Download Lecture Notes, Feb 24, 2016

Lecture VIII ( Feb 24, 2016)
Review from Last Class
FIG. 1: H-atom spectrum
MYSTERIES OF MATTER: Two puzzles of classical Theory
Reference: Section 4 of unit 5 of the Ref[1]
(1) ATOMIC SPECTRUM:
Early in the 20th century, it was known that everyday matter consists of atoms and that atoms
contain positive and negative charges. Furthermore, each type of atom, that is, each element, has
a unique spectrum- a pattern of wavelengths the atom radiates or absorbs if sufficiently heated. A
particularly important spectrum, the spectrum of atomic hydrogen, is shown in Figure (1). The
art of measuring the wavelengths, spectroscopy, had been highly developed, and scientists had
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QuantumTheory-Stable Atom
size of the orbit must corresponds to whole number of
wave lengths of electron
Classical Theory-Unstable Atom
C
A
B
FIG. 2: Unstable atom (left) and de Broglie waves (right)
generated enormous quantities of precise data on the wavelengths of light emitted or absorbed by
atoms and molecules.
The spectrum can be described by a formula,
1
1
1
= R( 2 − 2 )
λ
n
m
(1)
Where λ is the wave length, R is some constant and n, m are integers.
In spite of the elegance of spectroscopic measurement, it must have been uncomfortable for
scientists to realize that they knew essentially nothing about the structure of atoms, much less why
they radiate and absorb certain colors of light. Solving this puzzle ultimately led to the creation
of quantum mechanics, but the task took about 20 years.
(2) Instability of an ATOM:
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FIG. 3: Bohr orbits and de Broglie waves.
In planetary model of atom, note that unlike planets, the electrons orbiting around the
nucleus are charged particles. According to Maxwell’s theory, a charged going around a circle
should radiate and loose energy and fall into the nucleus. That means that the atom is unstable.
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BOHR THEORY ( OLD QUANTUM THEORY)
In trying to solve the above two puzzles, Bohr proposed revolutionary ideas, now known as
OLD QUANTUM THEORY.
BOHR MODEL
• In contrast to classical physics, electrons in an atom can exits only in certain orbits, n =
1, 2, 3.... as shown in Fig. (3) The integer n is called quantum number.
• For H-atom, the energies of these electrons is En = −13.6/n2 13.6 comes from a
constant that is determined by Planck constant and mass and charge of the electron (
13.6 = Ke4 me /h2 , where K is a constant that determines the force on the electron due
to proton).
• Occasionally, an atom somehow jumps from one energy state to another by radiating
the energy difference. If an atom jumps from state n to state m, , it will radiate energy
Em − En .
According to Einstein’s theory, we would say that the atom emits a photon of energy
Em − En , when it makes a quantum jump. The reverse is possible: An atom in a lower
energy state can absorb a photon with the correct energy and make a transition to the higher
state. The frequency of emitted or absorbed light when the atom jumps from the orbit n to
m will be f =
Em −En
h
This formula was able to predict the spectrum of H-atom as shown in Fig. (1).
• De Broglie wave picture explains discrete orbits of Bohr model. We note that an orbit
of radius r will be such that within the circumference of the orbit , integer number of de
Broglie’s wave must fit in, that is 2πr = nλ. Compare this with the waves in a string where
in the length L integer number of half wave lengths fit, that is, L = n λ2 .
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In summary,
2πr = nλ
h
nλ =
p
(2)
(3)
Using Newton’s equations of motion, we can calculate the condition for circular orbits and
this along with( 3) gives the following formulas, that show how the radius, and the de Broglie
wave length depends upon n, called the quantum number
• At absolute zero, or at very very low temperature, the atom will be in the lowest energy
state, called the ground state of the atom.
Correspondence Principle
“Quantum laws reduce to classical laws in some limit”.
In Bohr model that happens when n is very large. In this case En becomes almost continuous,
that is what happens in macroscopic world.
NOTE: Other than H-atoms:
What happens when we have more than one electron... We will discuss it later as it turns out
that each “state” can accommodate only two electrons (Pauli Exclusion Principle)
Quantum Mechanics
Heisenberg and Schödinger independently developed two distinct but equivalent frameworks
to describe microscopic world that replaced Newton’s classical equations with new equations of
quantum theory. The new theory is called Quantum Mechanics. As we will see, the equation of
quantum mechanics is completely different from classical mechanics.
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Schr”odinger Equation
In December 1925 while on vacation, Schro”odinger ( Univ of Zurich physics professor ),
looked at de Broglie’s thesis. He work out a single equation, explaining the behavior of particles
in terms of de Broglie waves. The lead player in the equation is a quantity called Ψ ( pronounced
”sigh” ) which is called the wave function.
• Instead of describing particle by its position and velocity, in Schr”odinger’s equation, the
particle is described by wave function Ψ.
• Even in classical physics, waves are described by wave functions, which gives the amplitude,
wavelength and shape of the wave. However, Schr”odinger wave function is not a “real”
quantity. In other words, unlike water or sound or even electro-magnetic waves, matter
waves are not described by ordinary real numbers. Since it is not real , called complex
number, we cannot determine the shape of the wave from its wave function.
• Known as the Schrödinger equation, this partial differential equation for a non-relativistic
particle of mass m in a potential V is given by
2
−~ 2
∂
∇ + V (r, t) Ψ(r, t)
i~ Ψ(r, t) =
∂t
2m
(4)
Here r in general represents all the spatial variables (x, y, z). i is a complex number equal to
√
−1. The function Ψ(r, t), known as the wave function of the particle, encodes complete
information about the state of the particle at a location r at time t.
• It was Max Born, who successfully interpreted the wave function Ψ as the probability
amplitude of the wave associated with the particle and was awarded the Nobel prize in 1954.
Unlike classical wave functions, Ψ is a complex number and is not altogether a measurable
quantity. Therefore, unlike water waves, or waves in a string, or electromagnetic waves,
where the wave function is an observable entity describing oscillations of the medium or
electromagnetic fields, the wave function for a matter wave is an abstract quantity . It is
the absolute square , namely |Ψ(r, t)|2 , that is a physical entity describing the probability of
finding the particle at location r at time t. While the probability amplitude encodes all the
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information about the state of the particle, taking the absolute value (the modulus) destroys
some information (called the phase). This subtle distinction is the ultimate source of all
quantum mechanical “weirdness”.
[1] http://www.learner.org/courses/physics/
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