Lecture Notes on Quantum Mechanics-Part I Yunbo Zhang Institute of Theoretical Physics,Shanri University Abstract 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 Schrodinger Equation. 1
Lecture Notes on Quantum Mechanics - Part I Yunbo Zhang Institute of Theoretical Physics, Shanxi University Abstract 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. 1
Contents I.Introduction:Matter Wave and Its Motion 3 A.de Broglie's hypothesis 3 B.Stationary Schrodinger equation C.Conclusion 6 II.Statistical Interpretation of Wave Mechanics A.Pose of the problem B.Wave packet-a possible way out? 9 C.Born's statistical interpretation 11 D.Probability 11 1.Example of discrete variables 11 2.Example of continuous variables 14 E.Normalization III.Momentum and Uncertainty Relation 18 A.Expectation value of dynamical quantities B.Examples of uncertainty relation 20 IV.Principle of Superposition of states 22 A.Superposition of 2-states 22 B.Superposition of more than 2 states 23 C.Measurement of state and probability amplitude 23 1.Measurement of state 23 2.Measurement of coordinate 24 3.Measurement of momentum 24 D.Expectation value of dynamical quantity 名 V.Schrodinger Equation 27 A.The quest for a basic equation of quantum mechanics 27 B.Probability current and probability conservation 29 C.Stationary Schrodinger equation 31 2
Contents I. Introduction: Matter Wave and Its Motion 3 A. de Broglie’s hypothesis 3 B. Stationary Schr¨odinger equation 4 C. Conclusion 6 II. Statistical Interpretation of Wave Mechanics 7 A. Pose of the problem 8 B. Wave packet - a possible way out? 9 C. Born’s statistical interpretation 11 D. Probability 11 1. Example of discrete variables 11 2. Example of continuous variables 14 E. Normalization 16 III. Momentum and Uncertainty Relation 18 A. Expectation value of dynamical quantities 18 B. Examples of uncertainty relation 20 IV. Principle of Superposition of states 22 A. Superposition of 2-states 22 B. Superposition of more than 2 states 23 C. Measurement of state and probability amplitude 23 1. Measurement of state 23 2. Measurement of coordinate 24 3. Measurement of momentum 24 D. Expectation value of dynamical quantity 26 V. Schr¨odinger Equation 27 A. The quest for a basic equation of quantum mechanics 27 B. Probability current and probability conservation 29 C. Stationary Schr¨odinger equation 31 2
VI.Review on basic concepts in quantum mechanics I.INTRODUCTION:MATTER WAVE AND ITS MOTION The emergence and development of quantum mechanics began in early years of the pre- vious 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. A.de Broglie's hypothesis Inspiration:Parallelism between light and matter Wave:frequency,w,wavelength入，wave vectork.… Particle:velocity v,momentum p,energy s...... .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 e=hv=hw g-费林，k=贸 .Matter particles should also possess another side of nature-the wave nature e=hv=hw P=1=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 e is given by 入=h/p=h/W2me The state of (micro)-particle should be described by a wave function.Here are some examples of state functions: 3
VI. Review on basic concepts in quantum mechanics 36 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. 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: 3
1.Free particle of definite momentum and energy is described by a monochromatic trav- eling wave of definite wave vector and frequency =cs(臣-2an+） =A'cos (kx-wt+po) =cs(房r-方t+o） Replenish an imaginary part 约=sm(r-方t+0) we get the final form of wave function 少=十i的=A'cipe严e-et=Aeipre-te which is the wave picture of motion of a free particle.In 3D we have =Aetfre-fet 2.Hydrogen atom in the ground state will be shown later to be 9-() ee-t This wave function shows that the motion of electron is in a"standing wave"state This is the wave picture of above state. B.Stationary Schrodinger 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 a new mechanics. wave nature can not be omitted namely quantum mechanics Particle Dynamics←→Geometric Optics 业 Quantum Mechanics←凸Wave Optics
1. Free particle of definite momentum and energy is described by a monochromatic traveling wave of definite wave vector and frequency ψ1 = A 0 cos µ 2π λ x − 2πνt + ϕ0 ¶ = A 0 cos (kx − ωt + ϕ0) = A 0 cos µ 1 ~ px − 1 ~ εt + ϕ0 ¶ Replenish an imaginary part ψ2 = A 0 sin µ 1 ~ px − 1 ~ εt + ϕ0 ¶ , we get the final form of wave function ψ = ψ1 + iψ2 = A 0 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) = µ 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. 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 4
A FIG.1:Maupertuis'Principle. 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 propagationHelmholtz equation Matter waveparticle dynamics Principle of least action wave propagation Presently unknown Now consider light wave propagation in a non-homogeneous medium light path= hn()ds Fermat's principle 6n问ds=0 For a particle moving in a potential field V(,the principle of least action reads 5amns=6v6me-阿w=0 The corresponding light wave equation (Helmholtz equation)is (-)0=0 which after the separation of variables reduces to + Here we note that w is a constant.Thus we arrived at a result of comparison as follows 5
A B FIG. 1: Maupertuis’ Principle. 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 = Z B A n(~r)ds Fermat’s principle δ Z B A n(~r)ds = 0 For a particle moving in a potential field V (~r), the principle of least action reads δ Z B A √ 2mT ds = δ Z B A p 2m(E − V (~r))ds = 0 The corresponding light wave equation (Helmholtz equation) is µ ∇2 − 1 c 2 ∂ 2 ∂t2 ¶ u (~r, t) = 0 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 5
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