Transcript Molekylfysik - Leiden Univ

1. Quantum theory: introduction and principles
1.1 The failures of classical physics
1.2 Wave-particle duality
1.3 The Schrödinger equation
1.4 The Born interpretation of the wavefunction
1.5 Operators and theorems of the quantum theory
1.6 The Uncertainty Principle
 = c

1.1 The failures of classical physics
A. Black-body radiation
at max
Observations:
1) Wien displacement law:
2) Stefan-Boltzmann law:
( = E/V)
Tentative explanation via the classical mechanics
Equipartition of the energy:
Per degree of freedom: average energy = kT
(26 meV at 25°C).
The total energy is equally “partitioned” over all the
available modes of motion.
Rayleigh and Jeans used the equipartition principle and
consider that the electromagnetic radiation is emitted from a
collection of excited oscillators, which can have any given
energy by controlling the applied forces (related to T). It led
to the Rayleigh-Jeans law for the energy density  as a
function of the wavelength .
It does not fit to the experiment. From this law,
every objects should emit IR, Vis, UV, X-ray
radiation. There should be no darkness!! This is
called the Ultraviolet catastrophe.
Introduction of the quantization of energy to solve
the black-body problem
Max Planck: quantization of energy.
E = n h only for n= 0,1,2, ...
h is the Planck constant
Cross-check of the theory: from the Planck distribution,
one can easily find the experimental Wien displacement
and the Stefan-Boltzmann law.
 the quantization of energy exists!
C. Atomic and molecular spectra
Excitation
energy
Photon
emission
Fe
h=hc/
The emission and absorption of radiation always
occurs at specific frequencies: another proof of the
energy quantization.
NB: wavenumber
~ 

c

1

Photon
absorption
1.2 Wave-particle duality
A. The particle character of
electromagnetic radiation:
 The photoelectric effect
e- (Ek)
h
metal
The photon h ↔ particle-like projectile
conservation of energy
½mv2 = h - 
= metal workfunction, the minimum energy required to
remove an electron from the metal to the infinity.
Threshold does not depend on intensity of incident radiation.
NB: The photoelectron spectroscopy (UPS, XPS) is based on
this photoelectric effect.
B. The wave character of the particles:
 Electron diffraction
Diffraction is a characteristic property of waves. With X-ray, Bragg showed that a
constructive interference occurs when =2d sin. Davidsson and Germer showed also
interference phenomenon but with electrons!
Particles are characterized by a wavefunction
 A link between the particle
(p=mv) and the wave () natures
V

 An appropriate potential difference creates
electrons that can diffract with the lattice of the nickel
d
1.3 The Schrödinger Equation
From the wave-particle duality, the concepts of classical physics (CP) have to be
abandoned to describe microscopic systems. The dynamics of microscopic systems will
be described in a new theory: the quantum theory (QT).
 A wave, called wavefunction (r,t), is associated to each object. The well-defined
trajectory of an object in CP (the location, r, and momenta, p = m.v, are precisely known
at each instant t) is replaced by (r,t) indicating that the particle is distributed through
space like a wave. In QT, the location, r, and momenta, p, are not precisely known at
each instant t (see Uncertainty Principle).
 In CP, all modes of motions (rot, trans, vib) can have any given energy by controlling
the applied forces. In the QT, all modes of motion cannot have any given energy, but can
only be excited at specified energy levels (see quantization of energy).
The Planck constant h can be a criterion to know if a problem has to be addressed in
CP or in QT. h can be seen has a “quantum of an action” that has the dimension of
ML2T-1 (E= h where E is in ML2T-2 and  is in T-1). With the specific parameters of a
problem, we built a quantity having the dimension of an action (ML2T-1). If this quantity
has the order of magnitude of h (10-34 Js), the problem has to be treated within the QT.
Hamiltonian function H = T + V.
T is the kinetic energy and V is the potential energy.
correspondence principles are proposed to pass from the classical mechanics
to the quantum mechanics
x  x

 


grad   
 
i
i  x y z 
2
2
2 
2
2 2
2  
p       2  2  2 
y
z 
 x
p

t
V ( x, y , z , t )  V ( x, y , z , t )
E  i
Classical mechanics
H  T V  E
p2
H
 V ( x, y, z, t )
2m
Quantum mechanics
H  i
 Schrödinger
 t Equation
2 2
H 
  V ( x, y , z , t )
2m

 The Schrödinger Equation (SE) shows that the operator H
H  i
and iħ/t give the same results when they act on the
t
wavefunction. Both are equivalent operators corresponding to the
2 2
H 
  V ( x, y , z , t )
total energy E.
2m
 In the case of stationary systems, the potential V(x,y,z) is time independent. The
wavefunction can be written as a stationary wave: (x,y,z,t)= (x,y,z) e-it (with E=ħ).
This solution of the SE leads to a density of probability |(x,y,z,t)|2= |(x,y,z)|2, which is
independent of time. The Time Independent Schrödinger Equation is:
 2 2




V
(
x
,
y
,
z
)

 ( x, y, z )  E  ( x, y, z ) or
 2m

H   E
NB: In the following, we only
envisage the time independent
version of the SE.
 The Schrödinger equation is an eigenvalue equation, which has the typical form:
(operator)(function)=(constant)×(same function)
 The eigenvalue is the energy E. The set of eigenvalues are the only values that the
energy can have (quantization).
 The eigenfunctions of the Hamiltonian operator H are the wavefunctions  of the
system.
 To each eigenvalue corresponds a set of eigenfunctions. Among those, only the
eigenfunctions that fulfill specific conditions have a physical meaning.
1.4 The Born interpretation of the wavefunction
 Physical meaning of the wavefunction:
probability of finding the particle in an
infinitesimal volume d=dxdydz at some point
r is proportional to |(r)|2d
Example of a 1-dimensional system
 |(r)|2 = (r)*(r) is a probability density.
It is always positive!
wavefunction may have negative or complex values
Node
A. Normalization Condition
 The solution of the differential equation of Schrödinger is defined within a constant N.
If  is a known solution of H=E, then =N is a also solution for the same E.
H=E  H(N)= E(N)  N(H)=N(E)  H=E
 The sum of the probability of finding the particle over all infinitesimal volumes d of the
space is 1: Normalization condition.
 We have to determine the constant N, such that the solution =N of the SE is
normalized.
*
*
2
*

d


1

(
N

'
)(
N

'
)
d


1

N

'

'
d  1  N 



1
*

'

'
d

B. Other mathematical conditions
*
 (r)  ; r  if not: no physical meaning for the normalization condition
 d   ???
 (r) should be single-valued r  if not: 2 probability for the same point!!
 The SE is a second-order differential equation: (r) and d(r)/dr should be continuous
C. The kinetic energy and the wavefunction
2 2
H  T V  
V
2
2m  x
2
 2 2 
2
*  
 d  
 T     

d
2 
2

2m
x
 2m  x 
*
T
We can expect a particle to have a high kinetic
energy if the average curvature of its
wavefunction is high.
The kinetic energy is then a kind of average over the curvature of the wavefunction: we
get a large contribution to the observed value from the regions where the wavefunction is
sharply curved (2 / x2 is large) and the wavefunction itself is large (* is large too).
Example: the wave function in a periodic system: electrons in a metal
1.5 Operators and principles of quantum mechanics
A. Operators in the quantum theory (QT)
An eigenvalue equation, f = f, can be associated to each operator . In the QT, the
operators are linear and hermitian.
 Linearity:
 is linear if:
(c f)= c  f (c=constant)
(f+)=  f+ 
and
 NB: “c” can be defined to fulfill the normalization condition
 Hermiticity:
A linear operator is hermitian if:

f * d    * f *d
where f and  are finite, uniform, continuous and the integral for the normalization converge.
 The eigenvalues of an hermitian operator are real numbers (= *)
 When the operator of an eigenvalue equation is hermitian, 2 eigenfunctions (fj, fk)
corresponding to 2 different eigenvalues (j,  k) are orthogonal.
 f j j f j
 f k  k f k

f j f k d  0
*
B. Principles of Quantum mechanics
 1. To each observable or measurable property <> of the system corresponds a linear
and hermitian operator , such that the only measurable values of this observable are the
eigenvalues j of the corresponding operator.
f = f
 2. Each hermitian operator  representing a physical property is “complete”.
Def: An operator  is “complete” if any function (finite, uniform and continuous) (x,y,z)
can be developed as a series of eigenfunctions fj of this operator.
 ( x, y, z)   C j f j ( x, y, z)
j
 3. If (x,y,z) is a solution of the Schrödinger equation for a particle, and if we want to
measure the value of the observable related to the complete and hermitian operator  (that is
not the Hamiltonian), then the probability to measure the eigenvalue k is equal to the
square of the modulus of fk’s coefficient, that is |Ck|2, for an othornomal set of
eigenfunctions {fj}.
Def: The eigenfunctions are orthonormal if

f j f k d  
*
ij
 Cj 1
2
NB: In this case:
j
 4. The average value of a large number of observations is given by the expectation value
<> of the operator  corresponding to the observable of interest. The expectation value of
an operator  is defined as:
 
*

   d
*

  d
For normalized
wavefunction
     *  d   C j  j
2
j
See p305 in the book
 5. If the wavefunction =f1 is the eigenfunction of the operator  (f = f), then the
expectation value of  is the eigenvalue 1.
     *  d   *1  d  1  * d  1
1.6 The Uncertainty Principle
1. When two operators are commutable (and with the Hamiltonian operator), their
eigenfunctions are common and the corresponding observables can be determined
simultaneously and accurately.
2. Reciprocally, if two operators do not commute, the corresponding observable cannot
be determined simultaneously and accurately.
If (12- 21) = c, where “c” is a constant, then an uncertainty relation takes place for
the measurement of these two observables:
1  2 

where 1   12    1  2

1/ 2
c
2
Uncertainty
Principle
Example of the Uncertainty Principle
1. For a free atom and without taking into account the spin-orbit coupling, the angular
orbital moment L2 and the total spin S2 commute with the Hamiltonian H. Hence, an exact
value of the eigenvalues L of L2 and S of S2 can be measured simultaneously. L and S are
good quantum numbers to characterize the wavefunction of a free atom  see Chap 13
“Atomic structure and atomic spectra”.
2. The position x and the momentum px (along the x axis). According to the
correspondence principles, the quantum operators are: x and ħ/i( / x). The
commutator can be calculated to be:
  

,
x




i

x
i


p x  x 

2
The consequence is a breakdown of the classical mechanics laws: if there is a complete
certainty about the position of the particle (x=0), then there is a complete uncertainty
about the momentum (px=).

If a system stays in a state during a time
2
t, the energy of this system cannot be determined more accurately than with an error E.
3. The time and the energy:
t E 
This incertitude is of major importance for all spectroscopies:  see Chap 16, 17, 18