LECTURES ON QUANTUM MECHANICS  - Introduction

by Reinaldo Baretti Machín

Para preguntas y sugerencias escriba a :

Lectures on Quantum Mechanics -Introduction

Chapter 1 - Some experimental facts

Chapter 2 - The Old Quantum Theory

Chapter 3 - The Postulates of Quantum Mechanics

Chapter 4- Potential Barriers

Chapter 5 - Potential Wells

Chapter 6 -  Periodic Potentials

Chapter 7 - Angular Momentum

Chapter 8 - The Hydrogen Atom

Chapter 9 - The Electron Spin

Chapter 10 -The two electron atom

Chapter 11-The H2 molecule

References:

1. Principles of quantum mechanics; nonrelativistic wave mechanics with illustrative applications.

2. Introduction to Quantum Mechanics with Applications to Chemistry by Linus Pauling and E. Bright Wilson Jr.

3. by L. D. Landau and L. M. Lifshitz

4.  http://perimeterinstitute.ca/personal/cfuchs/

Introduction:

Planck constant  -h- (6.63e-34 joules-second) appeared in physics in the year 1900 in the explanation of  black body radiation. He assumed that an oscillator with frequency ν could acquire energy values given by  ε = n h ν where n=0,1,2,3….This was the first  use of the hypothesis of  quantization in physics .

This was a problem that classical mechanics , statistical physics and the electromagnetic theory of the epoch were not successful in explaining. The dimensions of h can be viewed as  those of angular momentum  kgm2/s or  energy - time (joules-second) or what it is the same  position  - linear momentum   (m kg m/s) .

Many other physical phenomena required the use of h in some quantization scheme to gain a reasonable explanation. Some of these are the photoelectric effect, theory of the spectral lines of atoms , the Compton effect, the production of X-rays etc.

Two other important physical  parameters were also ascertained  at the time the discovery of  h , the charge and mass of the electron e and m , and the size of atomic nuclei.

 Some physical constants SI system CGS system c 2.99E8 m/s 2.99E10 cm/s h 6.63E-34 J-s 6.63E-27 J-s e 1.60E-19 C 4.80 E−10 esu melectron 9.11E-31 kg 9.11E-28 kg Kelectrostatic force constant 8.99 E9 Nm2/C2 1dyn cm2/esu2

For example an electron under the influence of the nuclei with charge + Ze

the potential energy would be   - Z e2/ r in CGS  units and the kinetic energy is    (1/2) m v2 . We can  constructed from combinations of   h , m , (Ze)1/2 scales of energy , length , velocity , such that they reflect an order of magnitude of atomic processes.  When a quantization theory is applied  propperly it will reproduce this quantities.

Dimensional relations are suggestive of new relations and what could be the order of magnitude of physical quantities at the atomic level.

For example    h ~ angular momentum ~length-linear momentum~ r p ~ r m v . If p is well defined then a certain length

λ is defined by  λ≡ h/p . Actually de Broglie suggested that a particle with momentum p has wavelength given by h/p. Diffraction phenomena shown by electron , neutrons and molecules confirmed this postulate.

The constant h appears in a fundamental  limitation on measurements as shown by Heisenberg. In one dimension the uncertainty ∆x in a measurement multiplied by the uncertainty in momentum ∆p satisfy

∆x∆p  ≥ h /(4π) .

Another uncertainty is suggested from the dimensionality of h , that is , the product  of  energy-time.  It will be shown later the duration the uncertainty in the energy ∆E multiplied  by ∆t , the duration of the measurement interval, satisfy  ∆E  ∆t ≥  (h/(4π) .

A critique of the tendency to overemphasize the limitations dictated by the uncertainty principle  is given in Refs (4- 5).

Now consider some atomic relations. Take one electron in the field of the nuclei of charge Ze.

Equating the centripetal force to the electrostatic attraction we get

m v2/r = Z e2/ r2 (in CGS units)  or    r= m Z e2/p2  where p=mv.  Using p = h/r one gets

the “distance” of the electron is

r ~ h2 /( m(Ze2) .                                     (1)

This value proves to be higher than what is called the Bohr radius.

It is shown later that  Bohr’s radius , is a  quantity smaller than (1) with Z=1. Defined by  aBohr = h2 /[4π2 m(e2)]  corresponds to the  first orbit of the hydrogen atom. Nevertheless eq(1) gives a glimpse of the order of magnitude.

The combination  h/2π is called h bar (here h’= 1.06E-34 J-s ).

The result (1) is improved using de Broglie argument that the length to be used in conjunction with h and p is the circumference 2πr. Then

p = h/(2πr) leds to   r = aBohr = h2 /[4π2 m(e2)]

=5.30E-011 m

, when Z=1.

The total energy of the hydrogenic atom

E = (1/2) mv2 – Z e2 /r . But (1/2) m v2 = (1/2) Z e2 /r , thus the total energy is

E = -(1/2) Ze2/r  .             (2)

Introduce r = h’2/(mZe2)   in (2) to obtain the  “ground state” energy of an electron in a hydrogenic atom with nuclear charge + Ze

E =  -(1/2) mZ2 e4/(h’2)      .       (3)

= -2.17E-018  Z2  joules

= -13.6 Z2 eV

An electron in an atomic state can produce a magnetic moment due to its orbital motion. One can think of a current multiplied by a tiny area ,which is of the order of

current x area = (e/period) π r2 = ( e m v/2πr) π r2

~ mvr (e/2m) ~ hbar e/(2m)~

~angular momentum  e/(2m)                    (4)

~ 9.31 E-24  joules/tesla  . This quantity is called the bohr magneton  μB = eh’/(2m).

It turns out that the electron has also an intrinsic magnetic moment , not related to orbital motion. Oddly enough the intrinsic dipole moment of the electron is

where the factor g S ≈ 2  and s = (1/2) h’  . The (1/2) factor is canceled by gS .

The impact of h’ and the quantization manifestation is also present in molecules , solids , liquids and nuclear phenomena. The theoretical complexity increases as the number of particles involved increases and also if the length scale diminishes drastically as is the case of nuclear phenomena. In the nuclear range an additional fundamental force ,the nuclear interaction is added to the electromagnetic force.

Fortran Code for calculations

c calculations of orders of magnitude

c data in cgs

real m

data h, e ,m /6.63E-27,4.80E-10,9.11E-28/

pi=2.*asin(1.)

hbar=h/(2.*pi)

abohr=hbar**2/(m*e**2)

eground= -(1./2.)*m*e**4/hbar**2

print*,'abohr(m)=',abohr*1.E-2

print*,'eground(j) , eV= ', eground*1.E-7,eground*1.E-7/1.60E-19

stop

end