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Chapter 23 WAVE EQUATION
• Introduction• Wave-particle duality• Quantized modes• Schrödinger equation• One-dimensional solutions• Three-dimensional solutions• Dirac Equation• Conclusions
INTRODUCTION
de Broglie’s 1923 hypothesis of wave-particle duality,
was the key idea that led to the development of quan-
tum mechanics. The basis of quantum mechanics was
developed in an incredible spurt of advances during
1926. It is fascinating to see how quickly things devel-
oped and how much trouble even the greatest minds
had in accepting some of these new ideas. Many of the
scientists involved had a spark of genius and then faded
from sight. Schrödinger’s development of his wave
equation was a tremendous step forward. Heisenberg
developed quantum mechanics via his matrix mechan-
ics methods which use abstract mathematical methods
that do not provide the intuition that wave mechan-
ics can provide. However, practicing physicists actu-
ally use the Heisenberg matrix mechanics because it is
more powerful. The story of the development of quan-
tum mechanics is interesting. Schrödinger had a rep-
utation of being a playboy, or as Einstein called him,
a clever rogue. Many of these physicists migrated to
the US in the late 1930’s, because of the rise of the
Nazis in Germany, and they contributed significantly
to advancing science in the US as well as development
of nuclear power and the bomb. Heisenberg was a
member of the German Youth movement and stayed
in Germany playing a role in the German nuclear bomb
project during World War 2.
The application of quantum mechanics requires de-
velopment of a wave equation. In contrast to classical
physics, where the wave equation can be derived from
first principles, this discussion will resort to plausibil-
ity arguments to develop a useful wave equation. This
chapter will review the wavefunction and the Copen-
hagen interpretation of matter waves before discussing
the non-relativistic Schrödinger wave equation and ap-
plications. This will be followed by a brief description
of the relativistic Dirac equation.
WAVE-PARTICLE DUALITY
Wavefunction
In wave mechanics one deals with the amplitude of
the matter waves. The amplitude of the matter wave
is usually written as a greek letter psi, Ψ which is a
function of x,y,z, and time. This is analogous to waves
on a string where, at any instant in time, we can write
down the amplitude of the string displacement as a
function of location on the string. The mathematics
of waves is simpler using complex numbers, rather than
sine and cosine functions to describe wavefunctions.
= (cos + sin )
It is much easier to add or subtract exponents of the
exponentials rather than take products of sines and
cosines.
The real part of projects out , while the
imaginary part projects out the corresponding sin
function. The simplest travelling wave in one dimen-
sion, of the form
Ψ = cos(− )
can be rewritten in the exponential form as:
Ψ = (−)
The wavefunction can comprise a range of values for
and For matter waves, and are given by the
de Broglie relations, that is, the relativistic energy
E = ~
and the component of the linear momentum
= ~
Copenhagen interpretation
The Copenhagen Interpretation of wave-particle du-
ality is that the intensity of the wave, |Ψ( )|2gives the probability distribution function () of
finding the particle at any location, x,y,z, at time t.
That is
() = |Ψ()|2
Quantum mechanics is probabilistic, that is, it does
not determine the exact behavior of the particle, it
only tells you the probability of finding the particle
with certain conditions. Assuming that the probability
175
of finding the particle somewhere is unity, implies that
the integral of the probability over all space is unity.Z
|Ψ|2 = 1
As discussed last time, the idea that nature is basically
probabilistic, even for a single particle, was philosphi-
cally unacceptable to many physicists. It ran counter
to the deterministic approach so successfully applied in
classical physics. Last lecture discussed the double-slit
interference pattern as an example where the predicted
intensity of the wave gave the probability distribution
of detecting individual photons.
Expectation values
Wave motion implies that exact specifications for a
particle cannot be determined simultaneously, only the
probability distribution is measureable. Thus one can
only determine expectation values, that is average val-
ues, for observables. For example the average value of
position x is given by evaluating the probability dis-
tribution with x,. That is, the expectation value of x
is;
hi =Z
|Ψ( )|2
Similarly one can evaluate the expectation value for
x2; 2®=
Z2 |Ψ( )|2
These can be used to calculate the standard deviation
of the spread of the average value That is;
() =p( 2 −( )2)
Note that () is a non-zero positive number for a
non-localized probability distribution.
The expectation values of time, linear momentum,
angular momentum, energy, or other possible observ-
ables can be evaluated if the wavefunction is known,
providing a method for relating physical observables
in quantum physics.
Uncertainty principle
As discussed last lecture, the Uncertainty Principle ap-
plies to all forms of wave motion, be it sound waves,
water waves, electromagnetic waves, or de Broglie mat-
ter waves. Namely for the wave intensity, the standard
deviations for measurements of conjugate variables,
that is related variables, have a minimum product of
the uncertainties:
() · () ≥ 12
() · () ≥ 12
Figure 1 Quantized standing waves on a violin string
Thus it is not possible to simultaneously measure both
conjugate variables with unlimited precision. That is,
to obtain a well defined frequency requires that the
wave packet must include many sinusoidal oscillations
of a wave. The concept of wave-particle duality en-
visions that the behaviour of a particle is described
by the behaviour of the corresponding wave packet.
Since the wave packet implies uncertainty principle
limitations, then for matter waves, since E = ~ and = ~ one is forced to accept that one cannot si-multaneously measure the energy and time, or position
and associated momentum with unlimited precision.
That is, the products of the measured uncertainities
are limited by the Hiesenberg Uncertainty Principle:
() · () > ~2
() · () > ~2
() · () > ~2
() · () > ~2
QUANTIZED MODES
For all forms of wavemotion, confinement of a wave to a
restricted range of position leads to two consequences,
(a) quantized modes, and, because of the Uncertainty
Principle, (b) a minimum zero-point motion.
Consider the one dimensional case. For a local-
ized region of x, discrete values of the corresponding
wavenumber =2occur. For example, the discrete
solutions on a violin string of length L have possible
wavelengths, =2where is a positive integer,
≥ 1. That is, the wavenumber has quantized so-
lutions =where is the corresponding inte-
ger quantum number. That is, solutions are separated
by multiples of ∆ =. The separation between
adjacent quantized solutions goes to zero, that is, it
becomes continuous, when →∞
If the wave has a fixed velocity , then since = ,
quantized values of correspond to quantized frequen-
176
cies . For matter waves, quantized values of im-
plies quantized = ~, and consequently quantizedvalues of kinetic energy.
A one-dimensional wave leads to a single integer
quantum number n characterizing the solutions. A
three-dimensional box can have three independent stand-
ing waves for each independent degree of freedom, i.e.
the x, y, and z directions. Standing wave systems for
each of these three independent spatial degrees of free-
dom has an associated quantum number characterizing
the number of half wavelengths fitting along that di-
mension. Thus there is one quantum number for every
degree of freedom of a quantal system.
Last lecture is was shown that the Heisenberg Un-
certainty principle implies that for any particle con-
fined within a localized region, there is a corresponding
zero-point kinetic energy which is the minimum energy
allowed for a particle of mass m confined within this
region. Thus even the lowest state of a system having
a localized particle, has a minimum kinetic energy.
These conclusions are properties of waves in gen-
eral and are manifest by matter waves as discussed.
More specific discussion of quantum theory requires
development of a wave equation.
SCHRÖDINGER EQUATION
Although Heisenberg’s matrix mechanics was devel-
oped first, the Schrödinger equation was the first use-
ful quantum wave equation. However, the Schrödinger
equation is applicable only to non-relativistic cases. As
opposed to classical physics, where the wave equa-
tion can be derived from first principles, in quantum
physics the wave equation is obtained by a plausibility
argument based on classical physics and the de Broglie
hypotheses. Inserting the de Broglie relations
E = ~
= ~
into the simplest form of a travelling wave in one di-
mension gives:
Ψ0 = (−) = ~ (−E)
The relativistic energy E = +2 where is the
kinetic energy. The wave function can be written as
Ψ0 = ~ (−E)−
~
2 = Ψ−~
2
where
Ψ = ~ (−)
The exponential term −~
2 can be ignored, since
it only introduces an unimportant phase factor that
has no influence on the probability, that is,
|Ψ0|2 = |Ψ|2
Consider the wavefunction Ψ( ) in one dimen-
sion. Note that differentials of Ψ( ) are:
Ψ
= −
~Ψ
That is:
Ψ = ~Ψ
Also:2Ψ
2= −
2
~2Ψ
That is:
2Ψ = −~22Ψ
2
Time-dependent Schrödinger equation
Schrödinger developed a non-relativistic wave equation
based on the fact that:
=2
2+
where U is the potential energy. Since¡
¢2is 10−5 for
electron orbits in atoms, and 10−3 for protons in thenucleus, the non-relativistic equation is a reasonable
approximation for solving many problems in atomic
and nuclear physics.
Substituting the above differential expressions for
E and 2 = 2 + 2 + 2, gives Schrödinger’s time-
dependent equation in three dimensions:
Ψ = ~Ψ
= − ~
2
2
µ2Ψ
2+
2Ψ
2+
2Ψ
2
¶+ Ψ
(Time dependent Schrodinger equation)
Note that the following facts were used in develop-
ing this equation:
• de Broglie-Einstein relations E = ~ −→p = ~−→k
• Non-relativistic relation = 2
2+
• Ignored consequences of the phase factor − ~
2
In general the wavefunction Ψ( ) is obtained
by solving this time-dependent equation for some known
potential ( )
Time-independent Schrödinger equation.
A frequently encountered problem is where the po-
tential is time independent, ( ) for which the
quantized energy solutions are time independent. For
such cases the right-hand side of the time-dependent
Schrödinger equation is a function of only the spatial
coordinates while the left-hand side is a function only
of time. For such a system the two sides are indepen-
dent and the spatial and time parts of the wavefunction
can be separated:
Ψ( ) = ( )−~
177
The left-hand side of the equation equals:
~Ψ
= ~
−~
= Ψ
while the right-hand side gives:
= − ~2
2
µ2
2+
2
2+
2
2
¶+
(Time independent Schrodinger equation)
where the time-dependent term of Ψ, that is, −~
on both sides cancels. This equation is easier to solve
than is Schrödinger’s time-dependent equation.
The solution of the Schrödinger equation must sat-
isfy the following restrictions.
Wavefunction normalization
The wavefunction must be correctly normalized,
that is the probability of finding the particle over all
space must be unity, that is:Z
||2 = 1 (Wavefunction normalization.)
This fixes the normalization of the wavefunction.
Continuity
The wavefunction must be a well-behaved function
for which Ψ,
22
22
and 22
all exist. This implies
that:
is continuous−→∇ is continuous
That is,
and
all are continuous, if the po-
tential energy function is not infinite.
ONE-DIMENSIONAL SOLUTIONS
Schrödinger’s equation is easiest to understand by con-
sidering problems in one-dimension.
Infinite square well
Consider a mass m contained in a one-dimensional box
with = 0 for 0 ≤ ≤ and = ∞ outside this
region. The time independent Schrödinger equation
for 0 ≤ ≤ is:
− ~2
2
2
2=
since = 0 The wavefunction must be zero outside
of the box because then the potential = ∞ Thus
() must equal zero at the walls of the box, = 0
and = . Therefore, as for waves on a violin string,
the solutions of = sin() for 0 ≤ ≤ are:
= sin() =
r2
sin³
´
Figure 2 Infinite one-dimensional box. The spatial
dependence of and excitation energies E are shown
for = 1 2 3.
where n is an integer quantum number. The constant
A has been normalized by requiringR 02 = 1
while the wavenumber is given by:
=2
=
that is,
=2
Substitution of into the Schrödinger equation
gives corresponding energies:
=~2
22 = 2
~22
22
These are the allowed energies E corresponding to the
wavefunction for each quantum number Figure
2 shows the energies, which are proportional to 2
and the corresponding wavefunctions. Note that the
lowest state has n=1 giving an energy 1 =~22
22
This is greater than the minimum zero-point energy
given by the Heisenberg uncertainty principle.
Finite square well
Consider the potential well shown in figure 3, where
= 0 for 0 ≤ ≤ = outside of this region of
. Thus the Schrödinger equation can be written as:
− ~2
2
2
2= (0 ≤ ≤ )
− ~2
2
2
2= ( − ) ( ≤ 0 or ≥ )
178
Figure 3 Particle in a finite square potential well.
Consider possible standing-wave solutions of the
form:
= +−
Inserting this into the Schrödinger equation gives
inside the potential well, 0 ≤ ≤
~2
22 =
and outside the well:
~2
220 = −
Thus the solution inside the well is a sinusoidal stand-
ing wave with wavenumber:
=
√2
~(Wavenumber inside well)
That is:
= +− = sin()
The wavenumber outside of the well is given by:
0 =
p2( − )
~(Wavenumber outside well.)
Note that if − is positive, that is exceeds the
depth of the potential, then the solution is sinusoidal
with a smaller wavenumber, that is longer wavelength,
than inside the potential well. If − is negative,
that is, thenp2( − ) is an imaginary
number. That is:
0 =p2( −)
~=
Thus the solution outside the well now is:
= − +
This corresponds to an exponential decay as shown in
figure 3.
The final solutions inside and outside of the well are
now obtained using the continuity and normalization
requirements, that is:
Figure 4 The potential energy, energy and wavefunction
distributions for a binding of a diatomic molecule.
• Matching the inside and outside values of atthe well boundaries = 0 and = .
• Matching the inside and outside gradients at
the well boundaries = 0 and = .
• Requiring that R∞−∞ ||2 = 1Note that for , we seem to violate classical
physics by having a finite probability of finding the
particle with negative kinetic energy. However, it can
be shown that, because of the Heisenberg uncertainty
principle, it is not possible to measure this negative
kinetic energy outside of the potential well.
This exponential decay of the wavefunction in classically-
forbidden regions is a common feature in quantum
physics. For example, a plot of the potential energy
versus separation distance due to binding of a diatomic
molecule is close to that of a one-dimensional harmonic
oscillator shown in figure 4. Solutions in this well are
illustrated in figure 4. For a large quantum number
the greatest probability occurs with the diatomic hav-
ing maximum separation distance just as is obtained
for a harmonic oscillator such as the pendulum.
THREE-DIMENSIONAL SOLUTIONS
Solution of the time-independent Schrödinger equation
in three dimensions leads to solutions in three inde-
pendent directions, each of which has its own quan-
tum number. It is useful to first consider the three-
dimensional infinite square well potential since it il-
lustrates important characteristics of solutions of the
Schrödinger equation.
179
THREE DIMENSIONAL INFINITE SQUARE
WELL
Consider the three-dimensional infinite square well po-
tential where () = 0 for 0 0 ,
0 Outside this cubical square well () =
∞ The wave function solutions to this potential must
be zero at the infinite boundaries. Thus the solutions
inside the box are of the form
Ψ() = sin 1 sin 2 sin 3
where the constant A is given by normalization of the
wavefunction. Inserting this solution into Schrödinger’s
equation gives energies inside the box of
=~2
2
¡21 + 22 + 23
¢As for the infinite one-dimensional square well, the po-
tential must be zero at the walls, therefore the only
possible solutions are quantized with
1 = 1
2 = 2
3 = 3
where ≥ 1 and integer. Therefore
=~22
22
¡21 + 22 + 23
¢The lowest energy, that is, ground state, for this
cubical infinite square well is
111 =3~22
22
The first excited state can be made three different
ways, each having the same energy. That is
211 = 121 = 112 =6~22
22
where, for example, the 2,1,1 wavefunction is
Ψ211 = sin2
sin
sin
Note that these three solutions are degenerate, that
is have they same excitation energy, if the box is the
same size in the three directions. This often happens
in quantum physics. This degeneracy is removed if
the box has different sizes in the three directions as
shown in figure 5. This happens in deformed nuclei
that are football shaped, that is, where one axis may
be significantly longer than the other two axes.
Figure 5 Energy-level diagrams for (a) a cubic infinite
square well and (b) noncubic infinite square well
SPHERICALLY-SYMMETRIC POTENTIALS
Several forces in physics are spherically symmetric,
that is the force depends only on the radial distance,
not angle. This is true of the gravitational force, Coulomb
force and nuclear force. For such forces it is bet-
ter to use the orthogonal spherical coordinates, , ,
and rather than , since then the spherically-
symmetric potential energy is only a function of the
one variable . The angles and are the latitude
and longitude on the surface of a sphere at a given
radius. Instead of associating quantum numbers de-
scribing the and directions, one uses quantum
numbers characterizing waves that are solutions in the
, and directions.
The Schrödinger equation in spherical coordinates
is:
− ~2
2
∙1
2
µ2
¶+1
2
½1
sin
µsin
¶+
1
sin2
2
2
¾¸+ =
For a spherically-symmetric potential, the solution sep-
arates into the form;
() = ()Θ ()Φ()
The corresponding quantum numbers are called;
for for , and for . Standing waves in and
requires that the pattern repeat after one complete
revolution around the surface of the sphere. Note that
if is spherically symmetric, then the angular part of
Schrödinger’s equation, is contained completely within
the curly brackets, and the solutions are independent
of the radial form of the potential well.
Schrödinger showed that the orbital angular mo-
mentum−→L = −→r × −→p of a particle described by the
180
Figure 6 Spherical coordinates.
angular wavefunction Θ ()Φ() has an angular mo-
mentum of:
= ~p( + 1)
where is called the orbital angular momentum quan-
tum number.
Also Schrodinger showed that the projection of this
orbital angular momentum on the axis forΘ ()Φ()
equals:
= ~
where is called the magnetic quantum number. Note
that the magnetic quantum number can take values
between ± That is;
− ≤ ≤ +
Next lecture will use Schrödinger’s equation in this
form to solve the hydrogen atom. Schrödinger derived
Bohr’s quantization of the orbital angular momentum
from his wave equation without resorting to ad hoc as-
sumptions. For such spherically-symmetric potentials
the binding energies are independent of and the
energy depends only on the radial quantum number ,
and due to centripetal effects. In general the solu-
tion of the radial part of the Schödinger equation can
be difficult and usually requires use of a computer.
DIRAC EQUATION
Schrödinger tried unsuccessfully to develop a relativis-
tic wave equation but failed. It was Dirac, in 1928,
who finally derived the ultimate relativistic wave equa-
tion. It is complicated to use and physicists only use
it when the non-relativistic approach is inapplicable.
What is remarkable about the Dirac equation is that
it predicts:
• electron spin,• negative-energy states
Dirac had to conclude that the negative energy
states, illustrated in the figure, were in fact completely
Figure 7 The Dirac negative sea.
filled with electrons. A hole in this negative sea of elec-
trons will behave like a positive electron. Raising an
electron from this negative sea produces a free electron
plus a positive hole, which we call a positron. Dirac re-
alized immediately that the positive hole has to have
the mass of the electron, that is, it is not a proton.
Oppenheimer, who later led the US nuclear bomb
project, showed that the negative-energy solutions ac-
tually correspond to positive-energy states of positive
particles. Dirac’s concept of a sea of electrons filling all
of the negative states was an unnecesary edifice, one
really just has positive and negative electrons. Thus
the Dirac equation predicts four states of the electron:,
spin up and spin down for both positive and negative
electrons. This seemingly wild idea of positive elec-
trons was confirmed in 1932 when Anderson discov-
ered the positron. He observed electrons deflected in
both directions in a cloud chamber that was between
the poles of a large magnet.
The positron is called the antiparticle of the elec-
tron. Soon after Anderson’s discovery, Dirac realized
that there should also be an antiparticle to the proton.
In fact, an antiparticle exists for every type of particle.
The masses of particles, and their corresponding an-
tiparticle, have been measured to be identical to a few
parts per billion, while the charge and magnetic prop-
erties are identical in value but opposite in sign. A
particle and its antiparticle can annihilate each other
and the energy is released as electromagnetic energy
or some other particle-antiparticle pair as long as en-
ergy conservation is satisfied. Using Einstein’s equa-
tion E = 2 to compute that the rest mass of an
electron corresponds to 0.511 MeV of energy. To create
an electron-positron pair requires at least twice 0.511
MeV, that is greater than 1.022 MeV of energy. We
observe this process all the time in nuclear physics. For
example, the 22 nucleus decays by positron emission
22 =22 + + + + 1542
The emitted positron stops in material and then an-
nihilates an electron to give off two 511 keV gamma
181
rays
+ + − = (0511 ) + (0511 )
These photons can be detected in our gamma-ray de-
tectors. A proton has a rest mass of 938.27 MeV so
one needs twice this to create a proton-antiproton pair.
Antiprotons were not observed until the Bevatron ac-
clerator was built at Berkeley in the 1950’s. If a pro-
ton annihilates an antiproton, they release an energy
of 1,876.5 GeV plus the kinetic energy. Some people
surmize that maybe antigalaxies exist. Think what
would happen to our galaxy if it collided with another
galaxy made of antimatter, the energy released would
be truly astronomical. If an antiworld piece of matter
the size of the earth hit the earth the annihilation en-
ergy would be 1042 J, that is the energy radiated by
the sun during the past 30 million years.
The Dirac theory was expanded in the 1940’s to
include electromagnetism, this extension of quantum
theory is called quantum electrodynamics, QED. It
is a remarkably successful theory. For example, the
magnetic moment of the proton is predicted to 10 dec-
imal places and has been measured to be this value
with a similar precision. The Dirac equation is more
difficult to solve than the non-relativistic Schrodinger
equation. Fortunately, relativistic effects are generally
negligible for atomic and molecular systems and thus
the Schrödinger equation is an adequate approxima-
tion for most applications of wave mechanics.
CONCLUSIONS
The Schrödinger equation led to an amazing advance
in our knowledge of quantum mechanics and the hy-
drogen atom. The next advances needed are the in-
troduction of spin and the Pauli Exclusion Principle.
These allow an explanation of both single and multi-
electron atoms, molecular systems and nuclei. The
Dirac equation correctly handles wave mechanics into
the relativistic regime. It also introduces the concept
of spin and the Pauli Exclusion Principle. Next lec-
ture will discuss atomic structure and the remarkable
consequences of the relativistic wave equation.
Reading assignment: Giancoli, Chapter 39.
182