0,0,t)= Vrage e-fu 1 1.Dynamical equation governing the motion of micro-particle is by itself a equation containing imaginary number 2.The wave function describing the state of micro-particle must fit the general theory frame of quantum theory (operator formalism)-requirement of homogeneity of space This means,the symmetry under a translation in spacera,where a is aconstant vector,is applicable in all isolated systems.Every region of space is equivalent to every other,or physical phenomena must be reproducible from one location to another. A.Pose of the problem What kind of wave it is? Optics:Electromagnetic wave wave propagating 卫，0=功e(-2aw=%,可 Eo-amplitude-field strength Intensity Eenergy density 。Acoustic wave U(x,t)=oUoe(号-2r) Uo-amplitude-mechanical displacement Intensity Uenergy density ·Wave function (x,t)=Ae(r-2红叫）=Aer-B到)
Wave function is a complex function of its variables ψ(x, t) = Ae i ~ (px−Et) ψ (r, θ, φ, t) = 1 p πa3 0 e − r a0 e − i ~ E1t 1. Dynamical equation governing the motion of micro-particle is by itself a equation containing imaginary number 2. The wave function describing the state of micro-particle must fit the general theory frame of quantum theory (operator formalism) - requirement of homogeneity of space. This means, the symmetry under a translation in space r → r+a, where a is a constant vector, is applicable in all isolated systems. Every region of space is equivalent to every other, or physical phenomena must be reproducible from one location to another. A. Pose of the problem What kind of wave it is? • Optics: Electromagnetic wave E(x, t) = ˆy0E0e i( 2π λ x−2πνt) = ˆy0E0 wave propagating z }| { e i(kx−ωt) E0 − amplitude → field strength Intensity E 2 0 → energy density • Acoustic wave U(x, t) = ˆy0U0e i( 2π λ x−2πνt) U0 − amplitude → mechanical displacement Intensity U 2 0 → energy density • Wave function ψ(x, t) = Aei( 2π λ x−2πνt) = Ae i ~ (px−Et) 8
pic】 FIG.3:Electromagnetic Wave Propagation. Scalar wave,Amplitude-A→？ Intensity-A2? Early attempt:Intensitymaterial density,particle mass distributed in wave.But wave is endless in space,how can it fit the idea of a particle which is local. B.Wave packet-a possible way out? particle=wave packet-rain drop -an endless train,How can it be connected with particle picture? Two examples of localized wave packets .Superposition of waves with wave number between (ko-Ak)and (ko+k)-square packet A,k0-△k<k<k0+△k P()= 0, elsewhere (x)= V交o)ek rk知+△k Aeikrdk 2Am(△keh V2 Amplitude 9
FIG. 3: Electromagnetic Wave Propagation. Scalar wave, Amplitude - A →? Intensity - A 2 →? Early attempt: Intensity → material density, particle mass distributed in wave. But wave is endless in space, how can it fit the idea of a particle which is local. B. Wave packet - a possible way out? particle = wave packet - rain drop e ik0x− an endless train, How can it be connected with particle picture? Two examples of localized wave packets • Superposition of waves with wave number between (k0 − ∆k) and (k0 + ∆k) - square packet ϕ(k) = A, k0 − ∆k < k < k0 + ∆k 0, elsewhere ψ(x) = 1 √ 2π Z ϕ(k)e ikxdk = 1 √ 2π Z k0+∆k k0−∆k Aeikxdk = 1 √ 2π 2A sin (∆kx) x | {z } Amplitude e ik0x 9
M N AkAx 1 FIG.4:Wave packet formed by plane waves with different frequency. Plane wave:lk /ave packet:ko and FIG.5:Spread of a wave packet Both wave number and spatial position have a spread-uncertainty relation △x~T/△k, △x·△k≈T △x·△p≈rh .Superposition of waves with wave number in a Gaussian packet =(） e-a(k-bo)2 Problem 1 The Fourier transformation of(k)is also Gaussian.It is the best one can do to localize a particle in position and momentum spaces at the same time.Find the root mean square(RMS)deviation△x·△p=? 10
FIG. 4: Wave packet formed by plane waves with different frequency. FIG. 5: Spread of a wave packet. Both wave number and spatial position have a spread - uncertainty relation ∆x ∼ π/∆k, ∆x · ∆k ≈ π ∆x · ∆p ≈ π~ • Superposition of waves with wave number in a Gaussian packet ϕ(k) = µ 2α π ¶1/4 e −α(k−k0) 2 Problem 1 The Fourier transformation of ϕ(k) is also Gaussian. It is the best one can do to localize a particle in position and momentum spaces at the same time. Find the root mean square (RMS) deviation ∆x · ∆p =? 10