1. Quantum theory: introduction and principles

= c

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

1.1 The failures of classical physics

A. Black-body radiation

at max

Observations:

1) Wien displacement law:

2) Stefan-Boltzmann law:

( = E/V)

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 .

Tentative explanation via the classical mechanics

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 spectraExcitation

energy Photon emission

h=hc/

FePhoton absorption

The emission and absorption of radiation always occurs at specific frequencies: another proof of the energy quantization.

NB: wavenumber

1~ c

1.2 Wave-particle duality

A. The particle character of electromagnetic radiation:

The photoelectric effect

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.

e- (Ek)h

metal

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!

d

V

Particles are characterized by a wavefunction

An appropriate potential difference creates electrons that can diffract with the lattice of the nickel

A link between the particle (p=mv) and the wave () natures

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

),,,(2

2

tzyxVm

pH

),,,(),,,(

2

2

2

2

2

22222

tzyxVtzyxVt

iE

zyxp

zyxigrad

ip

xx

),,,(2

22

tzyxVm

H

EVTH t

iH

Classical mechanics Quantum mechanics

Schrödinger Equation

),,,(2

22

tzyxVm

H

tiH

The Schrödinger Equation (SE) shows that the operator H and iħ/t give the same results when they act on the wavefunction. Both are equivalent operators corresponding to the total energy E.

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-it (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

22

zyxEzyxzyxVm

EH or

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.

NB: In the following, we only envisage the time independent version of the SE.

1.4 The Born interpretation of the wavefunction

Example of a 1-dimensional system

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

|(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.

dNdNdNNd

*

*2**

''

11''1)')('(1

B. Other mathematical conditions

(r) ; r if not: no physical meaning for the normalization condition

(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

???* d

C. The kinetic energy and the wavefunction

Vxm

VTH

2

22

2

dxm

dxm

T2

2*

2

2

22*

22

T

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).

We can expect a particle to have a high kinetic energy if the average curvature of its wavefunction is high.

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) and (f+)= f+

NB: “c” can be defined to fulfill the normalization condition

Hermiticity:

A linear operator is hermitian if:

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.

dfdf ***

jjj ff

kkk ff 0* dff kj

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.

j

jj zyxfCzyx ),,(),,(

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

NB: In this case:

ijkj dff *

j

jC 12

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:

5. If the wavefunction =f1 is the eigenfunction of the operator (f = f), then the expectation value of is the eigenvalue 1.

1*

11** ddd

For normalized wavefunction

d

d*

*

j

jjCd 2*

See p305 in the book

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 (12- 21) = c, where “c” is a constant, then an uncertainty relation takes place for the measurement of these two observables:

where

221

c

2/121

211

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:

ix

xi

,2

xpx

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=).

3. The time and the energy: If a system stays in a state during a time

t, the energy of this system cannot be determined more accurately than with an error E.

This incertitude is of major importance for all spectroscopies: see Chap 16, 17, 18

2

Et