Lesson 7.3 Quantum Mechanics
Suggested Reading
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Zumdahl Chapter 7 Section 7.5
Essential Question
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What is quantum mechanics?
Learning Objectives
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Describe the basic terms and ideas in the quantum mechanical model
of the atom.
Apply the de Broglie equation.
Explain the Heisenberg Uncertainty Principle.
Introduction
Bohr's model firmly established the concept of atomic energy levels.
However, it could not account for the details of atomic structure in more
complicated atoms. Further understanding of atomic structure required
other theoretical developments. Current ideas about atomic structure
depend on the principles of quantum mechanics. Quantum mechanics is a
theory that applies to very small, submicroscopic particles such as
electrons. The first important step in the development of the quantum
mechanical model of the atom was the discovery of the de Broglie relation
(usually pronounced " de broy").
The de Broglie Relation
In 1923 the French physicist Louis de Broglie reasoned that if light exhibits
both wave and particle properties, then perhaps particles of matter show
characteristics of waves. He therefore postulated that a particle of matter of
mass m and speed v has an associated wavelength, by analogy with light.
This equation is called the de Broglie relation.
If matter has wave properties, why are they not commonly observed?
Calculations using the de Broglie relation show that a baseball moving at
about 27 m/s (60 mi/hr) has a wavelength of about 10-34 m, a value so
incredible small that it can't be detected.
If the waves of matter are so small, then how do we know they exist?
The wave properties of electrons have been observed by experiment. The
wave property of electrons was first demonstrated in 1927 by U.S
physicists C. Davisson and L.H. Germer and by the British physicist
George Thomsom (son of J.J. Thomsom). They showed that a beam of
electrons, just like x-rays, can be diffracted by a crystal. The German
physicists Ernst Ruska used this wave property to construct the first
electron microscope, which we now use to examine very small things like
cells.
Example: Applying the de Broglie Relation
Calculate the wavelength (in meters) of the wave associated with a 1.00 kg
mass moving a 1.00 km/hr.
Solution:
First we must convert the speed of the mass from km/hr to m/s.
Note the cancellation of units in the problem above to give m. By thinking of
the J in terms of the SI units of kg ⋅ m2/s2, we can see how the other units
cancel leaving only m. Watch your units in these types of problems!
Example 2: Applying the de Broglie Relation
What is the wavelength (in picometers) associated with an electron, whose
mass is 9.11 x 10-31 kg, traveling at a speed of 4.19 x 106 m/s?
It is not uncommon for atomic distances to be expressed in picometers.
Recall that 1 pm = 10-12 m.
Wave Functions
De Broglie's relation is limited, because it only applies to particles in a
force-free environment. Thus, it can't be applied to an electron in an atom,
where the electron is subject to the attractive force of the nucleus. But in
1926, Erwin Shrodinger devised the theory of quantum mechanics to
overcome this. Quantum mechanics describes the wave properties of
electrons in atoms by using mathematical expressions called wave
functions, such as the one pictured to the right. As you can see, the
quantitative aspects of wave functions are very complex, and beyond the
scope of this class. However, we can look at some the most important
conclusions of the theory from a qualitative point of view.
Quantum mechanics alters the way we think about motion. Normally, we
think in terms of a continuous path for moving objects, such as when you
through a ball or when the electron orbits around the nucleus in the Bohr
model. In continuous movement, the path of the moving object is given by
the position and velocity (momentum) of the moving object at various times.
Quantum mechanics changes this view because it does not permit us to
think of an electron as having a precise orbit in an atom. To describe such
an orbit, we would have to know the exact position and momentum of the
electron at various times. However, in 1927, Werner Heisenberg showed
from quantum mechanics that it is impossible to know simultaneously,
which absolute precision, both the position and momentum of an electron.
Heisenberg's uncertainty principle is a relation that states that there is a
fundamental limitation to just how precisely we can know both the position
and momentum of a particle at a given time.
What quantum mechanics allows us to do is make statistical statements
about where the electron is in the atom. Although we can not say that the
electron will be at a particula posiiton at a given time, we can say that the
electron is likely (or not likely) to be at a particular position at a given
time. Information about an electron in a given energy level is given by
Shrodinger's wave function, which is denoted by the Greek letter psi, Ψ.
The square of solution to a wave function,Ψ2, gives the probability
distribution of the electron at a particular energy level in an atom.
When the results of these functions are graphed we can see peaks that
correspond to certain distances from the nucleus. These peaks represent
the area where the electrons are most likely to be found. You can see from
the graph, that an electron in the n=1 energy level (1s), is most likely to be
found near the nucleus, while those in higher energy levels will most likely
be further away.
In the quantum mechanical model of the atom, the atom does not have a
definite boundary like it does in the Bohr model since the locations of
electrons are defined by probability rather than precise orbits.
The image to the right shows the probability density of an electron in a
hydrogen atom. Even though the probability of finding the electron at a
point near the nucleus is high, the volume of space there is small.
Therefore, there is a higher probability of finding the electron within a
"shell" some distance from the nucleus. Quantum mechanics has shown
that this distance just happens to be equal to the radius that Bohr
calculated for an electron orbit in his model!
Watch this You Tube video
https://www.youtube.com/watch?v=IOYyCHGWJq4