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