Introduction to Quantum Physics, or Why Energies Are Quantized

In The Quantum Dot, Turton explains that electrons in atoms can only have certain allowed, or quantized, energies.  To fully understand why energy is quantized, we must introduce quantum mechanics and the ideas of wave functions.  You will not be expected to “do” quantum mechanics or solve the Schrödinger equation.  You will, however, need to understand how the wave function relates to probability.  You should also understand how the wave nature of matter implies quantized energies.
 

Heisenberg shook the physics community upside down with his now-famous uncertainty principle.  Heisenberg recognized that measuring a property of an object affects that object, altering other properties.  For example, when I measure the length of a computer screen, I look at light which bounces off the screen and reaches my eye.  The light which bounces off the screen exerts a force on the screen as it reflects.  Since visible light has a relatively low energy, and the computer screen is relatively large, the effect of the force on the screen in this macroscopic experiment is negligible.

When I try to pin down the position of an atom, however, the measurement has a more significant effect.  To resolve, or see, the atom, I can bounce light off it.  (Here I use the term “light” to mean any sort of electromagnetic radiation, including gamma rays).  As you will learn later in the semester, resolving a smaller object requires a higher energy of radiation.  When my high-energy radiation bounces off the atom, it gives it a kick and sends it shooting away.  I may have measured the atom’s position, but in doing so I have changed its momentum (momentum is mass times velocity).  So I cannot know both its position and its momentum at the same time, no matter how accurate my experiment.  The smaller my uncertainty in position gets, the greater my uncertainty in momentum becomes.

All of this uncertainty has profound effects on our understanding of kinematics.  No longer is nature viewed as deterministic, since velocity and position cannot be simultaneously measured.  Instead of knowing the location of a particle at any given moment, we know only the probability that it is at a certain location.  We can no longer discuss a particle’s trajectory but can only discuss the path of its most likely value, or expectation value.
 
 

In 1905, Einstein successfully explained puzzling experimental results by proposing that light behaves like a particle when it interacts.  Light comes in discrete quanta, now called photons, having energy

where h is a constant of nature, called Planck’s constant, and f is the frequency of the light.  The wave nature of light, however, was also well-documented by that time.  Light traveling through thin slits diffracts and produces interference patterns, just like a wave of water does.  But Einstein, extending earlier work by Planck, claimed that when light is absorbed by a material, its behavior is particle-like.  Light exhibits both wave and particle properties.

In 1924 Louis de Broglie suggested that matter particles, such as the electron, could also have wave properties.  His suggestion was based on symmetry, rather than experiment, but it was experimentally verified in 1927 by C. J. Davisson and L. H. Germer and by George Thomson.  The wavelength l of an object is related to its momentum p by

where h is again Planck’s constant.  Even though the electron interacts like a point particle, transferring energy to solely one point, it travels like a wave and can produce an interference pattern.  This seemingly schizophrenic behavior of matter and light is now called the wave-particle duality.

The wave-particle duality ties in nicely with the concept of uncertainty.  A traveling object exhibits wave properties and is therefore spread out.  When it interacts, however, the particle nature takes over, so it only interacts at one point.  You can visualize this process by picturing a wave collapsing to a point when it strikes something.  This collapse can occur anywhere, with varying probabilities.  The probability that it collapses at a certain point is the same as the probability of finding the particle at that point when you measure.  Just as we argued above, we can not predict exactly where a particle will be; we can only give the probability that a particle will be at a certain location.
 

Once we know the expression y(x,t) for a water wave, we can determine the behavior of the wave by using y in Newton’s equations. To determine the behavior of particles with quantum mechanics, we must find the expression for the wave function Y(x,t) and use it in an appropriate equation.  Just as the expression for the height of the water wave given above describes the wave behavior of the water, knowing the expression for Y describes the wave behavior of particles like electrons.

Y(x,t) depends on both position and time, but the time dependence can be treated separately from the position dependence.  For the purposes of this course, the time dependence will always have the same form, and we can write Y(x,t)=c(x)exp(-i2pt/T).  Note that Y(x,t) represents the full wave function dependent on both position and time, while c(x) represents just the position-dependent wave function.

Erwin Schrödinger showed in 1926 that the wave functions c(x) for matter waves obey a differential equation that now bears his name.  Newton’s Laws relate the behavior of an object to the forces F acting on it.  Schrödinger’s equation relates the behavior of an object instead to the potential energy U it has.  This distinction is not significant, however, since force and potential energy are related by

E is the energy of the object, m is its mass, and h is Planck’s constant.

Schrödinger’s equation appears to be just as deterministic as Newton’s equations at first glance.  If we know the kinetic energy (which equals E-U) and initial conditions, we should be able to determine c(x) and therefore Y(x,t) at any later time.  Nature, however, throws us a curve ball.  Thanks to the uncertainty stated by Heisenberg, and the wave-particle duality, we cannot measure c(x).  The wave collapses to a particle when we measure it, and we can only measure the probability that the particle will be found at a certain place and time.  Mathematically, the probability P is related to the wave function c(x) as follows:

We call the quantity inside the integral, |Y(x,t)|² = |c(x)|², the probability density.  Sometimes the wave function c(x) is called the probability amplitude.
 
 

Imagine a particle in a box with infinitely-high walls.  In our macroscopic world, a ping-pong ball in such a box could have any kinetic energy, even zero.  And the ball could be placed anywhere in the box.  In the subatomic world, however, quantum mechanics takes over.  As we shall see, an electron in a box behaves quite differently than a ping-pong ball.

We will choose the potential energy of a particle inside our box to be zero.  Then the solution to Schödinger’s equation inside the well is sinusoidal.  (For those who are interested, the proof of this statement is contained at the end of this reading.)  The energy needed to leave our box is infinite, so the particle will never leave.  Thus the wave function of the particle must be zero outside the box.   Physical properties, such as the wave function or the probability, must be continuous.  This means they do not change instantaneously.  If the wave function is zero outside the box, it must be zero at the edges of the box and change continuously inside the box.  This restricts the wavelength of the wave function, as shown below.
 

In the drawings to the left, c(x) satisfies all of our conditions.  They are sinusoidal (shown here is ½ a cycle of a sine wave, and 2 cycles), and they are zero at either end of the box.  In the drawing to our right, c(x) is not zero at the right side of the box, so this wave function would not occur.  From observation, we can see that the only wavelengths ln allowed are related to the width a of the box by a = nln/2,    n=1,2,3, . . . If a particle has the wavelength associated with the number n in the above equation, we say it is in the nth state.  The wave function c(x) has the form c(x) = B sin(npx/a), where B is a constant.  You might recognize this expression as similar to standing waves on a string.

 

The wavefunction for the state n=4 is shown to the left above.  The probability density for this state is found by squaring the wavefunction; a graph of the probability density is shown directly to the left of this paragraph.  If x is a/4, a/2, or 3a/4, the probability density is zero.  This means that the particle will never be found at those three points if it is in the n=4 state!

DeBroglie related the wavelength of an object to its momentum.  If only certain wavelengths are allowed, then the particle can only have certain momenta p_n.  These momenta may be found from

But momentum is related to the kinetic energy of a particle:

where m is the mass of the particle.  Since the potential energy is zero inside the well, the kinetic energy equals the total energy E of the electron, and we can write

The electron in our box can only have the energies given by the above equation.  One significant effect of this quantization of energy is that the electron can not have zero kinetic energy, so it cannot be at rest in the box.

These quantum effects are not observed for a ping-pong ball in a box because the mass of the ping-pong ball is relatively large.  A large mass results in smaller energies for the same n and, in particular, smaller differences between allowed energies.  Thus any energy appears to be allowed, and the minimum energy cannot be distinguished from zero.

To summarize, the wave-particle duality of matter restricts the properties of a particle in a box.  Only certain energies are allowed, and the probability of finding the particle at a particular point in the box could be zero.  The situation described here is somewhat idealized, since infinite wells do not exist.  But infinite wells are very similar to finite wells.  And an atom represents a three-dimensional finite well to an electron.  Just as for our one-dimensional infinite well, only certain quantized energies are allowed.  And the behavior of the electron is described by a wavefunction associated with the energy of the electron.  Your reading in The Quantum Dot describes the relationship between quantized energies and conduction.

An Example

Suppose I have a proton in an infinite well.  The well has width 1.0 nanometer, and the momentum of the proton is 1.65 x 10^-24 kg m/s.  Find the wavelength of the proton, what state it is in, and its kinetic energy.

Some useful numbers:  mass of proton m_p= 1.67 x 10^-27 kg
Planck’s constant h = 6.6 x 10^-34 Js

l = h/p = (6.6 x 10^-34 Js)/(1.65 x 10^-24 kg m/s) = 4.0 x 10^-10 m = 0.40 nm.
a = nl/2, so
n = 2a/l = 2(1.0 x 10^-9 m)/(4.0 x 10^-9 m) = 5
K = h²n²/8a²m = (6.6 x 10^-34 Js)²(25)/8(1.0 x 10^-9 m)²(1.67 x 10^-27 kg) = 8.15 x 10^-22 J
 
 

Since energy E, mass m, p, and h are all constants, this equation has the form

The function with a second derivative proportional to itself but having an opposite sign is a cosine or sine (try it!).  So we can express our solution as either a sine or a cosine.  Because the boundary conditions of our particular problem require the solution to be zero at x=0, we choose a sine function.

Note that if I take the second derivative, I get -w² B sin (wx) = - w² c, as predicted.