1.1
Introduction to Waves
Quantum mechanics is a term used to describe micro and nano systems. The most obvious example is the interaction of
an electron with metallic and semiconductor materials. The energy of an electron can be transferred to energy
associated with electromagnetic radiation (photon) and vice versa. We will examine the physics behind this behavior
and the range of energy values where this occurs. There is a range of energy values (and wavelengths) associated with
different types of electromagnetic radiation, and this plays a key role in how semiconductor devices work.
Wave theory, wave behavior and the modeling of waves, is fundamental to Quantum Mechanics theory, and will be
explained thoroughly. A review of wave mechanics is provided in sections 1.2 and 1.3 (Part I). Wave mechanics in turn
will be used to develop quantum mechanics models, and this will be described in detail in section 3.2 (Part III). There are
several quantum mechanics models, but we will focus on Schrödinger’s equation since it is the easiest to understand.
The solution to Schrödinger’s equation for a given set of conditions, results in a mathematic model of the energy and
position of an electron at any given time.
The term wave-particle duality will be discussed in detail, and plays an integral role in quantum mechanics. Wave
functions are used to model electromagnetic waves and electrons, although there are distinct differences in the
meaning of the mathematical models.
Another key term in quantum mechanics is “quantized” energy. As we solve Schrödinger’s equation, we will focus on
exact solutions that focus on the quantized energy values of electrons. Individual photons are thought to interact with
individual electrons, where both exhibit both particle and wave behavior.
1.1.1
Examples of Waves
Objective 1: Recall that electromagnetic energy is quantized and equal to hf (or hc/λ).
Objective 2: Recall that all electromagnetic waves travel at speed equal to c (3 x 108 cm/s) in vacuum.
Objective 3: Recall that electromagnetic waves behave as particles called photons.
Objective 4: Recall that electrons exhibit particle and wave-like behavior.
There are several examples of waves in nature.
•
Water waves – movement of water at different heights created by wind or disturbance
•
Sound waves – movement of fluid (air or water) through changes in pressure.
•
Electromagnetic waves – photons with quantized energy (E) and frequency (f) corresponding to radio waves,
microwaves, infrared waves, visible light waves, ultraviolet waves, X-rays, and gamma rays. All electromagnetic
waves propagate at 3 x 108 m/s in vacuum.
•
Matter waves (Electrons) – charged particles traveling through space with velocity, v, and corresponding kinetic
energy ½ mv2. The velocity of an electron can be related to its wavelength using de Broglie’s relation λ = h/mv.
In this course, we will mainly be concerned with electromagnetic waves (photons) and their interaction with matter
(specifically electrons).
All electromagnetic waves or photons travel at the speed of light in vacuum and have energy equal to hf, where f = c/λ,
and
h is Planck’s constant equal to 6.63 x 10-34 J s,
c is the speed of light and is equal to 3 x 108 m/s (or 299,792,458 m/s) in vacuum
λ is the wavelength (described in section 1.2)
It should be emphasized that the different regions of the electromagnetic spectrum are associated with different
wavelength and energy values, and all photons travel at the speed of light in vacuum. Table 1.1 includes the ranges of
wavelengths corresponding to the different regions in the electromagnetic spectrum. Based on the equation f = c/λ, the
wave frequency can be related to the wavelength, and from the equation E = hf, the wave energy can be related to the
frequency. This relationship will be expanded in section 1.2.
Table 1.1 Wavelength values for Electromagnetic Spectrum*
Radio Waves
Micro Waves
Infrared Waves
Visible Waves
Ultraviolet Waves
X-Rays
Gamma Waves
* Approximate values
> 10 cm
10 cm – 0.01 cm
700 nm – 0.01 cm
400 – 700 nm
400 nm – 1 nm
0.01 nm – 1 nm
< 0.01 nm
We will also examine the wave nature of electrons. A distinction will be made between electromagnetic waves and
matter waves. Quantum mechanics will give us the most probable position of an electron and the possible electron
energy levels that exist in a specified volume or space. If we know the values of the possible energy levels, and the
approximate number of electrons that possess energy values at the different levels, then we can predict possible
transitions in electron energy (Fig. 1.1). When an electron moves from a higher energy level to a lower energy level, a
photon is emitted with a wavelength that is inversely proportional to the energy difference between transitions (E =
hc/λ). Transfer of energy occurs with discrete energy exchanges consistent with the Laws of Conservation.
Photon
Figure 1.1 Emission of a photon as a result of electron transition from a higher energy to a lower energy
The inverse can happen as well, where light incident on a material will excite an electron from a lower energy level to a
higher energy level. If the photon is of sufficient energy, then a phenomenon known as the photoelectric effect occurs.
The photoelectric effect is where light (photon) hitting a metal transfers sufficient energy to break the bond of an
electron and completely ejected it from the metal. The energy of the photon in this case is used to both remove the
electron from the metal and eject it with its own kinetic energy. The kinetic energy of the ejected electron will be equal
to the initial photon energy minus the energy required to remove the electron from the metal. This is proof that
photons behave as both waves and particles.
1.1.2
Types of Waves
Objective 1: Recall the difference between longitudinal and transverse waves.
Objective 2: Recall that electromagnetic waves are classified as transverse waves.
Objective 3: Recall that the number of photons in an electromagnetic wave is proportional to the amplitude.
Objective 4: Recall that the energy of an electromagnetic wave is proportional to the frequency.
Objective 5: Recall that in quantum mechanics, the amplitude of the wave function represents the probability of
finding an electron at a given location at time t.
The movement of waves can be classified as longitudinal, transverse, or a combination of both.
Longitudinal waves consist of particles moving back and forth in the direction of wave propagation.
Ex. Sound waves traveling in air (pressure disturbances traveling in the direction of sound)
Ex. The motion of a slinky from one end to another initiated by a force into the slinky.
Figure 1.2 Schematic of a longitudinal wave
Simulation of longitudinal waves as a function of amplitude and frequency can
http://www.surendranath.org/Applets/Waves/Lwave01/Lwave01Applet.html [ctrl click on link]
be
observed at
Transverse waves consist of particles traveling in the direction perpendicular to wave propagation.
Ex. Wave on a string
Ex. Electromagnetic waves
Figure 1.3 Schematic of a transverse wave
Simulation of transverse waves as a function of amplitude and frequency can be observed at
http://surendranath.tripod.com/Applets/Waves/Twave01/Twave01Applet.html [ctrl click on link]
Note that changing the frequency is completely independent of changes in amplitude. When examining the transfer of
energy between electromagnetic waves and electrons, the transfer of energy is a function of the frequency. The
amplitude is related to the number of photons in an electromagnetic wave of a certain energy or frequency.
In quantum mechanics, electrons are modeled by a wave function that is purely mathematical and not representative of
anything physical. In this case, the amplitude represents the probability of finding an electron at a given location at time
t.
If we combine a transverse wave traveling in the + x direction (incident wave) with a transverse wave traveling in the –x
direction (reflected wave), we observe a wave that appears to be immobile (resultant wave), and this kind of wave is
characterized as a standing wave. The incident wave interferes constructively with the reflected wave resulting in a
wave with higher amplitude. Imagine a wave on a string fixed at both ends, where one end is displaced slightly with
respect to the other, forcing the string to move in a wave pattern back and forth between the fixed ends. The resulting
wave appears to be a single wave with a fixed pattern. A simulation of a standing wave can be observed at
http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html [ctrl click on link].
Notice that by increasing the frequency, the wavelength decreases and the number of nodes (fixed points) increases. If
we take a snap shot of the wave at any time, we observe a static standing wave. The interesting observation is that at
any given time the number of cycles is a whole number. This is an example of quantized value. Because the ends are
fixed, there can only be a whole number of wave envelopes.
Figure 1.3 Schematic of a standing wave
1.1.3
Wave-Particle Duality
Objective 1: Recall that electromagnetic radiation exhibits both wave and particle-like properties.
Objective 2: Recall that electrons and all other particles exhibit both particle and wave-like properties.
Objective 3: Recall the 3 experiments that were used to prove the wave-particle duality of electromagnetic radiation
and particles.
Objective 4: Recall that wave-particle duality of particles is only important at the nano-scale.
The use of wave mechanics to describe quantum behavior is possible because all matter has wave-like properties and all
waves exhibit particle-like properties. This idea was formalized by Louis de Broglie in 1924 to describe the energy and
momentum of an electron and is termed wave-particle duality. The use of this term began when scientists were trying
to understand the phenomenon associated with quantum mechanics.
It was mentioned in section 1.1.1 that electromagnetic waves possess quantized energy values. This was the first initial
idea that led to the study of quantum mechanics. The idea that electromagnetic radiation behaves as a wave with
particle-like properties was the second important idea. Once this idea was accepted, the term photon was used to
describe electromagnetic radiation (in 1926 by Gilbert Lewis). These 2 ideas will be discussed further in section 1.4
(Photons). The combination of these two important ideas is what led to Schrödinger’s equation and other models that
describe quantum mechanics.
de Broglie proposed this theory before there was evidence to support his ideas, and the occurrence of wave-particle
duality is now known through various experiments such as,
•
•
•
Photoelectric effect - A photon targeting a metal surface will result in energy transfer to a single electron,
ejecting that electron from the surface of the metal (discovered in 1887 by Heinrich Hertz, modeled in 1905 by
Albert Einstein, model verified by Robert Millikan in 1916).
Compton Effect and X-ray diffraction - Electrons with sufficient energy (kinetic energy) that are decelerated as
they hit an atom result in the production of electromagnetic radiation or X-rays (with energy equal to kinetic
energy of the electron). If the electron has sufficient energy, this process will result in the production of an Xray photon and the ejection of an electron – Compton effect. (discovered in 1895 by Wilhelm Roentegen,
characterized in 1912 by Max von Laue, developed by W.L. and W.H. Bragg, modeled in 1923 by Arthur
Compton).
Electron diffraction (double slit experiment) – When electrons pass through 2 parallel slits, and are aimed at
photographic
film
behind
the
slits,
an
interference
pattern
is
observed.
http://nanohub.org/resources/4916/about [ctrl click on link]
These examples exhibit the quantum mechanical behavior of electrons and their interaction with electromagnetic
radiation. As a result of these experiments, wave-particle duality was observed, and the existence of the photon and its
quantized energy values was confirmed. de Broglie expanded wave-particle duality to electrons and proposed the
relationship between the momentum of an electron and its wavelength (p = h/λ). (Electrons are considered particles
because they have mass and charge. The diffraction of electrons by crystals is evidence that they exhibit wave-like
behavior.) Using de Broglie’s relationship, one can determine the wavelength of any particle as a function of its mass
and velocity (p = mv).
The movement of macroscopic particles through space results in very small wavelengths compared to their size, and
wavelengths for large particles are difficult to observe and measure. Analyzing their wave properties is not necessary in
order to describe their motion and thus their energy. The motion of macroscopic particles is described accurately with
classical physics using Newton’s Law of Motion.
Ex.
The wavelength of a golf ball: Quantum mechanics calculations for a 46 g (43 mm diameter) golf ball moving
through the air at 30 m/s results in a wavelength of 4.8 x 10-34 m. The value of the wavelength in this case is
very small compared to the size of the object and does not have an impact on its motion through the air.
Quantum Mechanics calculations do not provide additional information about the energy of the ball as it moves
through a specified volume.
On the other hand, the movement of nano-size objects exhibit wavelengths comparable to their size. The motion of
these particles can only be modeled or described using quantum mechanics. In order to model these small particles, we
need to be able to describe their motion as waves. The electron is a very small particle and we can study electrons if we
assume that an electron can behave as a particle or a wave.
Ex.
The wavelength of an electron: Quantum mechanics calculations for an electron with a mass of 9.11 x 10-31 kg
and a velocity of 107 m/s results in a wavelength of 7.3 x 10-11 m. This value is comparable to the size of a
hydrogen atom of 5.3 x 10-11 m. Consequently, it is understandable that the wavelength of the electron is an
important factor in modeling the motion and behavior of an electron. In this case, we need to perform
quantum mechanics calculations in order to determine the most probable position of the electron in a specified
volume and the possible quantized (specific) energy values the electron can acquire.
Homework
1.
A photon is emitted as a result of an electron transition from a higher energy level to a lower energy level. (a)
Indicate the type of photon that is emitted if its wavelength is equal to (a) 500 nm and (b) 0.1 nm.
2.
An X-ray can be produced by decelerating an electron as it hits a metal surface. The initial kinetic energy of the
electron is transferred to the energy associated with the X-ray (photon) production. What velocity does the
electron need to achieve in order to produce an X-ray with a wavelength of 0.07 nm. (Recall E = hc/λ and K.E. =
½ mv2)
3.
Classify the following statements as True or False.
_____ Photons behave as both waves and particles.
_____ The amplitude of a wave is related to the energy of a wave.
_____ If a wave has a large enough amplitude, it is able to eject an electron from a metal.
_____ A wave with a small wavelength will have more energy than a wave with a large
wavelength.
_____ The higher the frequency of a wave, the smaller the wavelength.
_____ The higher the frequency of a wave, the higher the energy of the wave.
_____ The number of photons in an electromagnetic wave is related to the amplitude of the
wave.
_____ All electromagnetic waves propagate at 3 x 108 m/s in vacuum.
_____ The transfer of energy can occur from photon to electron.
_____ If an electron goes from a lower energy level E1 to a higher energy level E2, then a photon is emitted with
energy equal to E2 – E1.
_____ The intensity of an electromagnetic wave is related to the amplitude of the wave, and the energy of the
wave is related to the frequency of the wave.
_____ The photoelectric effect demonstrates the particle-like properties of photons.
_____ The two-slit experiment demonstrates the particle-like properties of electrons.
_____ The Compton effect demonstrates the wave-like properties of photons.
_____ If we apply quantum mechanical calculations to the motion of a macro-scale object, the results will be
the same as those associated with classical mechanical calculations for the same scenario.
4.
Watch the video of the double slit experiment available on the nanoHUB.org website
(http://nanohub.org/resources/4916/about), and explain whether this experiment proves that
electrons have wave-like and/or particle-like properties.
References
1.
2.
3.
4.
5.
6.
D.A.B. Miller, Quantum Mechanics for Scientists and Engineers, Cambridge University Press, New York, 2008.
A. Beiser, Concepts of Modern Physics, McGraw Hill, New York, 2003.
nd
J.R. Taylor, C.D. Zafiratos, M.A. Dubson, Modern Physics for Scientists and Engineers (2 Ed.), Prentice Hall, New Jersey, 2004.
F.W. Sears, Zemansky, Young, Addison Wesley Education Publishers, 1991.
“Electromagnetic Spectrum,” Available: http://csep10.phys.utk.edu/astr162/lect/light/spectrum.html. [Accessed: 12/1/09].
“Transverse Waves,” Available: http://www.physics.ubc.ca/~outreach/phys420/p420_05/anthony/Transverse%20Waves.htm.
[Accessed: 12/1/09].
7. C. France, “Waves: Longitudinal Waves,” gcsescience.com, 2008, Available: http://www.gcsescience.com/a/pwav2.htm.
[Accessed: 12/1/09].
8. B.S. Reddy, “Transverse Waves,” surendranath.tripod.com, General Physics Java Applets, 2006, Available:
http://surendranath.tripod.com/Applets/Waves/Twave01/Twave01Applet.html. [Accessed: 12/1/09].
9. R. Hart, “Standing Longitudinal Waves,” 2003, Available: http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipewaves.html. [Accessed: 1/5/10].
10. “Standing Waves,”, Available: http://www.lightandmatter.com/html_books/7cp/ch06/figs/standing-waves.png. [Accessed:
1/5/10].
11. D. Vasileska (2008) “Particle-Wave Duality: an Animation,” nanohub.org, 11/2/09, Available:
http://nanohub.org/resources/4916. [Accessed: 12/1/09].