Lecture Notes on Quantum Mechanics-Part I IeThSnr ongthe Schrodinger Equation. Contents 1.wave ad I Motion 2 34 II.Statistical Interpretation of Wave Mechanics 556 8 891 IIL.Moment nd Ine 224 ty amplitude t of coordinate 556616778 D.Expectation value of dynamical quantity v.ehridmganatiac 202 VI.Review on basic concepts in quantum mechanics 25 L.INTRODUCTION:MATTER WAVE AND ITS MOTION The emergence and development of quantum mechanics began in early years of the previous century and accom-
Lecture Notes on Quantum Mechanics - Part I Yunbo Zhang Institute of Theoretical Physics, Shanxi University This is the first part of my lecture notes. I mainly introduce some basic concepts and fundamental axioms in quantum theory. One should know what we are going to do with Quantum Mechanics - solving the Schr¨odinger Equation. Contents I. Introduction: Matter Wave and Its Motion 1 A. de Broglie’s hypothesis 2 B. Stationary Schr¨odinger equation 3 C. Conclusion 4 II. Statistical Interpretation of Wave Mechanics 5 A. Pose of the problem 5 B. Wave packet - a possible way out? 6 C. Born’s statistical interpretation 7 D. Probability 8 1. Example of discrete variables 8 2. Example of continuous variables 9 E. Normalization 11 III. Momentum and Uncertainty Relation 12 A. Expectation value of dynamical quantities 12 B. Examples of uncertainty relation 14 IV. Principle of Superposition of states 15 A. Superposition of 2-states 15 B. Superposition of more than 2 states 16 C. Measurement of state and probability amplitude 16 1. Measurement of state 16 2. Measurement of coordinate 17 3. Measurement of momentum 17 D. Expectation value of dynamical quantity 18 V. Schr¨odinger Equation 19 A. The quest for a basic equation of quantum mechanics 19 B. Probability current and probability conservation 20 C. Stationary Schr¨odinger equation 22 VI. Review on basic concepts in quantum mechanics 25 I. INTRODUCTION: MATTER WAVE AND ITS MOTION The emergence and development of quantum mechanics began in early years of the previous century and accomplished at the end of the twentieth years of the same century. We will not trace the historical steps since it is a long story. Here we try to access the theory by a way that seems to be more ”natural” and more easily conceivable
2 A.de Broglie's hypothesis Inspiration:Parallelism between light and matter Wave:frequency ,w,wavelength A,wave vectork..... Particle:velocity v,momentum p,energy c of Yet,tl (corue ture .Matter particles should also posses another side of ture-the wave nature g=hv=hw 卫== A=h/p h/V2mE The state of(micro)-particle should be described by a wave function.Here are some examples of state functions: 1.Free particle of definite momentum and energy is described by a monochromatic traveling wave of definite wave vector and frequency a=cs(红x-2t+0） A'cos(kr-wt+o) =Am(后r-t+0) Replenish an imaginary part 物=A'si血（肛-方t+o we get the final form of wave function =V1+it=A'chevetpe-tu=Aetpre-tet which is the wave picture of free particle.In 3Dwe have 2.Hydrogen atom in the ground state will be shown later to be e-()”- This wave function shows that the motion of electron is in a "standing wave"state.This is the wave picture of above state
2 A. de Broglie’s hypothesis Inspiration: Parallelism between light and matter Wave: frequency ν, ω, wavelength λ, wave vector k · · · · · · Particle: velocity v, momentum p, energy ε · · · · · · • Light is traditionally considered to be a typical case of wave. Yet, it also shows (possesses) a corpuscle nature - light photon. For monochromatic light wave ε = hν = ~ω p = hν c = h λ = ~k, (k = 2π λ ) • Matter particles should also possess another side of nature - the wave nature ε = hν = ~ω p = h λ = ~k This is call de Broglie’s Hypothesis and is verified by all experiments. In the case of non-relativistic theory, the de Broglie wavelength for a free particle with mass m and energy ε is given by λ = h/p = h/√ 2mε The state of (micro)-particle should be described by a wave function. Here are some examples of state functions: 1. Free particle of definite momentum and energy is described by a monochromatic traveling wave of definite wave vector and frequency ψ1 = A ′ cos ( 2π λ x − 2πνt + φ0 ) = A ′ cos (kx − ωt + φ0) = A ′ cos ( 1 ~ px − 1 ~ εt + φ0 ) Replenish an imaginary part ψ2 = A ′ sin ( 1 ~ px − 1 ~ εt + φ0 ) , we get the final form of wave function ψ = ψ1 + iψ2 = A ′ e iφ0 e i ~ pxe − i ~ εt = Ae i ~ pxe − i ~ εt which is the wave picture of motion of a free particle. In 3D we have ψ = Ae i ~ ⃗p·⃗re − i ~ εt 2. Hydrogen atom in the ground state will be shown later to be ψ(r, t) = ( 1 πa3 0 )1/2 e − r a0 e − i ~ E1t This wave function shows that the motion of electron is in a ”standing wave” state. This is the wave picture of above state
3 B A FIG.1:Maupertuis'Principle B.Stationary Schradinger equation The parallelism between light and matter can go further Light: wave nature omitted geometric optics wave nature can not be omitted new mechanics namely quantum mechanics Quantum Mechanics二Wave Optics☐ Light wave geometric optics Fermat's principle wave propagati Matter wav particle dynam Principle of least action wave propagation I'resently unknown Now conide light wave propagation in a non-homogenousmedium B light path=n(问d Fermat's principle n=0 For a particle moving inapotential field V(),the principle of least action reads iV2mia=iV2me-v阿as=0 The corresponding light wave quation (Helmholtzequation)is (（-)保利-0
3 A B FIG. 1: Maupertuis’ Principle. B. Stationary Schr¨odinger equation The parallelism between light and matter can go further Light: wave nature omitted geometric optics wave nature can not be omitted wave optics Matter: wave nature omitted particle dynamics wave nature can not be omitted a new mechanics, namely quantum mechanics Particle Dynamics ⇐⇒ Geometric Optics ⇓ ⇓ Quantum Mechanics ? ⇐⇒ Wave Optics Here we make comparison between light propagation of monochromatic light wave and wave propagation of monochromatic matter wave Light wave geometric optics Fermat’s principle wave propagation Helmholtz equation Matter wave particle dynamics Principle of least action wave propagation Presently unknown Now consider light wave propagation in a non-homogeneous medium light path = ∫ B A n(⃗r)ds Fermat’s principle δ ∫ B A n(⃗r)ds = 0 For a particle moving in a potential field V (⃗r), the principle of least action reads δ ∫ B A √ 2mT ds = δ ∫ B A √ 2m(E − V (⃗r))ds = 0 The corresponding light wave equation (Helmholtz equation) is ( ∇2 − 1 c 2 ∂ 2 ∂t2 ) u (⃗r, t) = 0
which after the separation of variables reduces to w+-0 Here we note thatis a constant.Thus we arrived at a result of comparison as follows 3 y2m(E-V(F))ds=06 n()ds=0 Pruko☐名2+妥=司 威the e e广6cmea 720+A2m(E-V(川中=0 Substitute the known free particle sotion -er into the above equation.We find the unknown constant A equals to1/,therefore we have 2+0E-V=0 o the general cas.It inaform -+v= and bears the name"Stationary Schrodinger Equation" C.Conclusion quation. rom ti side of the motion of micro-particl.V2mvm(V(trented a refraction index of matter wave forhat ialdtricstV()What appe .Clasical mechanics:particles with total energy Ecan not arrive at places with E<V( r),t at the attenuated in its course of propagation
4 which after the separation of variables reduces to ∇2ψ + n 2ω 2 c 2 ψ = 0. Here we note that ω is a constant. Thus we arrived at a result of comparison as follows δ ∫ B A √ 2m(E − V (⃗r))ds = 0 ⇐⇒ δ ∫ 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 = √ 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? • 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
Double Slit FIG.:Double II.STATISTICAL INTERPRETATION OF WAVE MECHANICS aves 3 made is hypothesis ectrons.r ons.etc. around stal Wave function is a complex function of its variables (红，t)=Ae-B到) v(r0.00-Viogze-t 1.Dynamical equation governing the motion of micro-particle is by itself a equation containing imaginary number This able the app be of space is ible fr A.Pose of the problem What kind of wave it is? .Optics:Electromagnetic wave wave propagating E(红，t)=oe(-2叫=%-网 Eo-amplitudefield strength Intensity E-energy density ·Acoustic wave U(,t)foloe(2)
5 FIG. 2: Double slit experiments. 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. Wave function is a complex function of its variables ψ(x, t) = Ae i ~ (px−Et) ψ (r, θ, ϕ, t) = 1 √ 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)