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