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A First Look at Quantum Physics

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1 A First Look at Quantum Physics
2006 Quantum Mechanics Prof. Y. F. Chen

2 A First Look at Quantum Physics
Historical Note at the end of the 19th century, the overwhelming success of classical physics – CM, EM, TD made people believe the ultimate description of nature has been achieved. at the turn of the 20th century, classical physics was challenged by Relativity & microphysics. the series of breakthroughs: (1) Max Planck → the energy of a quantum:the energy exchange between an EM wave & matter occurs only in integer multiples of 2006 Quantum Mechanics Prof. Y. F. Chen

3 A First Look at Quantum Physics
Historical Note (2) Einstein → photon:light itself is made of discrete bits of energy; an explanation to the photoelectric problem. (3) Neils Bohr → model of hydrogen atom:atoms can be found only in discrete states of energy & atoms with radiation takes place only in discrete amounts of ν. Rutherford’s model Bohr’s model 2006 Quantum Mechanics Prof. Y. F. Chen

4 A First Look at Quantum Physics
Historical Note (4) Compton → scattering X-rays with e-:the X-ray photons behave like particles with momenta 2006 Quantum Mechanics Prof. Y. F. Chen

5 A First Look at Quantum Physics
Essential Relativity for a free particle of rest mass m moving at speed υ, the total energy E, momentum p, and kinetic energy T can be written in the relativistically correct forms where using & 2006 Quantum Mechanics Prof. Y. F. Chen

6 A First Look at Quantum Physics
Essential Relativity in QM the momentum is a more natural variable than γ, a useful relation can be given by , the rest energies of various atomic particles will often be quoted in energy units; for the electron and proton the rest energies are given by the non-relativistic limit of E.g , where , is easily seen to be 2006 Quantum Mechanics Prof. Y. F. Chen

7 A First Look at Quantum Physics
Essential Relativity the ultra-relativistic limit when , can be approximated to be , which is also seen to be consistent with the energy-momentum relation for photons, namely (i) e- in atoms:when the relativistic effects become non-negligible. (ii) deuteron: for the simplest nuclear system; compared with →deuteron can be considered as non-relativistic system 2006 Quantum Mechanics Prof. Y. F. Chen

8 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant spectral energy density of blackbody radiation at different temp. the peak of the radiation spectrum occurs at freq that is proportional to the temp. Wien’s displacement law: ideal blackbody spectral distribution only depends on temperature 2006 Quantum Mechanics Prof. Y. F. Chen

9 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant blackbody radiation: (1) Rayleigh’s energy density distribution: when the cavity is in thermal equilibrium, the EM energy density in to is given by according to the equipartition theorem of classical thermodynamics, all oscillators in the cavity have the same mean energy: → is integrate over all freq, the integral diverges → this result is absurd → called the ultraviolet catastrophe   2006 Quantum Mechanics Prof. Y. F. Chen

10 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant blackbody radiation: (2) Plank’s energy density distribution: avoiding the ultraviolet catastrophe, Planck considered that the energy exchange between radiation & matter must be discrete: the spectrum of the blackbody radiation reveals the quantization of radiation, notably the particle behavior of EM waves 2006 Quantum Mechanics Prof. Y. F. Chen

11 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant photoelectric effect: (1) 2006 Quantum Mechanics Prof. Y. F. Chen

12 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant photoelectric effect: (2) when a metal is irradiation with light, electrons may get emitted (3) it was fond that the magnitude of the photoelectric current thus generated is proportional to the intensity of the incident radiation, yet the speed of the electrons does not depend on the radiation’s intensity, but on its frequency. → the photoelectric effect provides compelling evidence for the corpuscular nature of the EM radiation 2006 Quantum Mechanics Prof. Y. F. Chen

13 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant Compton effects: Compton treated the incident radiation as a stream of particles-photons-colliding elastically with individual e-   2006 Quantum Mechanics Prof. Y. F. Chen

14 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant Compton effects: by momentum conservation & energy conservation → the Compton effect confirms that photons behave like particles; they collide with e- like material particles 2006 Quantum Mechanics Prof. Y. F. Chen

15 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant wave aspect of particles: de Broglie → the wave-particle duality is not restricted to radiation, but must be universal: all material particles should also display a dual wave-particle behavior: known as the de Broglie relation, connects the momentum of a particle with the wavelength & wave vector of the wave corresponding to this particle 2006 Quantum Mechanics Prof. Y. F. Chen

16 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant wave aspect of particles: Davission-Germer exp. confirmation of de Broglie’s hypothesis: the intensity max of the scattered e- corresponds to the Bragg formula 2006 Quantum Mechanics Prof. Y. F. Chen

17 Quantum Physics: as a Fundamental Constant
A First Look at Quantum Physics Quantum Physics: as a Fundamental Constant wave aspect of particles: de Broglie’s wavelength: For an Ni crystal, , 2006 Quantum Mechanics Prof. Y. F. Chen

18 Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics Semi-classical model – Bohr Model of H Atom Bohr’s assumption: (1) only a discrete set of circular stable orbit are allowed (2) the orbital angular momentum of the electron is an integer multiple of → (3-a) 2006 Quantum Mechanics Prof. Y. F. Chen

19 Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics Semi-classical model – Bohr Model of H Atom (3-b) 2006 Quantum Mechanics Prof. Y. F. Chen

20 Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics Semi-classical model – Bohr Model of H Atom (3-c) (3-d) 2006 Quantum Mechanics Prof. Y. F. Chen

21 Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics Semi-classical model – Bohr Model of H Atom discussion (i) classically, (ii) for a circular orbit, the attractive force = centrifugal force (iii) with , (iv) considering a transition from to , according to Einstein’s relation, , & the fractional of angular momentum is so small → with → from (ii), 2006 Quantum Mechanics Prof. Y. F. Chen

22 Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics Semi-classical model – Bohr Model of H Atom Bohr suggested that hold even for energy small quantum number. The allowed value of is the same for positive & negative values, this means that if a given value of the angular momentum is allowed, its negative must also be allowed. (a) if , then this criterion is satisfied, for (b) if , the allowed values are (c) with any other value of , however, this condition cannot be met. 2006 Quantum Mechanics Prof. Y. F. Chen

23 Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics Semi-classical model – Bohr Model of H Atom correspondence principle first given by Bohr: Bohr noted that the photons emitted in transitions between the quantized energy levels satisfy the Balmer formula, written is the form a classical particle undergoing circular acceleration would emit radiation at its orbital freq., which is given by “the connections & interpolations between the QM & classical description of physical are stressed in this course.” 2006 Quantum Mechanics Prof. Y. F. Chen

24 Semi-classical model – Bohr Model of H Atom
A First Look at Quantum Physics Semi-classical model – Bohr Model of H Atom correspondence principle & the classical period: (a) show that the correspondence principle can be generalized to show that the classical periodicity, , of a quantum system in the large limit is given by (b) using the expression for the quantized energies of a particle in a box length , find the classical period in state & compare it to the expectations based on the classical motion 2006 Quantum Mechanics Prof. Y. F. Chen

25 Semi-classical model – CM & QM
A First Look at Quantum Physics Semi-classical model – CM & QM (1) the relationship between CM & QM certain sense is similar to that which exist between geometric & wave optics (2) in QM the wave function of quasi-classical form; where is called action (3) the small parameter have is the ratio transition from QM to CM formally is described by the WKB-method at tends to 2006 Quantum Mechanics Prof. Y. F. Chen

26 Semi-classical model – CM & QM
A First Look at Quantum Physics Semi-classical model – CM & QM the analogy between Optics & Mechanics showed to be vary fruitful to produce very important physical insight the 1st analogy put geometrical optics in correspondence with CM the development of this analogy was the formulation of electron optics the formulation of electron optics is similar to EM geometrical optics provided to replace the motion of light rays & refractive index with electron rays and potential, respectively 2006 Quantum Mechanics Prof. Y. F. Chen

27 Semi-classical model – CM & QM
A First Look at Quantum Physics Semi-classical model – CM & QM the 2nd analogy is extended to the wave level, going from Optics to Mechanics by de Broglie & Schrodinger, obtaining the wave mechanics & subsequently the QM from CM to QM:Schrodinger eq. has been recognized as the non-relativistic limit of a more general wave mechanical formulation induced by the correspondence with optics. The non-relativistic limit of Klein-Gordon eq. is just the Schrodinger eq 2006 Quantum Mechanics Prof. Y. F. Chen

28 A First Look at Quantum Physics
Quantization Rules Wilson & Sommerfeld offered a scheme that included , & as special cases in essence their scheme consists in quantizing the action variable of classical mechanics phase integral for 1D, , since the particle goes from one limit of oscillation to the other and back 2006 Quantum Mechanics Prof. Y. F. Chen

29 Quantization Rules → the limit of oscillation are determined by .
A First Look at Quantum Physics Quantization Rules the limit of oscillation are determined by thus the quantities in the brackets vanish & so if , then the quantization of the action, J, is usually referred to as “the Bohr-Sommerfeld quantum condition.” 2006 Quantum Mechanics Prof. Y. F. Chen

30 Quantization Rules Ex: Harmonic oscillator ,
A First Look at Quantum Physics Quantization Rules Ex: Harmonic oscillator , if , then , Plank Quantization rule Ex: for an electron moving in a circular orbit of radius r. 2006 Quantum Mechanics Prof. Y. F. Chen

31 From Classical Waves to de Broglie Waves
From Classical Waves To de Brogile Waves From Classical Waves to de Broglie Waves 2006 Quantum Mechanics Prof. Y. F. Chen

32 From Classical Waves To de Brogile Waves
The Classical Wave Eq. the eq. of motion for the transverse displacement of a piece of a stretched string : (1)T is tension (2)r is the mass density (3)Y(x,t) is the amplitude of the string at position x & time t usually in the form , with wave speed Similarly→ Maxwell’s eq.: , with light speed 2006 Quantum Mechanics Prof. Y. F. Chen

33 The Classical Wave Eq. classical wave eq. (1)wave speed :v
From Classical Waves To de Brogile Waves The Classical Wave Eq. classical wave eq. (1)wave speed :v (2)solution: wave number k & angular frequency w → related to wavelength l & freq. f k = 2p/l w = 2pf k & w must satisfy w = v k. 2006 Quantum Mechanics Prof. Y. F. Chen

34 Wave Packets and de Broglie Matter waves
From Classical Waves To de Brogile Waves Wave Packets and de Broglie Matter waves Matter Waves: de Broglie → a material particle can be described by a wave function Waves Packets:light pulse > l ; light pulse < apparatus → light beam acts like particle localized in the pulse → constructed by a group of waves of slightly different l → interfere constructively over a small region localized wave packet: 2006 Quantum Mechanics Prof. Y. F. Chen

35 Wave Packets and de Broglie Matter waves
From Classical Waves To de Brogile Waves Wave Packets and de Broglie Matter waves t=0, → , → 1D: → x=0, =0 where Fig.2.1 Dk=1, x=-4p~4p 2006 Quantum Mechanics Prof. Y. F. Chen

36 Motion of Wave Packets non-dispersion propagation wave packet:
From Classical Waves To de Brogile Waves Motion of Wave Packets non-dispersion propagation wave packet: → w is proportional to k → , wave packet travels with constant dispersion propagation → k = 2p n(l)/l , (1) (2) group velocity → (3) 2006 Quantum Mechanics Prof. Y. F. Chen

37 The Spreading of a Wave Packets
From Classical Waves To de Brogile Waves The Spreading of a Wave Packets (1) de Broglie: → (2) Einstein: determine the time-evolution of the wave packet corresponding to a free particle with an initial Gaussian packet → substituting into → use (1) (2) → obtain , centered at where width increase with time 2006 Quantum Mechanics Prof. Y. F. Chen

38 The Spreading of a Wave Packets
From Classical Waves To de Brogile Waves The Spreading of a Wave Packets intensity corresponding to the wave packet where width 2006 Quantum Mechanics Prof. Y. F. Chen

39 The Schrödinger Wave Equation
2006 Quantum Mechanics Prof. Y. F. Chen

40 The Schrödinger Wave Eq.
The Schrödinger Wave Equation The Schrödinger Wave Eq. Classical Mechanics : Wave Mechanics = Geometrical optics: Wave optics classical mechanics ←→ Newton’s theory = geometrical optics wave (quantum) mechanics ←→ Huygen’s theory = wave optics quantum phenomena ←→ diffraction & interference classical mechanics quantum mechanics 2006 Quantum Mechanics Prof. Y. F. Chen

41 Time-independent Schrödinger Wave Eq.
The Schrödinger Wave Equation Time-independent Schrödinger Wave Eq. wave eq.: , solution is assumed to be sinusoidal, → Helmholtz eq. de Broglie relation → → time-indep. Schrödinger eq.: the appearance of  → Schrödinger imposed the “quantum condition” on the wave eq. of matter Erwin Schrödinger 2006 Quantum Mechanics Prof. Y. F. Chen

42 Time-dependent Schrödinger Wave Eq.
The Schrödinger Wave Equation Time-dependent Schrödinger Wave Eq. Einstein relation: also represents the particle energy Schrödinger found a 1st-order derivative in time consistent with the time-indep. Schrödinger eq. 2006 Quantum Mechanics Prof. Y. F. Chen

43 The Probability Interpretation
The Schrödinger Wave Equation The Probability Interpretation the probability density of finding the particle : wave function = field distribution its modulus square = probability density distribution ∵the particle must be somewhere, ∴total integrated = 1 (the wave function for the probability interpretation needs to be normalized. ) 2006 Quantum Mechanics Prof. Y. F. Chen

44 The Probability Interpretation
The Schrödinger Wave Equation The Probability Interpretation N identically particles, all described by the number of particles found in the interval at t : 2006 Quantum Mechanics Prof. Y. F. Chen

45 The Probability Current Density
The Schrödinger Wave Equation The Probability Current Density a time variation of in a region is conserved by a net change in flux into the region. → satisfy a continuity eq. : by analogy with charge conservation in electrodynamics, ←→ conservation of probability 2006 Quantum Mechanics Prof. Y. F. Chen

46 Role of the Phase of the Wave Function
The Schrödinger Wave Equation Role of the Phase of the Wave Function is related to the phase gradient of the wave function. , where = phase of the wave function the larger varies with space, the greater 2006 Quantum Mechanics Prof. Y. F. Chen

47 Role of the Phase of the Wave Function
The Schrödinger Wave Equation Role of the Phase of the Wave Function reveals that the is irrotational only when has no any singularities, which are the points of Conversely, the singularities of play a role of vortices to cause to be rotational. 2006 Quantum Mechanics Prof. Y. F. Chen

48 Wave Functions in Coordinate and Momentum Spaces
The Schrödinger Wave Equation Wave Functions in Coordinate and Momentum Spaces with normalized: is the probability of finding the momentum of the particle in in the neighborhood of p at time t 2006 Quantum Mechanics Prof. Y. F. Chen

49 Operators and Expectation values of Physical Variables
The Schrödinger Wave Equation Operators and Expectation values of Physical Variables expectation value of r & p: , find an expression for <p> in coordinate space: → <p> can be represented by the differential operator any function of p , & any function of r , can be given by: 2006 Quantum Mechanics Prof. Y. F. Chen

50 Operators and Expectation values of Physical Variables
The Schrödinger Wave Equation Operators and Expectation values of Physical Variables CM:all physical quantities can be expressed in terms of coordinates & momenta. QM: all physical quantities can be given by any physical operator in quantum mechanics needs to a Hermitian operator. 2006 Quantum Mechanics Prof. Y. F. Chen

51 Time Evolution of Expectation values & Ehrenfest’s Theorem
The Schrödinger Wave Equation Time Evolution of Expectation values & Ehrenfest’s Theorem the operators used in QM needs to be consistent with the requirement that their expectation values generally satisfy the laws of CM the time derivative of x can be given by: → integration by parts: → the classical relation between velocity and p holds for the expectation values of wave packets. 2006 Quantum Mechanics Prof. Y. F. Chen

52 Time Evolution of Expectation values & Ehrenfest’s Theorem
The Schrödinger Wave Equation Time Evolution of Expectation values & Ehrenfest’s Theorem Ehrenfes’s theorem:the time derivative of p can be given by → has a form like Newton’s 2nd law, written for expectation values for any operator A, the time derivative of <A> can be given by: (1) where is the Hamiltonian operator (2) the eq. is of the extreme importance for time evolution of expectation values in QM 2006 Quantum Mechanics Prof. Y. F. Chen

53 Stationary States & General Solutions of the Schrödinger Eq.
The Schrödinger Wave Equation Stationary States & General Solutions of the Schrödinger Eq. superposition of eigenstates:based on the separation of t & r & (1) E = eigenvalue (2) = eigenfunction (3) stationary states:if the initial state is represented by → , independent of t 2006 Quantum Mechanics Prof. Y. F. Chen

54 Formalism of Quantum Mechanics
Prof. Y. F. Chen

55 Formalism of Quantum Mechanics
linear DE.:the main foundation of QM consists in the Schrödinger eq. The formalism of QM deals with linear operators & wave functions that form a Hilbert space ch4 will focus on the Hermitian operators & the superposition properties of linear DE. in Hilbert space 2006 Quantum Mechanics Prof. Y. F. Chen

56 Definition of Inner Product & Hilbert Space
Formalism of Quantum Mechanics Definition of Inner Product & Hilbert Space inhomogeneous linear differential: (1) = linear differential operator acting upon (2) l = eigenvalue & = eigenfunction (3) is a weight function any function in this vector space can be expanded as        , =a set of linearly indep. basis functions inner product: 2006 Quantum Mechanics Prof. Y. F. Chen

57 Definition of Inner Product & Hilbert Space
Formalism of Quantum Mechanics Definition of Inner Product & Hilbert Space orthogonal:if , then & are orthogonal. the norm of : a basis of orthnormal, linearly independent basis functions satisfies 2006 Quantum Mechanics Prof. Y. F. Chen

58 Gram-Schmidt Orthogonalization
Formalism of Quantum Mechanics Gram-Schmidt Orthogonalization Gram-Schmidt orthogonalization: = linearly independent, not orthonormal basis = orthonormal basis produced by the Gram-Schmidt orthogonalization, in which is to be normalized 2006 Quantum Mechanics Prof. Y. F. Chen

59 Gram-Schmidt Orthogonalization
Formalism of Quantum Mechanics Gram-Schmidt Orthogonalization although the Gram-Schmidt procedure constructed an orthonormal set, are not unique. There is an infinite number of possible orthonormal sets. construct the first three orthonormal functions over the range : 2006 Quantum Mechanics Prof. Y. F. Chen

60 Gram-Schmidt Orthogonalization
Formalism of Quantum Mechanics Gram-Schmidt Orthogonalization it can be shown that          where is the nth-order Legendre polynomials the eq. for Gram-Schmidt orthogonalization tend to be ill-conditioned because of the subtractions. A method for avoiding this difficulty is to use the polynomial recurrence relation   2006 Quantum Mechanics Prof. Y. F. Chen

61 Definition of Self-Adjoint (Hermitian Operators)
Formalism of Quantum Mechanics Definition of Self-Adjoint (Hermitian Operators) the adjoint/Hermitian conjugate of a matrix A: from inner product space, the definition of the adjoint: the adjoint of an operator in inner product function: 2006 Quantum Mechanics Prof. Y. F. Chen

62 Definition of Self-Adjoint (Hermitian Operators)
Formalism of Quantum Mechanics Definition of Self-Adjoint (Hermitian Operators) self-adjoint/Hermitian operator: →(1) →(2) measurement of the physical quantity  : (1) → , is real (2) is not necessarily an eigenfunction of   2006 Quantum Mechanics Prof. Y. F. Chen

63 The Properties of Hermitian Operators
Formalism of Quantum Mechanics The Properties of Hermitian Operators (1) the eigenvalues of an hermitian operator are real (2) the eigenfunctions of an hermitian operator are orthogonal (3) the eigenfunctions of an hermitian operator form a complete set proof (1) & (2): × ×         integrating complex conjugate 2006 Quantum Mechanics Prof. Y. F. Chen

64 The Properties of Hermitian Operators
Formalism of Quantum Mechanics The Properties of Hermitian Operators proof (1) & (2): → if i=j, then → →   is real if i≠j, then → & are orthogonal ∵ the eigenfunctions of an hermitian operator form a complete set ∴any function 2006 Quantum Mechanics Prof. Y. F. Chen

65 The Sturm-Liouville Eq.
Formalism of Quantum Mechanics The Sturm-Liouville Eq. general form of SL eq.: with ,where p(x), q(x), and r(x) are real functions of x Ex. Legendre’s eq.:                 & eigenvalues l(l+1) linear operator that are self-adjoint can be written in the form: linear operator=Hermitian over [a,b] satisfies BCs: 2006 Quantum Mechanics Prof. Y. F. Chen

66 The Sturm-Liouville Eq.
Formalism of Quantum Mechanics The Sturm-Liouville Eq. BCs:(1) → the wave with fixed ends (2) → the wave with free ends (3) → the periodic wave show that subject to the BCs, the SL operator is Hermitian over [a, b]: putting into 2006 Quantum Mechanics Prof. Y. F. Chen

67 The Sturm-Liouville Eq.
Formalism of Quantum Mechanics The Sturm-Liouville Eq. integrating by parts for the first term & using the BCs → the SL operators is Hermitian over the prescribed interval 2006 Quantum Mechanics Prof. Y. F. Chen

68 Transforming an Eq. into SL Form
Formalism of Quantum Mechanics Transforming an Eq. into SL Form any eq can be put into self-adjoin form by introducing in place of proof:Let → to satisfy the requirement of SL eq. form for 2006 Quantum Mechanics Prof. Y. F. Chen

69 Transforming an Eq. into SL Form
Formalism of Quantum Mechanics Transforming an Eq. into SL Form rewrite eq. as the SL form for : with 2006 Quantum Mechanics Prof. Y. F. Chen

70 Quantum Harmonic Oscillator
2006 Quantum Mechanics Prof. Y. F. Chen

71 Quantum Harmonic Oscillator
1D S.H.O.:linear restoring force , k is the force constant & parabolic potential . harmonic potential’s minimum at = a point of stability in a system A particle oscillating in a harmonic potential 2006 Quantum Mechanics Prof. Y. F. Chen

72 Quantum Harmonic Oscillator
Ex:the positions of atoms that form a crystal are stabilized by the presence of a potential that has a local min at the location of each atom ∵ the atom position is stabilized by the potential, a local min results in the first derivative of the series expansion = 0 → a local min in V(x) is only approximated by the quadratic function of a H.O. 2006 Quantum Mechanics Prof. Y. F. Chen

73 Schrödinger Wave Eq. for 1D Harmonic Oscillator
Quantum Harmonic Oscillator Schrödinger Wave Eq. for 1D Harmonic Oscillator for the H.O. potential        , the time-indep Schrödinger wave eq.: use(1) & (2) making the substitution → called Hermite functions. 2006 Quantum Mechanics Prof. Y. F. Chen

74 Quantum Harmonic Oscillator
Hermite Functions One important class of orthogonal polynomials encountered in QM & laser physics is the Hermite polynomials, which can be defined by the formula the first few Hermite polynomials are: in general:    . 2006 Quantum Mechanics Prof. Y. F. Chen

75 Quantum Harmonic Oscillator
Hermite Functions the Hermite polynomials come from the generating function:                  . → Taylor series:                   . substituting into   : → recurrence relation: 2006 Quantum Mechanics Prof. Y. F. Chen

76 Hermite Functions substituting into : → recurrence relation: with &
Quantum Harmonic Oscillator Hermite Functions substituting into   : → recurrence relation: with & → 2nd-order ordinary differential equation for eigenvalues of the 1D quantum H.O.: 2006 Quantum Mechanics Prof. Y. F. Chen

77 Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator the eigenfunctions of 1D H.O.: with the help of , find normalization constant , → (i) in CM, the oscillator is forbidden to go beyond the potential, beyond the turning points where its kinetic energy turns negative. (ii) the quantum wave functions extend beyond the potential, and thus there is a finite probability for the oscillator to be found in a classically forbidden region   2006 Quantum Mechanics Prof. Y. F. Chen

78 Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator 2006 Quantum Mechanics Prof. Y. F. Chen

79 Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator the classical probability of finding the particle inside a region   : . the velocity can be expressed as a function of  : 2006 Quantum Mechanics Prof. Y. F. Chen

80 Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator (i) the difference between the two probabilities for n=0 is extremely striking ∵there is no zero-point energy in CM (ii) the quantum and classical probability distributions coincide when the quantum number n becomes large (iii) this is an evidence of Bohr’s correspondence principle 2006 Quantum Mechanics Prof. Y. F. Chen

81 Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator (1) classically, the motion of the H.O. is in such a manner that the position of the particle changes from one moment to another. (2) however, although there is a probability distribution for any eigenstate in QM, this distribution is indep of time → stationary states (3) even so, the Ehrenfest theorem reveals that a coherent superposition of a number of eigenstates, i.e., so-called “wave packet state”, will lead to the classical behavior 2006 Quantum Mechanics Prof. Y. F. Chen

82 Stationary States of 1D Harmonic Oscillator
Quantum Harmonic Oscillator Stationary States of 1D Harmonic Oscillator show : using the generation function , we can have ∵ the orthogonality property, the integration leads to as a consequence, we can obtain 2006 Quantum Mechanics Prof. Y. F. Chen

83 The Poisson Distribution
Quantum Harmonic Oscillator The Poisson Distribution given a mean rate of occurrence r of the events in the relevant interval, the Poisson distribution gives the probability that exactly n events will occur for a small time interval the probability of receiving a call is the probability of receiving no call during the same tiny interval is given by the probability of receiving exactly n calls in the total interval is given by 2006 Quantum Mechanics Prof. Y. F. Chen

84 The Poisson Distribution
Quantum Harmonic Oscillator The Poisson Distribution rearranging , dividing through by , and letting , the differential recurrence eq. can be found and written as for : which can be integrated to lead to with the fact that the probability of receiving no calls in a zero time interval must be equal to unity: 2006 Quantum Mechanics Prof. Y. F. Chen

85 The Poisson Distribution
Quantum Harmonic Oscillator The Poisson Distribution substituting into for : , repeating this process, can be found to be the sum of the probabilities is unity: the mean of the Poisson distribution:   2006 Quantum Mechanics Prof. Y. F. Chen

86 The Poisson Distribution
Quantum Harmonic Oscillator The Poisson Distribution in other words, the Poisson distribution with a mean of is given by: 2006 Quantum Mechanics Prof. Y. F. Chen

87 Schrödinger Coherent States of the 1D H.O.
Quantum Harmonic Oscillator Schrödinger Coherent States of the 1D H.O. The Schrödinger coherent wave packet state can be generalized as with it can be found that the norm square of the coefficient is exactly the same as the Poisson distribution with the mean of 2006 Quantum Mechanics Prof. Y. F. Chen

88 Schrödinger Coherent States of the 1D H.O.
Quantum Harmonic Oscillator Schrödinger Coherent States of the 1D H.O. substituting & into using 2006 Quantum Mechanics Prof. Y. F. Chen

89 Schrödinger Coherent States of the 1D H.O.
Quantum Harmonic Oscillator Schrödinger Coherent States of the 1D H.O. as a result, the probability distribution of the coherent state is given by: it can be clearly seen that the center of the wave packet moves in the path of the classical motion 2006 Quantum Mechanics Prof. Y. F. Chen

90 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators with , & the operator acting on the eigenstate 2006 Quantum Mechanics Prof. Y. F. Chen

91 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators in a similar way, the operator acting on the eigenstate 2006 Quantum Mechanics Prof. Y. F. Chen

92 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators & consequently, it is convenient to define 2 new operators: 2006 Quantum Mechanics Prof. Y. F. Chen

93 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators the operator is the increasing (creation) operator: this means that operating with on the n-th stationary states yields a state, which is proportional to the higher (n +1)-th state the operator is the lowering (annihilation) operator: this means that operating with on the n-th stationary states yields a state, which is proportional to the higher (n -1)-th state   2006 Quantum Mechanics Prof. Y. F. Chen

94 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators in terms of & , the operators & can be expressed as: & we can find the commutator of these 2 ladder operators: which is the so-called canonical commutation relation 2006 Quantum Mechanics Prof. Y. F. Chen

95 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators is the hermitian conjugate : proof: 2006 Quantum Mechanics Prof. Y. F. Chen

96 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators with , & the operator acting on the eigenstate 2006 Quantum Mechanics Prof. Y. F. Chen

97 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators in a similar way, the operator acting on the eigenstate 2006 Quantum Mechanics Prof. Y. F. Chen

98 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators & consequently, it is convenient to define 2 new operators: 2006 Quantum Mechanics Prof. Y. F. Chen

99 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators the operator is the increasing (creation) operator: this means that operating with on the n-th stationary states yields a state, which is proportional to the higher (n +1)-th state the operator is the lowering (annihilation) operator: this means that operating with on the n-th stationary states yields a state, which is proportional to the higher (n -1)-th state   2006 Quantum Mechanics Prof. Y. F. Chen

100 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators in terms of & , the operators & can be expressed as: & we can find the commutator of these 2 ladder operators: which is the so-called canonical commutation relation 2006 Quantum Mechanics Prof. Y. F. Chen

101 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators is the hermitian conjugate : proof: 2006 Quantum Mechanics Prof. Y. F. Chen

102 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators with & using the commutation relation define the so-called number operator: → the H.O. Hamiltonian takes the form: 2006 Quantum Mechanics Prof. Y. F. Chen

103 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators the eigenstates of can be found to be coherent states : coherent states have the minimum uncertainty 2006 Quantum Mechanics Prof. Y. F. Chen

104 Creation & Annihilation Operators
Quantum Harmonic Oscillator Creation & Annihilation Operators as a consequence, we obtain the minimum uncertainty state: 2006 Quantum Mechanics Prof. Y. F. Chen

105 One-Dimensional Scattering of Waves
2006 Quantum Mechanics Prof. Y. F. Chen

106 One-Dimensional Scattering of Waves
in this chapter we will explore the phenomena of lD scattering to show that transmission is possible even when the quantum particle has insufficient energy to surmount the barrier the transfer matrix method will be utilized to analyze the one-dimensional propagation of quantum waves 2006 Quantum Mechanics Prof. Y. F. Chen

107 The Transfer Matrix Method
One-Dimensional Scattering of Waves The Transfer Matrix Method consider a particle of energy E and mass m to be incident from the left on arbitrarily shaped, 1D, smooth & continuous potential Such a problem can be solved by : (1) dividing the potential into a piecewise constant function (2) using the transfer matrix method to calculate the probability of the particle emerging on the right-hand side of the barrier 2006 Quantum Mechanics Prof. Y. F. Chen

108 The Transfer Matrix Method
One-Dimensional Scattering of Waves The Transfer Matrix Method Figure 6.1 Sketch of the quantum scattering at the jth interface between 2 successive constant values of the piecewise potential & the wave propagating through the constant potential until reaching the next interface at a distance after crossing the jth interface 2006 Quantum Mechanics Prof. Y. F. Chen

109 The Transfer Matrix Method
One-Dimensional Scattering of Waves The Transfer Matrix Method the dynamics of the quantum particle is described by the Schrödinger eq., which is given in the jth region by: the general solutions: where & correspond to waves traveling forward and backward in jth region, respectively 2006 Quantum Mechanics Prof. Y. F. Chen

110 The Transfer Matrix Method
One-Dimensional Scattering of Waves The Transfer Matrix Method the relationship between the coefficients & are determined by applying the boundary conditions at the interface: as a result, it can be found that  & is referred to be the scattering matrix & & 2006 Quantum Mechanics Prof. Y. F. Chen

111 The Transfer Matrix Method
One-Dimensional Scattering of Waves The Transfer Matrix Method we can find that propagation between potential steps separated by distance carries phase information only so that a propagation matrix is defined as the successive operation of the scattering & propagation matrices leads to 2006 Quantum Mechanics Prof. Y. F. Chen

112 The Transfer Matrix Method
One-Dimensional Scattering of Waves The Transfer Matrix Method for the general case of N potential steps, the transfer matrix for each region can be multiplied out to obtain the total transfer matrix ∵ the quantum particle is introduced from the left, the initial condition is given by if no backward particle can be found on the right side of the total potential → 2006 Quantum Mechanics Prof. Y. F. Chen

113 The Transfer Matrix Method
One-Dimensional Scattering of Waves The Transfer Matrix Method as a consequence, the transmission & reflection coefficients are given by those can be used to calculate the transmission & reflection probability of a quantum particle through an arbitrary 1D potential & 2006 Quantum Mechanics Prof. Y. F. Chen

114 One-Dimensional Scattering of Waves
The Potential Barrier consider a particle of energy E and mass m that are sent from the left on a potential barrier Figure 6.2 Sketch of the quantum scattering of a 1D rectangular barrier of energy VB 2006 Quantum Mechanics Prof. Y. F. Chen

115 The Potential Barrier With the total matrix Q is given by where &
One-Dimensional Scattering of Waves The Potential Barrier With the total matrix Q is given by where & it simplified as 2006 Quantum Mechanics Prof. Y. F. Chen

116 The Potential Barrier transmission probability in the case
One-Dimensional Scattering of Waves The Potential Barrier transmission probability in the case in terms of energy E and potential 2006 Quantum Mechanics Prof. Y. F. Chen

117 The Potential Barrier transmission probability in the case
One-Dimensional Scattering of Waves The Potential Barrier transmission probability in the case occurs whenever: with the condition corresponds to resonances in transmission that occur when quantum waves back-scattered from the step change in barrier potential at positions & interfere and exactly cancel each other, resulting in zero reflection from the potential barrier 2006 Quantum Mechanics Prof. Y. F. Chen

118 The Potential Barrier transmission probability in the case
One-Dimensional Scattering of Waves The Potential Barrier transmission probability in the case (1) when , the transmission probability T → 1 the particles are nearly not affected by the barrier & have total transmission (2) in the limit case , we have 2006 Quantum Mechanics Prof. Y. F. Chen

119 The Potential Barrier transmission probability in the case
One-Dimensional Scattering of Waves The Potential Barrier transmission probability in the case the wave number becomes imaginary, with if → 2006 Quantum Mechanics Prof. Y. F. Chen

120 The Potential Barrier transmission probability in the case
One-Dimensional Scattering of Waves The Potential Barrier transmission probability in the case Figure 6.3 Transmission probability as a function of particle energy for and several widths 2006 Quantum Mechanics Prof. Y. F. Chen

121 Scattering of a Wave Package State
One-Dimensional Scattering of Waves Scattering of a Wave Package State in terms of & , the total wave function can be given by where is the Heaviside unit step func. , , the matrix element & are determined from the efficient & can be found to be given by 2006 Quantum Mechanics Prof. Y. F. Chen

122 Scattering of a Wave Package State
One-Dimensional Scattering of Waves Scattering of a Wave Package State where and the identities & are used to express the equation in a general form 2006 Quantum Mechanics Prof. Y. F. Chen


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