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Lecture Notes on Quantum Mechanics - Part I

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Lecture Notes on Quantum Mechanics - Part I 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¨odingerEquation. Contents I. Introduction: Matter Wave and Its Motion 1 A. de Broglie's hypothesis 2 B. Stationary Schr¨odingerequation 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¨odingerEquation 19 A. The quest for a basic equation of quantum mechanics 19 B. Probability current and probability conservation 20 C. Stationary Schr¨odingerequation 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 accom- plished 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 ν; !; wavelength λ, wave vector k ······ Particle: velocity v; momentum p; energy " ······ • 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 " = hν = ~! hν h 2π p = = = ~k; (k = ) c λ λ • Matter particles should also possess another side of nature - the wave nature " = hν = ~! h p = = ~k λ 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 " is given by p λ = h=p = h= 2m" 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 ( ) 2π = A0 cos x − 2πνt + ' 1 λ 0 = A0 cos (kx − !t + ' ) ( 0 ) 0 1 − 1 = A cos ~px ~"t + '0 Replenish an imaginary part ( ) 0 1 − 1 2 = A sin ~px ~"t + '0 ; we get the final form of wave function i i i i 0 i'0 ~ px − ~ "t ~ px − ~ "t = 1 + i 2 = A e e e = Ae e which is the wave picture of motion of a free particle. In 3D we have i · − i = Ae ~ ~p ~re ~ "t 2. Hydrogen atom in the ground state will be shown later to be ( )1=2 1 − r − i E t a0 ~ 1 (r; t) = 3 e e πa0 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 Schr¨odingerequation 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 + + 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 Z B light path = n(~r)ds A Fermat's principle Z B δ n(~r)ds = 0 A For a particle moving in a potential field V (~r), the principle of least action reads Z Z B p B p δ 2mT ds = δ 2m(E − V (~r))ds = 0 A A The corresponding light wave equation (Helmholtz equation) is ( ) 1 @2 r2 − u (~r; t) = 0 c2 @t2 4 which after the separation of variables reduces to n2!2 r2 + = 0: c2 Here we note that ! is a constant. Thus we arrived at a result of comparison as follows R p R B − () B δ A 2m(E V (~r))ds = 0 δ A n(~r)ds = 0 + + ()? r2 n2!2 Presently unknown + c2 = 0 + Presumed to be of the form r2 + An2 = 0 p 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 r2 + A [2m(E − V (~r))] = 0 Substitute the known free particle solution i · = e ~ ~p ~r 1 E = ~p2 2m V (~r) = 0 into the above equation. We find the unknown constant A equals to 1=~2, therefore we have 2m r2 + [E − V (~r)] = 0 ~2 for the general case. It is often written in a form ~2 − r2 (~r) + V (~r) (~r) = E (~r) 2m and bears the name "Stationary Schr¨odingerEquation" 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¨odingerEquation. Fromp the procedurep 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. 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 i (px−Et) (x; t) = Ae ~ 1 − r − i E t (r; θ; ϕ, t) = p e a0 e ~ 1 3 πa0 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 wavez propagating}| { i( 2π x−2πνt) i(kx−!t) E(x; t) =y ^0E0e λ =y ^0E0 e E0 − amplitude ! field strength 2 ! Intensity E0 energy density • Acoustic wave i( 2π x−2πνt) U(x; t) =y ^0U0e λ 6 FIG. 3: Electromagnetic Wave Propagation. U0 − amplitude ! mechanical displacement 2 ! Intensity U0 energy density • Wave function i 2π x−2πνt i (px−Et) (x; t) = Ae ( λ ) = Ae ~ Scalar wave, Amplitude - A !? Intensity - A2 !? 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 eik0x− 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 ( A; k − ∆k < k < k + ∆k '(k) = 0 0 0; elsewhere Z 1 (x) = p '(k)eikxdk 2π Z 1 k0+∆k = p Aeikxdk 2π k0−∆k 1 sin (∆kx) = p 2A eik0x | 2π {z x } Amplitude Both wave number and spatial position have a spread - uncertainty relation ∆x ∼ π=∆k; ∆x · ∆k ≈ π ∆x · ∆p ≈ π~ 7 FIG. 4: Wave packet formed by plane waves with different frequency. FIG. 5: Spread of a wave packet. • Superposition of waves with wave number in a Gaussian packet ( )1=4 2α − − 2 '(k) = e α(k k0) π Problem 1 The Fourier transformation of '(k) is also Gaussian.
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