**LECTURES ON QUANTUM
MECHANICS - Introduction **

by Reinaldo Baretti Machín

http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysics.htm

http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart2.htm

http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicsPart3.htm

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 6 - Periodic Potentials

Chapter 8 - The Hydrogen AtomChapter 10 -The two electron atom

Chapter 11-The H_{2} 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.
**
Quantum Mechanics: Non-Relativistic Theory,
Volume 3, Third Edition (Quantum Mechanics)
**
by L. D.
Landau and L. M. Lifshitz

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

5. Quantum Mechanics as Quantum Information (and only a little more)

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 kgm^{2}/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 |

m |
9.11E-31 kg |
9.11E-28 kg |

K |
8.99 E9 Nm |
1dyn cm |

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

the potential energy would
be - Z e^{2}/ r in CGS units and the kinetic energy is (1/2) m v^{2}
. 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 v^{2}/r = Z e^{2}/
r^{2} (in CGS units) or r= m Z e^{2}/p^{2} where
p=mv. Using p = h/r one gets

the “distance” of the electron is

r ~ h^{2} /( m(Ze^{2})
. (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 a_{Bohr
}= h^{2} /[4π^{2} m(e^{2})] 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 =
a_{Bohr }= h^{2} /[4π^{2} m(e^{2})]

=5.30E-011 m

, when Z=1.

The total energy of the hydrogenic atom

E = (1/2) mv^{2} –
Z e^{2} /r . But (1/2) m v^{2} = (1/2) Z e^{2} /r , thus
the total energy is

E = -(1/2) Ze^{2}/r
. (2)

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

E = -(1/2) mZ^{2} e^{4}/(h’^{2})
. (3)

= -2.17E-018 Z^{2} joules

= -13.6 Z^{2} 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 r^{2} = ( e m v/2πr) π r^{2}

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

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

~ 9.31 E-24 joules/tesla . This quantity is called the bohr magneton μ_{B}
= 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 g_{S} .

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